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    <title>Compositionality - Latest Publications</title>
    <description>Latest articles</description>
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    <pubDate>Mon, 06 Apr 2026 06:45:59 +0000</pubDate>
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    <link>https://compositionality.episciences.org</link>
    <author>Compositionality</author>
    <dc:creator>Compositionality</dc:creator>
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    <item>
      <title>Compositionality in algorithms for smoothing</title>
      <description><![CDATA[Backward Filtering Forward Guiding (BFFG) is a bidirectional algorithm proposed in Mider et al. [2021] and studied more in depth in a general setting in Van der Meulen and Schauer [2022]. In category theory, optics have been proposed for modelling systems with bidirectional data flow. We connect BFFG with optics by demonstrating that the forward and backwards map together define a functor from a category of Markov kernels into a category of optics, which is furthermore lax monoidal in the case when the guiding kernels coincide with the generative dynamics]]></description>
      <pubDate>Tue, 24 Mar 2026 15:00:49 +0000</pubDate>
      <link>https://doi.org/10.46298/compositionality-8-3</link>
      <guid>https://doi.org/10.46298/compositionality-8-3</guid>
      <author>Schauer, Moritz</author>
      <author>van der Meulen, Frank</author>
      <author>Wang, Andi Q.</author>
      <dc:creator>Schauer, Moritz</dc:creator>
      <dc:creator>van der Meulen, Frank</dc:creator>
      <dc:creator>Wang, Andi Q.</dc:creator>
      <content:encoded><![CDATA[Backward Filtering Forward Guiding (BFFG) is a bidirectional algorithm proposed in Mider et al. [2021] and studied more in depth in a general setting in Van der Meulen and Schauer [2022]. In category theory, optics have been proposed for modelling systems with bidirectional data flow. We connect BFFG with optics by demonstrating that the forward and backwards map together define a functor from a category of Markov kernels into a category of optics, which is furthermore lax monoidal in the case when the guiding kernels coincide with the generative dynamics]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Rank-based linkage I: triplet comparisons and oriented simplicial complexes</title>
      <description><![CDATA[Rank-based linkage is a new tool for summarizing a collection $S$ of objects according to their relationships. These objects are not mapped to vectors, and ``similarity'' between objects need be neither numerical nor symmetrical. All an object needs to do is rank nearby objects by similarity to itself, using a Comparator which is transitive, but need not be consistent with any metric on the whole set. Call this a ranking system on $S$. Rank-based linkage is applied to the $K$-nearest neighbor digraph derived from a ranking system. Computations occur on a 2-dimensional abstract oriented simplicial complex whose faces are among the points, edges, and triangles of the line graph of the undirected $K$-nearest neighbor graph on $S$. In $|S| K^2$ steps it builds an edge-weighted linkage graph $(S, \mathcal{L}, σ)$ where $σ(\{x, y\})$ is called the in-sway between objects $x$ and $y$. Take $\mathcal{L}_t$ to be the links whose in-sway is at least $t$, and partition $S$ into components of the graph $(S, \mathcal{L}_t)$, for varying $t$. Rank-based linkage is a functor from a category of ``out-ordered'' digraphs to a category of partitioned sets, with the practical consequence that augmenting the set of objects in a rank-respectful way gives a fresh clustering which does not ``rip apart'' the previous one. The same holds for single linkage clustering in the metric space context, but not for typical optimization-based methods. Orientation sheaves play in a fundamental role and ensure that partially overlapping data sets can be ``glued'' together. Open combinatorial problems are presented in the last section.]]></description>
      <pubDate>Fri, 20 Mar 2026 14:35:09 +0000</pubDate>
      <link>https://doi.org/10.46298/compositionality-8-2</link>
      <guid>https://doi.org/10.46298/compositionality-8-2</guid>
      <author>Darling, R. W. R.</author>
      <author>Grilliette, Will</author>
      <author>Logan, Adam</author>
      <dc:creator>Darling, R. W. R.</dc:creator>
      <dc:creator>Grilliette, Will</dc:creator>
      <dc:creator>Logan, Adam</dc:creator>
      <content:encoded><![CDATA[Rank-based linkage is a new tool for summarizing a collection $S$ of objects according to their relationships. These objects are not mapped to vectors, and ``similarity'' between objects need be neither numerical nor symmetrical. All an object needs to do is rank nearby objects by similarity to itself, using a Comparator which is transitive, but need not be consistent with any metric on the whole set. Call this a ranking system on $S$. Rank-based linkage is applied to the $K$-nearest neighbor digraph derived from a ranking system. Computations occur on a 2-dimensional abstract oriented simplicial complex whose faces are among the points, edges, and triangles of the line graph of the undirected $K$-nearest neighbor graph on $S$. In $|S| K^2$ steps it builds an edge-weighted linkage graph $(S, \mathcal{L}, σ)$ where $σ(\{x, y\})$ is called the in-sway between objects $x$ and $y$. Take $\mathcal{L}_t$ to be the links whose in-sway is at least $t$, and partition $S$ into components of the graph $(S, \mathcal{L}_t)$, for varying $t$. Rank-based linkage is a functor from a category of ``out-ordered'' digraphs to a category of partitioned sets, with the practical consequence that augmenting the set of objects in a rank-respectful way gives a fresh clustering which does not ``rip apart'' the previous one. The same holds for single linkage clustering in the metric space context, but not for typical optimization-based methods. Orientation sheaves play in a fundamental role and ensure that partially overlapping data sets can be ``glued'' together. Open combinatorial problems are presented in the last section.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Abstract Markov Random Fields</title>
      <description><![CDATA[Markov random fields are known to be fully characterized by properties of their information diagrams, or I-diagrams. In particular, for Markov random fields, regions in the I-diagram corresponding to disconnected vertex sets in the graph vanish. Recently, I-diagrams have been generalized to F-diagrams, for a larger class of functions F satisfying the chain rule beyond Shannon entropy, such as Kullback-Leibler divergence and cross-entropy. In this work, we generalize the notion and characterization of Markov random fields to this larger class of functions F and investigate preliminary applications. We define F-independences, F-mutual independences, and F-Markov random fields and characterize them by their F-diagram. In the process, we also define F-dual total correlation and prove that its vanishing is equivalent to F-mutual independence. We then apply our results to information functions F that are applied to probability mass functions. We show that if the probability distributions of a set of random variables are Markov random fields for the same graph, then we formally recover the notion of an F-Markov random field for that graph. We then study the Kullback-Leibler diagrams on specific Markov chains, leading to a visual representation of the second law of thermodynamics and a simple explicit derivation of the decomposition of the evidence lower bound for diffusion models.]]></description>
      <pubDate>Fri, 20 Mar 2026 07:02:05 +0000</pubDate>
      <link>https://doi.org/10.46298/compositionality-8-1</link>
      <guid>https://doi.org/10.46298/compositionality-8-1</guid>
      <author>Lang, Leon</author>
      <author>de Mulatier, Clélia</author>
      <author>Quax, Rick</author>
      <author>Forré, Patrick</author>
      <dc:creator>Lang, Leon</dc:creator>
      <dc:creator>de Mulatier, Clélia</dc:creator>
      <dc:creator>Quax, Rick</dc:creator>
      <dc:creator>Forré, Patrick</dc:creator>
      <content:encoded><![CDATA[Markov random fields are known to be fully characterized by properties of their information diagrams, or I-diagrams. In particular, for Markov random fields, regions in the I-diagram corresponding to disconnected vertex sets in the graph vanish. Recently, I-diagrams have been generalized to F-diagrams, for a larger class of functions F satisfying the chain rule beyond Shannon entropy, such as Kullback-Leibler divergence and cross-entropy. In this work, we generalize the notion and characterization of Markov random fields to this larger class of functions F and investigate preliminary applications. We define F-independences, F-mutual independences, and F-Markov random fields and characterize them by their F-diagram. In the process, we also define F-dual total correlation and prove that its vanishing is equivalent to F-mutual independence. We then apply our results to information functions F that are applied to probability mass functions. We show that if the probability distributions of a set of random variables are Markov random fields for the same graph, then we formally recover the notion of an F-Markov random field for that graph. We then study the Kullback-Leibler diagrams on specific Markov chains, leading to a visual representation of the second law of thermodynamics and a simple explicit derivation of the decomposition of the evidence lower bound for diffusion models.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Towards a theory of natural directed paths</title>
      <description><![CDATA[We introduce the abstract setting of presheaf category on a thick category of cubes. Precubical sets, symmetric transverse sets, symmetric precubical sets and the new category of (non-symmetric) transverse sets are examples of this structure. All these presheaf categories share the same metric and homotopical properties from a directed homotopy point of view. This enables us to extend Raussen's notion of natural $d$-path for each of them. Finally, we adapt Ziemiański's notion of cube chain to this abstract setting and we prove that it has the expected behavior on precubical sets. As an application, we verify that the formalization of the parallel composition with synchronization of process algebra using the coskeleton functor of the category of symmetric transverse sets has a category of cube chains with the correct homotopy type.]]></description>
      <pubDate>Wed, 07 Jan 2026 13:20:18 +0000</pubDate>
      <link>https://doi.org/10.46298/compositionality-7-6</link>
      <guid>https://doi.org/10.46298/compositionality-7-6</guid>
      <author>Gaucher, Philippe</author>
      <dc:creator>Gaucher, Philippe</dc:creator>
      <content:encoded><![CDATA[We introduce the abstract setting of presheaf category on a thick category of cubes. Precubical sets, symmetric transverse sets, symmetric precubical sets and the new category of (non-symmetric) transverse sets are examples of this structure. All these presheaf categories share the same metric and homotopical properties from a directed homotopy point of view. This enables us to extend Raussen's notion of natural $d$-path for each of them. Finally, we adapt Ziemiański's notion of cube chain to this abstract setting and we prove that it has the expected behavior on precubical sets. As an application, we verify that the formalization of the parallel composition with synchronization of process algebra using the coskeleton functor of the category of symmetric transverse sets has a category of cube chains with the correct homotopy type.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Cloven operadic categories: An approach to operadic categories with cardinalities in finite unordered sets</title>
      <description><![CDATA[We introduce and study operadic categories with cardinalities in finite sets and establish conditions under which their associated theories of operads and algebras are equivalent to the standard framework introduced in 2015 by Batanin and Markl. Our approach is particularly natural in applications to the operadic category of graphs and the related category of modular operads and their clones.]]></description>
      <pubDate>Fri, 17 Oct 2025 11:32:38 +0000</pubDate>
      <link>https://doi.org/10.46298/compositionality-7-5</link>
      <guid>https://doi.org/10.46298/compositionality-7-5</guid>
      <author>Markl, Martin</author>
      <dc:creator>Markl, Martin</dc:creator>
      <content:encoded><![CDATA[We introduce and study operadic categories with cardinalities in finite sets and establish conditions under which their associated theories of operads and algebras are equivalent to the standard framework introduced in 2015 by Batanin and Markl. Our approach is particularly natural in applications to the operadic category of graphs and the related category of modular operads and their clones.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Additive Invariants of Open Petri Nets</title>
      <description><![CDATA[We classify all additive invariants of open Petri nets: these are $\mathbb{N}$-valued invariants which are additive with respect to sequential and parallel composition of open Petri nets. In particular, we prove two classification theorems: one for open Petri nets and one for monically open Petri nets (i.e. open Petri nets whose interfaces are specified by monic maps). Our results can be summarized as follows. The additive invariants of open Petri nets are completely determined by their values on a particular class of single-transition Petri nets. However, for monically open Petri nets, the additive invariants are determined by their values on transitionless Petri nets and all single-transition Petri nets. Our results confirm a conjecture of John Baez (stated during the AMS' 2022 Mathematical Research Communities workshop).]]></description>
      <pubDate>Fri, 25 Jul 2025 14:33:04 +0000</pubDate>
      <link>https://doi.org/10.46298/compositionality-7-4</link>
      <guid>https://doi.org/10.46298/compositionality-7-4</guid>
      <author>Bumpus, Benjamin Merlin</author>
      <author>Libkind, Sophie</author>
      <author>Garcia, Jordy Lopez</author>
      <author>Sorkatti, Layla</author>
      <author>Tenka, Samuel</author>
      <dc:creator>Bumpus, Benjamin Merlin</dc:creator>
      <dc:creator>Libkind, Sophie</dc:creator>
      <dc:creator>Garcia, Jordy Lopez</dc:creator>
      <dc:creator>Sorkatti, Layla</dc:creator>
      <dc:creator>Tenka, Samuel</dc:creator>
      <content:encoded><![CDATA[We classify all additive invariants of open Petri nets: these are $\mathbb{N}$-valued invariants which are additive with respect to sequential and parallel composition of open Petri nets. In particular, we prove two classification theorems: one for open Petri nets and one for monically open Petri nets (i.e. open Petri nets whose interfaces are specified by monic maps). Our results can be summarized as follows. The additive invariants of open Petri nets are completely determined by their values on a particular class of single-transition Petri nets. However, for monically open Petri nets, the additive invariants are determined by their values on transitionless Petri nets and all single-transition Petri nets. Our results confirm a conjecture of John Baez (stated during the AMS' 2022 Mathematical Research Communities workshop).]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Universal pseudomorphisms, with applications to diagrammatic coherence for braided and symmetric monoidal functors</title>
      <description><![CDATA[This work introduces a general theory of universal pseudomorphisms and develops their connection to diagrammatic coherence. The main results give hypotheses under which pseudomorphism coherence is equivalent to the coherence theory of strict algebras. Applications include diagrammatic coherence for plain, symmetric, and braided monoidal functors. The final sections include a variety of examples.]]></description>
      <pubDate>Wed, 25 Jun 2025 09:29:04 +0000</pubDate>
      <link>https://doi.org/10.46298/compositionality-7-3</link>
      <guid>https://doi.org/10.46298/compositionality-7-3</guid>
      <author>Gurski, Nick</author>
      <author>Johnson, Niles</author>
      <dc:creator>Gurski, Nick</dc:creator>
      <dc:creator>Johnson, Niles</dc:creator>
      <content:encoded><![CDATA[This work introduces a general theory of universal pseudomorphisms and develops their connection to diagrammatic coherence. The main results give hypotheses under which pseudomorphism coherence is equivalent to the coherence theory of strict algebras. Applications include diagrammatic coherence for plain, symmetric, and braided monoidal functors. The final sections include a variety of examples.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Lifting couplings in Wasserstein spaces</title>
      <description><![CDATA[This paper makes mathematically precise the idea that conditional probabilities are analogous to path liftings in geometry. The idea of lifting is modelled in terms of the category-theoretic concept of a lens, which can be interpreted as a consistent choice of arrow liftings. The category we study is the one of probability measures over a given standard Borel space, with morphisms given by the couplings, or transport plans. The geometrical picture is even more apparent once we equip the arrows of the category with weights, which one can interpret as "lengths" or "costs", forming a so-called weighted category, which unifies several concepts of category theory and metric geometry. Indeed, we show that the weighted version of a lens is tightly connected to the notion of submetry in geometry. Every weighted category gives rise to a pseudo-quasimetric space via optimization over the arrows. In particular, Wasserstein spaces can be obtained from the weighted categories of probability measures and their couplings, with the weight of a coupling given by its cost. In this case, conditionals allow one to form weighted lenses, which one can interpret as "lifting transport plans, while preserving their cost".]]></description>
      <pubDate>Wed, 07 May 2025 08:34:10 +0000</pubDate>
      <link>https://doi.org/10.46298/compositionality-7-2</link>
      <guid>https://doi.org/10.46298/compositionality-7-2</guid>
      <author>Perrone, Paolo</author>
      <dc:creator>Perrone, Paolo</dc:creator>
      <content:encoded><![CDATA[This paper makes mathematically precise the idea that conditional probabilities are analogous to path liftings in geometry. The idea of lifting is modelled in terms of the category-theoretic concept of a lens, which can be interpreted as a consistent choice of arrow liftings. The category we study is the one of probability measures over a given standard Borel space, with morphisms given by the couplings, or transport plans. The geometrical picture is even more apparent once we equip the arrows of the category with weights, which one can interpret as "lengths" or "costs", forming a so-called weighted category, which unifies several concepts of category theory and metric geometry. Indeed, we show that the weighted version of a lens is tightly connected to the notion of submetry in geometry. Every weighted category gives rise to a pseudo-quasimetric space via optimization over the arrows. In particular, Wasserstein spaces can be obtained from the weighted categories of probability measures and their couplings, with the weight of a coupling given by its cost. In this case, conditionals allow one to form weighted lenses, which one can interpret as "lifting transport plans, while preserving their cost".]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Information Decomposition Diagrams Applied beyond Shannon Entropy: A Generalization of Hu's Theorem</title>
      <description><![CDATA[In information theory, one major goal is to find useful functions that summarize the amount of information contained in the interaction of several random variables. Specifically, one can ask how the classical Shannon entropy, mutual information, and higher interaction information relate to each other. This is answered by Hu's theorem, which is widely known in the form of information diagrams: it relates shapes in a Venn diagram to information functions, thus establishing a bridge from set theory to information theory. In this work, we view random variables together with the joint operation as a monoid that acts by conditioning on information functions, and entropy as a function satisfying the chain rule of information. This abstract viewpoint allows to prove a generalization of Hu's theorem. It applies to Shannon and Tsallis entropy, (Tsallis) Kullback-Leibler Divergence, cross-entropy, Kolmogorov complexity, submodular information functions, and the generalization error in machine learning. Our result implies for Chaitin's Kolmogorov complexity that the interaction complexities of all degrees are in expectation close to Shannon interaction information. For well-behaved probability distributions on increasing sequence lengths, this shows that the per-bit expected interaction complexity and information asymptotically coincide, thus showing a strong bridge between algorithmic and classical information theory.]]></description>
      <pubDate>Thu, 30 Jan 2025 10:57:55 +0000</pubDate>
      <link>https://doi.org/10.46298/compositionality-7-1</link>
      <guid>https://doi.org/10.46298/compositionality-7-1</guid>
      <author>Lang, Leon</author>
      <author>Baudot, Pierre</author>
      <author>Quax, Rick</author>
      <author>Forré, Patrick</author>
      <dc:creator>Lang, Leon</dc:creator>
      <dc:creator>Baudot, Pierre</dc:creator>
      <dc:creator>Quax, Rick</dc:creator>
      <dc:creator>Forré, Patrick</dc:creator>
      <content:encoded><![CDATA[In information theory, one major goal is to find useful functions that summarize the amount of information contained in the interaction of several random variables. Specifically, one can ask how the classical Shannon entropy, mutual information, and higher interaction information relate to each other. This is answered by Hu's theorem, which is widely known in the form of information diagrams: it relates shapes in a Venn diagram to information functions, thus establishing a bridge from set theory to information theory. In this work, we view random variables together with the joint operation as a monoid that acts by conditioning on information functions, and entropy as a function satisfying the chain rule of information. This abstract viewpoint allows to prove a generalization of Hu's theorem. It applies to Shannon and Tsallis entropy, (Tsallis) Kullback-Leibler Divergence, cross-entropy, Kolmogorov complexity, submodular information functions, and the generalization error in machine learning. Our result implies for Chaitin's Kolmogorov complexity that the interaction complexities of all degrees are in expectation close to Shannon interaction information. For well-behaved probability distributions on increasing sequence lengths, this shows that the per-bit expected interaction complexity and information asymptotically coincide, thus showing a strong bridge between algorithmic and classical information theory.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Homotopy Theoretic and Categorical Models of Neural Information Networks</title>
      <description><![CDATA[In this paper we develop a novel mathematical formalism for the modeling of neural information networks endowed with additional structure in the form of assignments of resources, either computational or metabolic or informational. The starting point for this construction is the notion of summing functors and of Segal's Gamma-spaces in homotopy theory. The main results in this paper include functorial assignments of concurrent/distributed computing architectures and associated binary codes to networks and their subsystems, a categorical form of the Hopfield network dynamics, which recovers the usual Hopfield equations when applied to a suitable category of weighted codes, a functorial assignment to networks of corresponding information structures and information cohomology, and a cohomological version of integrated information.]]></description>
      <pubDate>Fri, 06 Sep 2024 14:18:51 +0000</pubDate>
      <link>https://doi.org/10.46298/compositionality-6-4</link>
      <guid>https://doi.org/10.46298/compositionality-6-4</guid>
      <author>Manin, Yuri</author>
      <author>Marcolli, Matilde</author>
      <dc:creator>Manin, Yuri</dc:creator>
      <dc:creator>Marcolli, Matilde</dc:creator>
      <content:encoded><![CDATA[In this paper we develop a novel mathematical formalism for the modeling of neural information networks endowed with additional structure in the form of assignments of resources, either computational or metabolic or informational. The starting point for this construction is the notion of summing functors and of Segal's Gamma-spaces in homotopy theory. The main results in this paper include functorial assignments of concurrent/distributed computing architectures and associated binary codes to networks and their subsystems, a categorical form of the Hopfield network dynamics, which recovers the usual Hopfield equations when applied to a suitable category of weighted codes, a functorial assignment to networks of corresponding information structures and information cohomology, and a cohomological version of integrated information.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>A Framework for Universality in Physics, Computer Science, and Beyond</title>
      <description><![CDATA[Turing machines and spin models share a notion of universality according to which some simulate all others. Is there a theory of universality that captures this notion? We set up a categorical framework for universality which includes as instances universal Turing machines, universal spin models, NP completeness, top of a preorder, denseness of a subset, and more. By identifying necessary conditions for universality, we show that universal spin models cannot be finite. We also characterize when universality can be distinguished from a trivial one and use it to show that universal Turing machines are non-trivial in this sense. Our framework allows not only to compare universalities within each instance, but also instances themselves. We leverage a Fixed Point Theorem inspired by a result of Lawvere to establish that universality and negation give rise to unreachability (such as uncomputability). As such, this work sets the basis for a unified approach to universality and invites the study of further examples within the framework.]]></description>
      <pubDate>Thu, 29 Aug 2024 13:48:44 +0000</pubDate>
      <link>https://doi.org/10.46298/compositionality-6-3</link>
      <guid>https://doi.org/10.46298/compositionality-6-3</guid>
      <author>Gonda, Tomáš</author>
      <author>Reinhart, Tobias</author>
      <author>Stengele, Sebastian</author>
      <author>Coves, Gemma De les</author>
      <dc:creator>Gonda, Tomáš</dc:creator>
      <dc:creator>Reinhart, Tobias</dc:creator>
      <dc:creator>Stengele, Sebastian</dc:creator>
      <dc:creator>Coves, Gemma De les</dc:creator>
      <content:encoded><![CDATA[Turing machines and spin models share a notion of universality according to which some simulate all others. Is there a theory of universality that captures this notion? We set up a categorical framework for universality which includes as instances universal Turing machines, universal spin models, NP completeness, top of a preorder, denseness of a subset, and more. By identifying necessary conditions for universality, we show that universal spin models cannot be finite. We also characterize when universality can be distinguished from a trivial one and use it to show that universal Turing machines are non-trivial in this sense. Our framework allows not only to compare universalities within each instance, but also instances themselves. We leverage a Fixed Point Theorem inspired by a result of Lawvere to establish that universality and negation give rise to unreachability (such as uncomputability). As such, this work sets the basis for a unified approach to universality and invites the study of further examples within the framework.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks</title>
      <description><![CDATA[Regulatory networks depict promoting or inhibiting interactions between molecules in a biochemical system. We introduce a category-theoretic formalism for regulatory networks, using signed graphs to model the networks and signed functors to describe occurrences of one network in another, especially occurrences of network motifs. With this foundation, we establish functorial mappings between regulatory networks and other mathematical models in biochemistry. We construct a functor from reaction networks, modeled as Petri nets with signed links, to regulatory networks, enabling us to precisely define when a reaction network could be a physical mechanism underlying a regulatory network. Turning to quantitative models, we associate a regulatory network with a Lotka-Volterra system of differential equations, defining a functor from the category of signed graphs to a category of parameterized dynamical systems. We extend this result from closed to open systems, demonstrating that Lotka-Volterra dynamics respects not only inclusions and collapsings of regulatory networks, but also the process of building up complex regulatory networks by gluing together simpler pieces. Formally, we use the theory of structured cospans to produce a lax double functor from the double category of open signed graphs to that of open parameterized dynamical systems. Throughout the paper, we ground the categorical formalism in examples inspired by systems biology.]]></description>
      <pubDate>Sun, 12 May 2024 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-6-2</link>
      <guid>https://doi.org/10.32408/compositionality-6-2</guid>
      <author>Aduddell, Rebekah</author>
      <author>Fairbanks, James</author>
      <author>Kumar, Amit</author>
      <author>Ocal, Pablo S.</author>
      <author>Patterson, Evan</author>
      <author>Shapiro, Brandon T.</author>
      <dc:creator>Aduddell, Rebekah</dc:creator>
      <dc:creator>Fairbanks, James</dc:creator>
      <dc:creator>Kumar, Amit</dc:creator>
      <dc:creator>Ocal, Pablo S.</dc:creator>
      <dc:creator>Patterson, Evan</dc:creator>
      <dc:creator>Shapiro, Brandon T.</dc:creator>
      <content:encoded><![CDATA[Regulatory networks depict promoting or inhibiting interactions between molecules in a biochemical system. We introduce a category-theoretic formalism for regulatory networks, using signed graphs to model the networks and signed functors to describe occurrences of one network in another, especially occurrences of network motifs. With this foundation, we establish functorial mappings between regulatory networks and other mathematical models in biochemistry. We construct a functor from reaction networks, modeled as Petri nets with signed links, to regulatory networks, enabling us to precisely define when a reaction network could be a physical mechanism underlying a regulatory network. Turning to quantitative models, we associate a regulatory network with a Lotka-Volterra system of differential equations, defining a functor from the category of signed graphs to a category of parameterized dynamical systems. We extend this result from closed to open systems, demonstrating that Lotka-Volterra dynamics respects not only inclusions and collapsings of regulatory networks, but also the process of building up complex regulatory networks by gluing together simpler pieces. Formally, we use the theory of structured cospans to produce a lax double functor from the double category of open signed graphs to that of open parameterized dynamical systems. Throughout the paper, we ground the categorical formalism in examples inspired by systems biology.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Profunctor Optics, a Categorical Update</title>
      <description><![CDATA[Optics are bidirectional data accessors that capture data transformation patterns such as accessing subfields or iterating over containers. Profunctor optics are a particular choice of representation supporting modularity, meaning that we can construct accessors for complex structures by combining simpler ones. Profunctor optics have previously been studied only in an unenriched and non-mixed setting, in which both directions of access are modelled in the same category. However, functional programming languages are arguably better described by enriched categories; and we have found that some structures in the literature are actually mixed optics, with access directions modelled in different categories. Our work generalizes a classic result by Pastro and Street on Tambara theory and uses it to describe mixed V-enriched profunctor optics and to endow them with V-category structure. We provide some original families of optics and derivations, including an elementary one for traversals. Finally, we discuss a Haskell implementation.]]></description>
      <pubDate>Thu, 22 Feb 2024 23:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-6-1</link>
      <guid>https://doi.org/10.32408/compositionality-6-1</guid>
      <author>Clarke, Bryce</author>
      <author>Elkins, Derek</author>
      <author>Gibbons, Jeremy</author>
      <author>Loregian, Fosco</author>
      <author>Milewski, Bartosz</author>
      <author>Pillmore, Emily</author>
      <author>Román, Mario</author>
      <dc:creator>Clarke, Bryce</dc:creator>
      <dc:creator>Elkins, Derek</dc:creator>
      <dc:creator>Gibbons, Jeremy</dc:creator>
      <dc:creator>Loregian, Fosco</dc:creator>
      <dc:creator>Milewski, Bartosz</dc:creator>
      <dc:creator>Pillmore, Emily</dc:creator>
      <dc:creator>Román, Mario</dc:creator>
      <content:encoded><![CDATA[Optics are bidirectional data accessors that capture data transformation patterns such as accessing subfields or iterating over containers. Profunctor optics are a particular choice of representation supporting modularity, meaning that we can construct accessors for complex structures by combining simpler ones. Profunctor optics have previously been studied only in an unenriched and non-mixed setting, in which both directions of access are modelled in the same category. However, functional programming languages are arguably better described by enriched categories; and we have found that some structures in the literature are actually mixed optics, with access directions modelled in different categories. Our work generalizes a classic result by Pastro and Street on Tambara theory and uses it to describe mixed V-enriched profunctor optics and to endow them with V-category structure. We provide some original families of optics and derivations, including an elementary one for traversals. Finally, we discuss a Haskell implementation.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Traced Monads and Hopf Monads</title>
      <description><![CDATA[A traced monad is a monad on a traced symmetric monoidal category that lifts the traced symmetric monoidal structure to its Eilenberg-Moore category. A long-standing question has been to provide a characterization of traced monads without explicitly mentioning the Eilenberg-Moore category. On the other hand, a symmetric Hopf monad is a symmetric bimonad whose fusion operators are invertible. For compact closed categories, symmetric Hopf monads are precisely the kind of monads that lift the compact closed structure to their Eilenberg-Moore categories. Since compact closed categories and traced symmetric monoidal categories are closely related, it is a natural question to ask what is the relationship between Hopf monads and traced monads. In this paper, we introduce trace-coherent Hopf monads on traced monoidal categories, which can be characterized without mentioning the Eilenberg-Moore category. The main theorem of this paper is that a symmetric Hopf monad is a traced monad if and only if it is a trace-coherent Hopf monad. We provide many examples of trace-coherent Hopf monads, such as those induced by cocommutative Hopf algebras or any symmetric Hopf monad on a compact closed category. We also explain how for traced Cartesian monoidal categories, trace-coherent Hopf monads can be expressed using the Conway operator, while for traced coCartesian monoidal categories, any trace-coherent Hopf monad is an idempotent monad. We also provide separating examples of traced monads that are not Hopf monads, as well as symmetric Hopf monads that are not trace-coherent.]]></description>
      <pubDate>Sun, 29 Oct 2023 23:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-5-10</link>
      <guid>https://doi.org/10.32408/compositionality-5-10</guid>
      <author>Hasegawa, Masahito</author>
      <author>Lemay, Jean-Simon Pacaud</author>
      <dc:creator>Hasegawa, Masahito</dc:creator>
      <dc:creator>Lemay, Jean-Simon Pacaud</dc:creator>
      <content:encoded><![CDATA[A traced monad is a monad on a traced symmetric monoidal category that lifts the traced symmetric monoidal structure to its Eilenberg-Moore category. A long-standing question has been to provide a characterization of traced monads without explicitly mentioning the Eilenberg-Moore category. On the other hand, a symmetric Hopf monad is a symmetric bimonad whose fusion operators are invertible. For compact closed categories, symmetric Hopf monads are precisely the kind of monads that lift the compact closed structure to their Eilenberg-Moore categories. Since compact closed categories and traced symmetric monoidal categories are closely related, it is a natural question to ask what is the relationship between Hopf monads and traced monads. In this paper, we introduce trace-coherent Hopf monads on traced monoidal categories, which can be characterized without mentioning the Eilenberg-Moore category. The main theorem of this paper is that a symmetric Hopf monad is a traced monad if and only if it is a trace-coherent Hopf monad. We provide many examples of trace-coherent Hopf monads, such as those induced by cocommutative Hopf algebras or any symmetric Hopf monad on a compact closed category. We also explain how for traced Cartesian monoidal categories, trace-coherent Hopf monads can be expressed using the Conway operator, while for traced coCartesian monoidal categories, any trace-coherent Hopf monad is an idempotent monad. We also provide separating examples of traced monads that are not Hopf monads, as well as symmetric Hopf monads that are not trace-coherent.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Bayesian open games</title>
      <description><![CDATA[This paper generalises the treatment of compositional game theory as introduced by Ghani et al. in 2018, where games are modelled as morphisms of a symmetric monoidal category. From an economic modelling perspective, the notion of a game in the work by Ghani et al. is not expressive enough for many applications. This includes stochastic environments, stochastic choices by players, as well as incomplete information regarding the game being played. The current paper addresses these three issues all at once.]]></description>
      <pubDate>Tue, 03 Oct 2023 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-5-9</link>
      <guid>https://doi.org/10.32408/compositionality-5-9</guid>
      <author>Bolt, Joe</author>
      <author>Hedges, Jules</author>
      <author>Zahn, Philipp</author>
      <dc:creator>Bolt, Joe</dc:creator>
      <dc:creator>Hedges, Jules</dc:creator>
      <dc:creator>Zahn, Philipp</dc:creator>
      <content:encoded><![CDATA[This paper generalises the treatment of compositional game theory as introduced by Ghani et al. in 2018, where games are modelled as morphisms of a symmetric monoidal category. From an economic modelling perspective, the notion of a game in the work by Ghani et al. is not expressive enough for many applications. This includes stochastic environments, stochastic choices by players, as well as incomplete information regarding the game being played. The current paper addresses these three issues all at once.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Substructural fixed-point theorems and the diagonal argument: theme and variations</title>
      <description><![CDATA[This article re-examines Lawvere's abstract, category-theoretic proof of the fixed-point theorem whose contrapositive is a `universal' diagonal argument. The main result is that the necessary axioms for both the fixed-point theorem and the diagonal argument can be stripped back further, to a semantic analogue of a weak substructural logic lacking weakening or exchange.]]></description>
      <pubDate>Wed, 09 Aug 2023 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-5-8</link>
      <guid>https://doi.org/10.32408/compositionality-5-8</guid>
      <author>Roberts, David Michael</author>
      <dc:creator>Roberts, David Michael</dc:creator>
      <content:encoded><![CDATA[This article re-examines Lawvere's abstract, category-theoretic proof of the fixed-point theorem whose contrapositive is a `universal' diagonal argument. The main result is that the necessary axioms for both the fixed-point theorem and the diagonal argument can be stripped back further, to a semantic analogue of a weak substructural logic lacking weakening or exchange.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Monotones in General Resource Theories</title>
      <description><![CDATA[A central problem in the study of resource theories is to find functions that are nonincreasing under resource conversions - termed monotones - in order to quantify resourcefulness. Various constructions of monotones appear in many different concrete resource theories. How general are these constructions? What are the necessary conditions on a resource theory for a given construction to be applicable? To answer these questions, we introduce a broad scheme for constructing monotones. It involves finding an order-preserving map from the preorder of resources of interest to a distinct preorder for which nontrivial monotones are previously known or can be more easily constructed; these monotones are then pulled back through the map. In one of the two main classes we study, the preorder of resources is mapped to a preorder of sets of resources, where the order relation is set inclusion, such that monotones can be defined via maximizing or minimizing the value of a function within these sets. In the other class, the preorder of resources is mapped to a preorder of tuples of resources, and one pulls back monotones that measure the amount of distinguishability of the different elements of the tuple (hence its information content). Monotones based on contractions arise naturally in the latter class, and, more surprisingly, so do weight and robustness measures. In addition to capturing many standard monotone constructions, our scheme also suggests significant generalizations of these. In order to properly capture the breadth of applicability of our results, we present them within a novel abstract framework for resource theories in which the notion of composition is independent of the types of the resources involved (i.e., whether they are states, channels, combs, etc.).]]></description>
      <pubDate>Tue, 08 Aug 2023 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-5-7</link>
      <guid>https://doi.org/10.32408/compositionality-5-7</guid>
      <author>Gonda, Tomáš</author>
      <author>Spekkens, Robert W.</author>
      <dc:creator>Gonda, Tomáš</dc:creator>
      <dc:creator>Spekkens, Robert W.</dc:creator>
      <content:encoded><![CDATA[A central problem in the study of resource theories is to find functions that are nonincreasing under resource conversions - termed monotones - in order to quantify resourcefulness. Various constructions of monotones appear in many different concrete resource theories. How general are these constructions? What are the necessary conditions on a resource theory for a given construction to be applicable? To answer these questions, we introduce a broad scheme for constructing monotones. It involves finding an order-preserving map from the preorder of resources of interest to a distinct preorder for which nontrivial monotones are previously known or can be more easily constructed; these monotones are then pulled back through the map. In one of the two main classes we study, the preorder of resources is mapped to a preorder of sets of resources, where the order relation is set inclusion, such that monotones can be defined via maximizing or minimizing the value of a function within these sets. In the other class, the preorder of resources is mapped to a preorder of tuples of resources, and one pulls back monotones that measure the amount of distinguishability of the different elements of the tuple (hence its information content). Monotones based on contractions arise naturally in the latter class, and, more surprisingly, so do weight and robustness measures. In addition to capturing many standard monotone constructions, our scheme also suggests significant generalizations of these. In order to properly capture the breadth of applicability of our results, we present them within a novel abstract framework for resource theories in which the notion of composition is independent of the types of the resources involved (i.e., whether they are states, channels, combs, etc.).]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Degrees in random $m$-ary hooking networks</title>
      <description><![CDATA[The theme in this paper is a composition of random graphs and P\'olya urns. The random graphs are generated through a small structure called the seed. Via P\'olya urns, we study the asymptotic degree structure in a random $m$-ary hooking network and identify strong laws. We further upgrade the result to second-order asymptotics in the form of multivariate Gaussian limit laws. We give a few concrete examples and explore some properties with a full representation of the Gaussian limit in each case. The asymptotic covariance matrix associated with the P\'olya urn is obtained by a new method that originated in this paper and is reported in [25].]]></description>
      <pubDate>Tue, 08 Aug 2023 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-5-6</link>
      <guid>https://doi.org/10.32408/compositionality-5-6</guid>
      <author>Bhutani, Kiran R.</author>
      <author>Kalpathy, Ravi</author>
      <author>Mahmoud, Hosam</author>
      <dc:creator>Bhutani, Kiran R.</dc:creator>
      <dc:creator>Kalpathy, Ravi</dc:creator>
      <dc:creator>Mahmoud, Hosam</dc:creator>
      <content:encoded><![CDATA[The theme in this paper is a composition of random graphs and P\'olya urns. The random graphs are generated through a small structure called the seed. Via P\'olya urns, we study the asymptotic degree structure in a random $m$-ary hooking network and identify strong laws. We further upgrade the result to second-order asymptotics in the form of multivariate Gaussian limit laws. We give a few concrete examples and explore some properties with a full representation of the Gaussian limit in each case. The asymptotic covariance matrix associated with the P\'olya urn is obtained by a new method that originated in this paper and is reported in [25].]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Completeness of the ZH-calculus</title>
      <description><![CDATA[There are various gate sets used for describing quantum computation. A particularly popular one consists of Clifford gates and arbitrary single-qubit phase gates. Computations in this gate set can be elegantly described by the ZX-calculus, a graphical language for a class of string diagrams describing linear maps between qubits. The ZX-calculus has proven useful in a variety of areas of quantum information, but is less suitable for reasoning about operations outside its natural gate set such as multi-linear Boolean operations like the Toffoli gate. In this paper we study the ZH-calculus, an alternative graphical language of string diagrams that does allow straightforward encoding of Toffoli gates and other more complicated Boolean logic circuits. We find a set of simple rewrite rules for this calculus and show it is complete with respect to matrices over $\mathbb{Z}[\frac12]$, which correspond to the approximately universal Toffoli+Hadamard gateset. Furthermore, we construct an extended version of the ZH-calculus that is complete with respect to matrices over any ring $R$ where $1+1$ is not a zero-divisor.]]></description>
      <pubDate>Tue, 11 Jul 2023 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-5-5</link>
      <guid>https://doi.org/10.32408/compositionality-5-5</guid>
      <author>Backens, Miriam</author>
      <author>Kissinger, Aleks</author>
      <author>Miller-Bakewell, Hector</author>
      <author>van de Wetering, John</author>
      <author>Wolffs, Sal</author>
      <dc:creator>Backens, Miriam</dc:creator>
      <dc:creator>Kissinger, Aleks</dc:creator>
      <dc:creator>Miller-Bakewell, Hector</dc:creator>
      <dc:creator>van de Wetering, John</dc:creator>
      <dc:creator>Wolffs, Sal</dc:creator>
      <content:encoded><![CDATA[There are various gate sets used for describing quantum computation. A particularly popular one consists of Clifford gates and arbitrary single-qubit phase gates. Computations in this gate set can be elegantly described by the ZX-calculus, a graphical language for a class of string diagrams describing linear maps between qubits. The ZX-calculus has proven useful in a variety of areas of quantum information, but is less suitable for reasoning about operations outside its natural gate set such as multi-linear Boolean operations like the Toffoli gate. In this paper we study the ZH-calculus, an alternative graphical language of string diagrams that does allow straightforward encoding of Toffoli gates and other more complicated Boolean logic circuits. We find a set of simple rewrite rules for this calculus and show it is complete with respect to matrices over $\mathbb{Z}[\frac12]$, which correspond to the approximately universal Toffoli+Hadamard gateset. Furthermore, we construct an extended version of the ZH-calculus that is complete with respect to matrices over any ring $R$ where $1+1$ is not a zero-divisor.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Koszul duality for operadic categories</title>
      <description><![CDATA[The aim of this sequel to arXiv:1812.02935 is to set up the cornerstones of Koszul duality and Koszulity in the context of operads over a large class of operadic categories. In particular, for these operadic categories we will study concrete examples of binary quadratic operads, describe their Koszul duals and prove that they are Koszul. This includes operads whose algebras are the most important operad- and PROP-like structures such as the classical operads, their variants such as cyclic, modular or wheeled operads, and also diverse versions of PROPs such as properads, dioperads, 1/2PROPs, and still more exotic objects such as permutads and pre-permutads.]]></description>
      <pubDate>Thu, 15 Jun 2023 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-5-4</link>
      <guid>https://doi.org/10.32408/compositionality-5-4</guid>
      <author>Batanin, Michael</author>
      <author>Markl, Martin</author>
      <dc:creator>Batanin, Michael</dc:creator>
      <dc:creator>Markl, Martin</dc:creator>
      <content:encoded><![CDATA[The aim of this sequel to arXiv:1812.02935 is to set up the cornerstones of Koszul duality and Koszulity in the context of operads over a large class of operadic categories. In particular, for these operadic categories we will study concrete examples of binary quadratic operads, describe their Koszul duals and prove that they are Koszul. This includes operads whose algebras are the most important operad- and PROP-like structures such as the classical operads, their variants such as cyclic, modular or wheeled operads, and also diverse versions of PROPs such as properads, dioperads, 1/2PROPs, and still more exotic objects such as permutads and pre-permutads.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Operadic categories as a natural environment for Koszul duality</title>
      <description><![CDATA[This is the first paper of a series which aims to set up the cornerstones of Koszul duality for operads over operadic categories. To this end we single out additional properties of operadic categories under which the theory of quadratic operads and their Koszulity can be developped, parallel to the traditional one by Ginzburg and Kapranov. We then investigate how these extra properties interact with discrete operadic (op)fibrations, which we use as a powerful tool to construct new operadic categories from old ones. We pay particular attention to the operadic category of graphs, giving a full description of this category (and its variants) as an operadic category, and proving that it satisfies all the additional properties. Our present work provides an answer to a question formulated in Loday's last talk in 2012:``What encodes types of operads?''. In the second and third papers of our series we continue Loday's program by answering his second question: ``How to construct Koszul duals to these objects?'', and proving Koszulity of some of the most relevant operads.]]></description>
      <pubDate>Sun, 04 Jun 2023 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-5-3</link>
      <guid>https://doi.org/10.32408/compositionality-5-3</guid>
      <author>Batanin, Michael</author>
      <author>Markl, Martin</author>
      <dc:creator>Batanin, Michael</dc:creator>
      <dc:creator>Markl, Martin</dc:creator>
      <content:encoded><![CDATA[This is the first paper of a series which aims to set up the cornerstones of Koszul duality for operads over operadic categories. To this end we single out additional properties of operadic categories under which the theory of quadratic operads and their Koszulity can be developped, parallel to the traditional one by Ginzburg and Kapranov. We then investigate how these extra properties interact with discrete operadic (op)fibrations, which we use as a powerful tool to construct new operadic categories from old ones. We pay particular attention to the operadic category of graphs, giving a full description of this category (and its variants) as an operadic category, and proving that it satisfies all the additional properties. Our present work provides an answer to a question formulated in Loday's last talk in 2012:``What encodes types of operads?''. In the second and third papers of our series we continue Loday's program by answering his second question: ``How to construct Koszul duals to these objects?'', and proving Koszulity of some of the most relevant operads.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Categorical Vector Space Semantics for Lambek Calculus with a Relevant Modality</title>
      <description><![CDATA[We develop a categorical compositional distributional semantics for Lambek Calculus with a Relevant Modality !L*, which has a limited edition of the contraction and permutation rules. The categorical part of the semantics is a monoidal biclosed category with a coalgebra modality, very similar to the structure of a Differential Category. We instantiate this category to finite dimensional vector spaces and linear maps via "quantisation" functors and work with three concrete interpretations of the coalgebra modality. We apply the model to construct categorical and concrete semantic interpretations for the motivating example of !L*: the derivation of a phrase with a parasitic gap. The effectiveness of the concrete interpretations are evaluated via a disambiguation task, on an extension of a sentence disambiguation dataset to parasitic gap phrases, using BERT, Word2Vec, and FastText vectors and Relational tensors.]]></description>
      <pubDate>Mon, 15 May 2023 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-5-2</link>
      <guid>https://doi.org/10.32408/compositionality-5-2</guid>
      <author>McPheat, Lachlan</author>
      <author>Sadrzadeh, Mehrnoosh</author>
      <author>Wazni, Hadi</author>
      <author>Wijnholds, Gijs</author>
      <dc:creator>McPheat, Lachlan</dc:creator>
      <dc:creator>Sadrzadeh, Mehrnoosh</dc:creator>
      <dc:creator>Wazni, Hadi</dc:creator>
      <dc:creator>Wijnholds, Gijs</dc:creator>
      <content:encoded><![CDATA[We develop a categorical compositional distributional semantics for Lambek Calculus with a Relevant Modality !L*, which has a limited edition of the contraction and permutation rules. The categorical part of the semantics is a monoidal biclosed category with a coalgebra modality, very similar to the structure of a Differential Category. We instantiate this category to finite dimensional vector spaces and linear maps via "quantisation" functors and work with three concrete interpretations of the coalgebra modality. We apply the model to construct categorical and concrete semantic interpretations for the motivating example of !L*: the derivation of a phrase with a parasitic gap. The effectiveness of the concrete interpretations are evaluated via a disambiguation task, on an extension of a sentence disambiguation dataset to parasitic gap phrases, using BERT, Word2Vec, and FastText vectors and Relational tensors.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Toposes of Topological Monoid Actions</title>
      <description><![CDATA[We demonstrate that categories of continuous actions of topological monoids on discrete spaces are Grothendieck toposes. We exhibit properties of these toposes, giving a solution to the corresponding Morita-equivalence problem. We characterize these toposes in terms of their canonical points. We identify natural classes of representatives with good topological properties, `powder monoids' and then `complete monoids', for the Morita-equivalence classes of topological monoids. Finally, we show that the construction of these toposes can be made (2-)functorial by considering geometric morphisms induced by continuous semigroup homomorphisms.]]></description>
      <pubDate>Mon, 09 Jan 2023 23:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-5-1</link>
      <guid>https://doi.org/10.32408/compositionality-5-1</guid>
      <author>Rogers, Morgan</author>
      <dc:creator>Rogers, Morgan</dc:creator>
      <content:encoded><![CDATA[We demonstrate that categories of continuous actions of topological monoids on discrete spaces are Grothendieck toposes. We exhibit properties of these toposes, giving a solution to the corresponding Morita-equivalence problem. We characterize these toposes in terms of their canonical points. We identify natural classes of representatives with good topological properties, `powder monoids' and then `complete monoids', for the Morita-equivalence classes of topological monoids. Finally, we show that the construction of these toposes can be made (2-)functorial by considering geometric morphisms induced by continuous semigroup homomorphisms.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Urns &amp; Tubes</title>
      <description><![CDATA[Urn models play an important role to express various basic ideas in probability theory. Here we extend this urn model with tubes. An urn contains coloured balls, which can be drawn with probabilities proportional to the numbers of balls of each colour. For each colour a tube is assumed. These tubes have different sizes (lengths). The idea is that after drawing a ball from the urn it is dropped in the urn of the corresponding colour. We consider two associated probability distributions. The first-full distribution on colours gives for each colour the probability that the corresponding tube is full first, before any of the other tubes. The negative distribution on natural numbers captures for a number k the probability that all tubes are full for the first time after k draws. This paper uses multisets to systematically describe these first-full and negative distributions in the urns & tubes setting, in fully multivariate form, for all three standard drawing modes (multinomial, hypergeometric, and Polya).]]></description>
      <pubDate>Tue, 27 Dec 2022 23:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-4-4</link>
      <guid>https://doi.org/10.32408/compositionality-4-4</guid>
      <author>Jacobs, Bart</author>
      <dc:creator>Jacobs, Bart</dc:creator>
      <content:encoded><![CDATA[Urn models play an important role to express various basic ideas in probability theory. Here we extend this urn model with tubes. An urn contains coloured balls, which can be drawn with probabilities proportional to the numbers of balls of each colour. For each colour a tube is assumed. These tubes have different sizes (lengths). The idea is that after drawing a ball from the urn it is dropped in the urn of the corresponding colour. We consider two associated probability distributions. The first-full distribution on colours gives for each colour the probability that the corresponding tube is full first, before any of the other tubes. The negative distribution on natural numbers captures for a number k the probability that all tubes are full for the first time after k draws. This paper uses multisets to systematically describe these first-full and negative distributions in the urns & tubes setting, in fully multivariate form, for all three standard drawing modes (multinomial, hypergeometric, and Polya).]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Categorical Data Structures for Technical Computing</title>
      <description><![CDATA[Many mathematical objects can be represented as functors from finitely-presented categories $\mathsf{C}$ to $\mathsf{Set}$. For instance, graphs are functors to $\mathsf{Set}$ from the category with two parallel arrows. Such functors are known informally as $\mathsf{C}$-sets. In this paper, we describe and implement an extension of $\mathsf{C}$-sets having data attributes with fixed types, such as graphs with labeled vertices or real-valued edge weights. We call such structures "acsets," short for "attributed $\mathsf{C}$-sets." Derived from previous work on algebraic databases, acsets are a joint generalization of graphs and data frames. They also encompass more elaborate graph-like objects such as wiring diagrams and Petri nets with rate constants. We develop the mathematical theory of acsets and then describe a generic implementation in the Julia programming language, which uses advanced language features to achieve performance comparable with specialized data structures.]]></description>
      <pubDate>Tue, 27 Dec 2022 23:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-4-5</link>
      <guid>https://doi.org/10.32408/compositionality-4-5</guid>
      <author>Patterson, Evan</author>
      <author>Lynch, Owen</author>
      <author>Fairbanks, James</author>
      <dc:creator>Patterson, Evan</dc:creator>
      <dc:creator>Lynch, Owen</dc:creator>
      <dc:creator>Fairbanks, James</dc:creator>
      <content:encoded><![CDATA[Many mathematical objects can be represented as functors from finitely-presented categories $\mathsf{C}$ to $\mathsf{Set}$. For instance, graphs are functors to $\mathsf{Set}$ from the category with two parallel arrows. Such functors are known informally as $\mathsf{C}$-sets. In this paper, we describe and implement an extension of $\mathsf{C}$-sets having data attributes with fixed types, such as graphs with labeled vertices or real-valued edge weights. We call such structures "acsets," short for "attributed $\mathsf{C}$-sets." Derived from previous work on algebraic databases, acsets are a joint generalization of graphs and data frames. They also encompass more elaborate graph-like objects such as wiring diagrams and Petri nets with rate constants. We develop the mathematical theory of acsets and then describe a generic implementation in the Julia programming language, which uses advanced language features to achieve performance comparable with specialized data structures.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Structured versus Decorated Cospans</title>
      <description><![CDATA[One goal of applied category theory is to understand open systems. We compare two ways of describing open systems as cospans equipped with extra data. First, given a functor $L \colon \mathsf{A} \to \mathsf{X}$, a "structured cospan" is a diagram in $\mathsf{X}$ of the form $L(a) \rightarrow x \leftarrow L(b)$. If $\mathsf{A}$ and $\mathsf{X}$ have finite colimits and $L$ preserves them, it is known that there is a symmetric monoidal double category whose objects are those of $\mathsf{A}$ and whose horizontal 1-cells are structured cospans. Second, given a pseudofunctor $F \colon \mathsf{A} \to \mathbf{Cat}$, a "decorated cospan" is a diagram in $\mathsf{A}$ of the form $a \rightarrow m \leftarrow b$ together with an object of $F(m)$. Generalizing the work of Fong, we show that if $\mathsf{A}$ has finite colimits and $F \colon (\mathsf{A},+) \to (\mathsf{Cat},\times)$ is symmetric lax monoidal, there is a symmetric monoidal double category whose objects are those of $\mathsf{A}$ and whose horizontal 1-cells are decorated cospans. We prove that under certain conditions, these two constructions become isomorphic when we take $\mathsf{X} = \int F$ to be the Grothendieck category of $F$. We illustrate these ideas with applications to electrical circuits, Petri nets, dynamical systems and epidemiological modeling.]]></description>
      <pubDate>Wed, 31 Aug 2022 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-4-3</link>
      <guid>https://doi.org/10.32408/compositionality-4-3</guid>
      <author>Baez, John C.</author>
      <author>Courser, Kenny</author>
      <author>Vasilakopoulou, Christina</author>
      <dc:creator>Baez, John C.</dc:creator>
      <dc:creator>Courser, Kenny</dc:creator>
      <dc:creator>Vasilakopoulou, Christina</dc:creator>
      <content:encoded><![CDATA[One goal of applied category theory is to understand open systems. We compare two ways of describing open systems as cospans equipped with extra data. First, given a functor $L \colon \mathsf{A} \to \mathsf{X}$, a "structured cospan" is a diagram in $\mathsf{X}$ of the form $L(a) \rightarrow x \leftarrow L(b)$. If $\mathsf{A}$ and $\mathsf{X}$ have finite colimits and $L$ preserves them, it is known that there is a symmetric monoidal double category whose objects are those of $\mathsf{A}$ and whose horizontal 1-cells are structured cospans. Second, given a pseudofunctor $F \colon \mathsf{A} \to \mathbf{Cat}$, a "decorated cospan" is a diagram in $\mathsf{A}$ of the form $a \rightarrow m \leftarrow b$ together with an object of $F(m)$. Generalizing the work of Fong, we show that if $\mathsf{A}$ has finite colimits and $F \colon (\mathsf{A},+) \to (\mathsf{Cat},\times)$ is symmetric lax monoidal, there is a symmetric monoidal double category whose objects are those of $\mathsf{A}$ and whose horizontal 1-cells are decorated cospans. We prove that under certain conditions, these two constructions become isomorphic when we take $\mathsf{X} = \int F$ to be the Grothendieck category of $F$. We illustrate these ideas with applications to electrical circuits, Petri nets, dynamical systems and epidemiological modeling.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Coherence for adjunctions in a $3$-category via string diagrams</title>
      <description><![CDATA[We define a coherent adjunction in a strict $3$-category and we use string diagrams to show that any adjunction can be extended to a coherent adjunction in an essentially unique way.]]></description>
      <pubDate>Mon, 29 Aug 2022 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-4-2</link>
      <guid>https://doi.org/10.32408/compositionality-4-2</guid>
      <author>Araújo, Manuel</author>
      <dc:creator>Araújo, Manuel</dc:creator>
      <content:encoded><![CDATA[We define a coherent adjunction in a strict $3$-category and we use string diagrams to show that any adjunction can be extended to a coherent adjunction in an essentially unique way.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Lambek pregroups are Frobenius spiders in preorders</title>
      <description><![CDATA["Spider" is a nickname of special Frobenius algebras, a fundamental structure from mathematics, physics, and computer science. Pregroups are a fundamental structure from linguistics. Pregroups and spiders have been used together in natural language processing: one for syntax, the other for semantics. It turns out that pregroups themselves can be characterized as pointed spiders in the category of preordered relations, where they naturally arise from grammars. The other way around, preordered spider algebras in general can be characterized as unions of pregroups. This extends the characterization of relational spider algebras as disjoint unions of groups. The compositional framework that emerged with the results suggests new ways to understand and apply the basis structures in machine learning and data analysis.]]></description>
      <pubDate>Tue, 12 Apr 2022 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-4-1</link>
      <guid>https://doi.org/10.32408/compositionality-4-1</guid>
      <author>Pavlovic, Dusko</author>
      <dc:creator>Pavlovic, Dusko</dc:creator>
      <content:encoded><![CDATA["Spider" is a nickname of special Frobenius algebras, a fundamental structure from mathematics, physics, and computer science. Pregroups are a fundamental structure from linguistics. Pregroups and spiders have been used together in natural language processing: one for syntax, the other for semantics. It turns out that pregroups themselves can be characterized as pointed spiders in the category of preordered relations, where they naturally arise from grammars. The other way around, preordered spider algebras in general can be characterized as unions of pregroups. This extends the characterization of relational spider algebras as disjoint unions of groups. The compositional framework that emerged with the results suggests new ways to understand and apply the basis structures in machine learning and data analysis.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Language Modeling with Reduced Densities</title>
      <description><![CDATA[This work originates from the observation that today's state-of-the-art statistical language models are impressive not only for their performance, but also - and quite crucially - because they are built entirely from correlations in unstructured text data. The latter observation prompts a fundamental question that lies at the heart of this paper: What mathematical structure exists in unstructured text data? We put forth enriched category theory as a natural answer. We show that sequences of symbols from a finite alphabet, such as those found in a corpus of text, form a category enriched over probabilities. We then address a second fundamental question: How can this information be stored and modeled in a way that preserves the categorical structure? We answer this by constructing a functor from our enriched category of text to a particular enriched category of reduced density operators. The latter leverages the Loewner order on positive semidefinite operators, which can further be interpreted as a toy example of entailment.]]></description>
      <pubDate>Mon, 29 Nov 2021 23:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-3-4</link>
      <guid>https://doi.org/10.32408/compositionality-3-4</guid>
      <author>Bradley, Tai-Danae</author>
      <author>Vlassopoulos, Yiannis</author>
      <dc:creator>Bradley, Tai-Danae</dc:creator>
      <dc:creator>Vlassopoulos, Yiannis</dc:creator>
      <content:encoded><![CDATA[This work originates from the observation that today's state-of-the-art statistical language models are impressive not only for their performance, but also - and quite crucially - because they are built entirely from correlations in unstructured text data. The latter observation prompts a fundamental question that lies at the heart of this paper: What mathematical structure exists in unstructured text data? We put forth enriched category theory as a natural answer. We show that sequences of symbols from a finite alphabet, such as those found in a corpus of text, form a category enriched over probabilities. We then address a second fundamental question: How can this information be stored and modeled in a way that preserves the categorical structure? We answer this by constructing a functor from our enriched category of text to a particular enriched category of reduced density operators. The latter leverages the Loewner order on positive semidefinite operators, which can further be interpreted as a toy example of entailment.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Homotopy theory of Moore flows (I)</title>
      <description><![CDATA[Erratum, 11 July 2022: This is an updated version of the original paper in which the notion of reparametrization category was incorrectly axiomatized. Details on the changes to the original paper are provided in the Appendix. A reparametrization category is a small topologically enriched semimonoidal category such that the semimonoidal structure induces a structure of a semigroup on objects, such that all spaces of maps are contractible and such that each map can be decomposed (not necessarily in a unique way) as a tensor product of two maps. A Moore flow is a small semicategory enriched over the biclosed semimonoidal category of enriched presheaves over a reparametrization category. We construct the q-model category of Moore flows. It is proved that it is Quillen equivalent to the q-model category of flows. This result is the first step to establish a zig-zag of Quillen equivalences between the q-model structure of multipointed $d$-spaces and the q-model structure of flows.]]></description>
      <pubDate>Sun, 29 Aug 2021 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-3-3</link>
      <guid>https://doi.org/10.32408/compositionality-3-3</guid>
      <author>Gaucher, Philippe</author>
      <dc:creator>Gaucher, Philippe</dc:creator>
      <content:encoded><![CDATA[Erratum, 11 July 2022: This is an updated version of the original paper in which the notion of reparametrization category was incorrectly axiomatized. Details on the changes to the original paper are provided in the Appendix. A reparametrization category is a small topologically enriched semimonoidal category such that the semimonoidal structure induces a structure of a semigroup on objects, such that all spaces of maps are contractible and such that each map can be decomposed (not necessarily in a unique way) as a tensor product of two maps. A Moore flow is a small semicategory enriched over the biclosed semimonoidal category of enriched presheaves over a reparametrization category. We construct the q-model category of Moore flows. It is proved that it is Quillen equivalent to the q-model category of flows. This result is the first step to establish a zig-zag of Quillen equivalences between the q-model structure of multipointed $d$-spaces and the q-model structure of flows.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Compositionality of Rewriting Rules with Conditions</title>
      <description><![CDATA[We extend the notion of compositional associative rewriting as recently studied in the rule algebra framework literature to the setting of rewriting rules with conditions. Our methodology is category-theoretical in nature, where the definition of rule composition operations encodes the non-deterministic sequential concurrent application of rules in Double-Pushout (DPO) and Sesqui-Pushout (SqPO) rewriting with application conditions based upon $\mathcal{M}$-adhesive categories. We uncover an intricate interplay between the category-theoretical concepts of conditions on rules and morphisms, the compositionality and compatibility of certain shift and transport constructions for conditions, and thirdly the property of associativity of the composition of rules.]]></description>
      <pubDate>Wed, 21 Apr 2021 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-3-2</link>
      <guid>https://doi.org/10.32408/compositionality-3-2</guid>
      <author>Behr, Nicolas</author>
      <author>Krivine, Jean</author>
      <dc:creator>Behr, Nicolas</dc:creator>
      <dc:creator>Krivine, Jean</dc:creator>
      <content:encoded><![CDATA[We extend the notion of compositional associative rewriting as recently studied in the rule algebra framework literature to the setting of rewriting rules with conditions. Our methodology is category-theoretical in nature, where the definition of rule composition operations encodes the non-deterministic sequential concurrent application of rules in Double-Pushout (DPO) and Sesqui-Pushout (SqPO) rewriting with application conditions based upon $\mathcal{M}$-adhesive categories. We uncover an intricate interplay between the category-theoretical concepts of conditions on rules and morphisms, the compositionality and compatibility of certain shift and transport constructions for conditions, and thirdly the property of associativity of the composition of rules.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Categorical Stochastic Processes and Likelihood</title>
      <description><![CDATA[In this work we take a Category Theoretic perspective on the relationship between probabilistic modeling and function approximation. We begin by defining two extensions of function composition to stochastic process subordination: one based on the co-Kleisli category under the comonad (Omega x -) and one based on the parameterization of a category with a Lawvere theory. We show how these extensions relate to the category Stoch and other Markov Categories. Next, we apply the Para construction to extend stochastic processes to parameterized statistical models and we define a way to compose the likelihood functions of these models. We conclude with a demonstration of how the Maximum Likelihood Estimation procedure defines an identity-on-objects functor from the category of statistical models to the category of Learners. Code to accompany this paper can be found at https://github.com/dshieble/Categorical_Stochastic_Processes_and_Likelihood]]></description>
      <pubDate>Tue, 13 Apr 2021 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-3-1</link>
      <guid>https://doi.org/10.32408/compositionality-3-1</guid>
      <author>Shiebler, Dan</author>
      <dc:creator>Shiebler, Dan</dc:creator>
      <content:encoded><![CDATA[In this work we take a Category Theoretic perspective on the relationship between probabilistic modeling and function approximation. We begin by defining two extensions of function composition to stochastic process subordination: one based on the co-Kleisli category under the comonad (Omega x -) and one based on the parameterization of a category with a Lawvere theory. We show how these extensions relate to the category Stoch and other Markov Categories. Next, we apply the Para construction to extend stochastic processes to parameterized statistical models and we define a way to compose the likelihood functions of these models. We conclude with a demonstration of how the Maximum Likelihood Estimation procedure defines an identity-on-objects functor from the category of statistical models to the category of Learners. Code to accompany this paper can be found at https://github.com/dshieble/Categorical_Stochastic_Processes_and_Likelihood]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Closing the category of finitely presented functors under images made constructive</title>
      <description><![CDATA[For an additive category $\mathbf{P}$ we provide an explict construction of a category $\mathcal{Q}( \mathbf{P} )$ whose objects can be thought of as formally representing $\frac{\mathrm{im}( \gamma )}{\mathrm{im}( \rho ) \cap \mathrm{im}( \gamma )}$ for given morphisms $\gamma: A \rightarrow B$ and $\rho: C \rightarrow B$ in $\mathbf{P}$, even though $\mathbf{P}$ does not need to admit quotients or images. We show how it is possible to calculate effectively within $\mathcal{Q}( \mathbf{P} )$, provided that a basic problem related to syzygies can be handled algorithmically. We prove an equivalence of $\mathcal{Q}( \mathbf{P} )$ with the subcategory of the category of contravariant functors from $\mathbf{P}$ to the category of abelian groups $\mathbf{Ab}$ which contains all finitely presented functors and is closed under the operation of taking images. Moreover, we characterize the abelian case: $\mathcal{Q}( \mathbf{P} )$ is abelian if and only if it is equivalent to $\mathrm{fp}( \mathbf{P}^{\mathrm{op}}, \mathbf{Ab} )$, the category of all finitely presented functors, which in turn, by a theorem of Freyd, is abelian if and only if $\mathbf{P}$ has weak kernels. The category $\mathcal{Q}( \mathbf{P} )$ is a categorical abstraction of the data structure for finitely presented $R$-modules employed by the computer algebra system Macaulay2, where $R$ is a ring. By our generalization to arbitrary additive categories, we show how this data structure can also be used for modeling finitely presented graded modules, finitely presented functors, and some not necessarily finitely presented modules over a non-coherent ring.]]></description>
      <pubDate>Mon, 31 Aug 2020 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-2-4</link>
      <guid>https://doi.org/10.32408/compositionality-2-4</guid>
      <author>Posur, Sebastian</author>
      <dc:creator>Posur, Sebastian</dc:creator>
      <content:encoded><![CDATA[For an additive category $\mathbf{P}$ we provide an explict construction of a category $\mathcal{Q}( \mathbf{P} )$ whose objects can be thought of as formally representing $\frac{\mathrm{im}( \gamma )}{\mathrm{im}( \rho ) \cap \mathrm{im}( \gamma )}$ for given morphisms $\gamma: A \rightarrow B$ and $\rho: C \rightarrow B$ in $\mathbf{P}$, even though $\mathbf{P}$ does not need to admit quotients or images. We show how it is possible to calculate effectively within $\mathcal{Q}( \mathbf{P} )$, provided that a basic problem related to syzygies can be handled algorithmically. We prove an equivalence of $\mathcal{Q}( \mathbf{P} )$ with the subcategory of the category of contravariant functors from $\mathbf{P}$ to the category of abelian groups $\mathbf{Ab}$ which contains all finitely presented functors and is closed under the operation of taking images. Moreover, we characterize the abelian case: $\mathcal{Q}( \mathbf{P} )$ is abelian if and only if it is equivalent to $\mathrm{fp}( \mathbf{P}^{\mathrm{op}}, \mathbf{Ab} )$, the category of all finitely presented functors, which in turn, by a theorem of Freyd, is abelian if and only if $\mathbf{P}$ has weak kernels. The category $\mathcal{Q}( \mathbf{P} )$ is a categorical abstraction of the data structure for finitely presented $R$-modules employed by the computer algebra system Macaulay2, where $R$ is a ring. By our generalization to arbitrary additive categories, we show how this data structure can also be used for modeling finitely presented graded modules, finitely presented functors, and some not necessarily finitely presented modules over a non-coherent ring.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Infinite products and zero-one laws in categorical probability</title>
      <description><![CDATA[Markov categories are a recent category-theoretic approach to the foundations of probability and statistics. Here we develop this approach further by treating infinite products and the Kolmogorov extension theorem. This is relevant for all aspects of probability theory in which infinitely many random variables appear at a time. These infinite tensor products $\bigotimes_{i \in J} X_i$ come in two versions: a weaker but more general one for families of objects $(X_i)_{i \in J}$ in semicartesian symmetric monoidal categories, and a stronger but more specific one for families of objects in Markov categories. As a first application, we state and prove versions of the zero-one laws of Kolmogorov and Hewitt-Savage for Markov categories. This gives general versions of these results which can be instantiated not only in measure-theoretic probability, where they specialize to the standard ones in the setting of standard Borel spaces, but also in other contexts.]]></description>
      <pubDate>Mon, 10 Aug 2020 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-2-3</link>
      <guid>https://doi.org/10.32408/compositionality-2-3</guid>
      <author>Fritz, Tobias</author>
      <author>Rischel, Eigil Fjeldgren</author>
      <dc:creator>Fritz, Tobias</dc:creator>
      <dc:creator>Rischel, Eigil Fjeldgren</dc:creator>
      <content:encoded><![CDATA[Markov categories are a recent category-theoretic approach to the foundations of probability and statistics. Here we develop this approach further by treating infinite products and the Kolmogorov extension theorem. This is relevant for all aspects of probability theory in which infinitely many random variables appear at a time. These infinite tensor products $\bigotimes_{i \in J} X_i$ come in two versions: a weaker but more general one for families of objects $(X_i)_{i \in J}$ in semicartesian symmetric monoidal categories, and a stronger but more specific one for families of objects in Markov categories. As a first application, we state and prove versions of the zero-one laws of Kolmogorov and Hewitt-Savage for Markov categories. This gives general versions of these results which can be instantiated not only in measure-theoretic probability, where they specialize to the standard ones in the setting of standard Borel spaces, but also in other contexts.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Assignments to sheaves of pseudometric spaces</title>
      <description><![CDATA[An assignment to a sheaf is the choice of a local section from each open set in the sheaf's base space, without regard to how these local sections are related to one another. This article explains that the consistency radius -- which quantifies the agreement between overlapping local sections in the assignment -- is a continuous map. When thresholded, the consistency radius produces the consistency filtration, which is a filtration of open covers. This article shows that the consistency filtration is a functor that transforms the structure of the sheaf and assignment into a nested set of covers in a structure-preserving way. Furthermore, this article shows that consistency filtration is robust to perturbations, establishing its validity for arbitrarily thresholded, noisy data.]]></description>
      <pubDate>Mon, 01 Jun 2020 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-2-2</link>
      <guid>https://doi.org/10.32408/compositionality-2-2</guid>
      <author>Robinson, Michael</author>
      <dc:creator>Robinson, Michael</dc:creator>
      <content:encoded><![CDATA[An assignment to a sheaf is the choice of a local section from each open set in the sheaf's base space, without regard to how these local sections are related to one another. This article explains that the consistency radius -- which quantifies the agreement between overlapping local sections in the assignment -- is a continuous map. When thresholded, the consistency radius produces the consistency filtration, which is a filtration of open covers. This article shows that the consistency filtration is a functor that transforms the structure of the sheaf and assignment into a nested set of covers in a structure-preserving way. Furthermore, this article shows that consistency filtration is robust to perturbations, establishing its validity for arbitrarily thresholded, noisy data.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Distributive laws for Lawvere theories</title>
      <description><![CDATA[Distributive laws give a way of combining two algebraic structures expressed as monads; in this paper we propose a theory of distributive laws for combining algebraic structures expressed as Lawvere theories. We propose four approaches, involving profunctors, monoidal profunctors, an extension of the free finite-product category 2-monad from Cat to Prof, and factorisation systems respectively. We exhibit comparison functors between CAT and each of these new frameworks to show that the distributive laws between the Lawvere theories correspond in a suitable way to distributive laws between their associated finitary monads. The different but equivalent formulations then provide, between them, a framework conducive to generalisation, but also an explicit description of the composite theories arising from distributive laws.]]></description>
      <pubDate>Sun, 24 May 2020 22:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-2-1</link>
      <guid>https://doi.org/10.32408/compositionality-2-1</guid>
      <author>Cheng, Eugenia</author>
      <dc:creator>Cheng, Eugenia</dc:creator>
      <content:encoded><![CDATA[Distributive laws give a way of combining two algebraic structures expressed as monads; in this paper we propose a theory of distributive laws for combining algebraic structures expressed as Lawvere theories. We propose four approaches, involving profunctors, monoidal profunctors, an extension of the free finite-product category 2-monad from Cat to Prof, and factorisation systems respectively. We exhibit comparison functors between CAT and each of these new frameworks to show that the distributive laws between the Lawvere theories correspond in a suitable way to distributive laws between their associated finitary monads. The different but equivalent formulations then provide, between them, a framework conducive to generalisation, but also an explicit description of the composite theories arising from distributive laws.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Network Models from Petri Nets with Catalysts</title>
      <description><![CDATA[Petri networks and network models are two frameworks for the compositional design of systems of interacting entities. Here we show how to combine them using the concept of a "catalyst": an entity that is neither destroyed nor created by any process it engages in. In a Petri net, a place is a catalyst if its in-degree equals its out-degree for every transition. We show how a Petri net with a chosen set of catalysts gives a network model. This network model maps any list of catalysts from the chosen set to the category whose morphisms are all the processes enabled by this list of catalysts. Applying the Grothendieck construction, we obtain a category fibered over the category whose objects are lists of catalysts. This category has as morphisms all processes enabled by some list of catalysts. While this category has a symmetric monoidal structure that describes doing processes in parallel, its fibers also have premonoidal structures that describe doing one process and then another while reusing the catalysts.]]></description>
      <pubDate>Tue, 24 Dec 2019 23:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-1-4</link>
      <guid>https://doi.org/10.32408/compositionality-1-4</guid>
      <author>Baez, John C.</author>
      <author>Foley, John</author>
      <author>Moeller, Joe</author>
      <dc:creator>Baez, John C.</dc:creator>
      <dc:creator>Foley, John</dc:creator>
      <dc:creator>Moeller, Joe</dc:creator>
      <content:encoded><![CDATA[Petri networks and network models are two frameworks for the compositional design of systems of interacting entities. Here we show how to combine them using the concept of a "catalyst": an entity that is neither destroyed nor created by any process it engages in. In a Petri net, a place is a catalyst if its in-degree equals its out-degree for every transition. We show how a Petri net with a chosen set of catalysts gives a network model. This network model maps any list of catalysts from the chosen set to the category whose morphisms are all the processes enabled by this list of catalysts. Applying the Grothendieck construction, we obtain a category fibered over the category whose objects are lists of catalysts. This category has as morphisms all processes enabled by some list of catalysts. While this category has a symmetric monoidal structure that describes doing processes in parallel, its fibers also have premonoidal structures that describe doing one process and then another while reusing the catalysts.]]></content:encoded>
      <slash:comments>0</slash:comments>
    </item>
    <item>
      <title>Stinespring's construction as an adjunction</title>
      <description><![CDATA[Given a representation of a unital $C^*$-algebra $\mathcal{A}$ on a Hilbert space $\mathcal{H}$, together with a bounded linear map $V:\mathcal{K}\to\mathcal{H}$ from some other Hilbert space, one obtains a completely positive map on $\mathcal{A}$ via restriction using the adjoint action associated to $V$. We show this restriction forms a natural transformation from a functor of $C^*$-algebra representations to a functor of completely positive maps. We exhibit Stinespring's construction as a left adjoint of this restriction. Our Stinespring adjunction provides a universal property associated to minimal Stinespring dilations and morphisms of Stinespring dilations. We use these results to prove the purification postulate for all finite-dimensional $C^*$-algebras.]]></description>
      <pubDate>Thu, 19 Dec 2019 23:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-1-2</link>
      <guid>https://doi.org/10.32408/compositionality-1-2</guid>
      <author>Parzygnat, Arthur J.</author>
      <dc:creator>Parzygnat, Arthur J.</dc:creator>
      <content:encoded><![CDATA[Given a representation of a unital $C^*$-algebra $\mathcal{A}$ on a Hilbert space $\mathcal{H}$, together with a bounded linear map $V:\mathcal{K}\to\mathcal{H}$ from some other Hilbert space, one obtains a completely positive map on $\mathcal{A}$ via restriction using the adjoint action associated to $V$. We show this restriction forms a natural transformation from a functor of $C^*$-algebra representations to a functor of completely positive maps. We exhibit Stinespring's construction as a left adjoint of this restriction. Our Stinespring adjunction provides a universal property associated to minimal Stinespring dilations and morphisms of Stinespring dilations. We use these results to prove the purification postulate for all finite-dimensional $C^*$-algebras.]]></content:encoded>
      <slash:comments>0</slash:comments>
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    <item>
      <title>Fuzzy sets and presheaves</title>
      <description><![CDATA[This note presents a presheaf theoretic approach to the construction of fuzzy sets, which builds on Barr's description of fuzzy sets as sheaves of monomorphisms on a locale. A presheaf-theoretic method is used to show that the category of fuzzy sets is complete and co-complete, and to present explicit descriptions of classical fuzzy sets that arise as limits and colimits. The Boolean localization construction for sheaves and presheaves on a locale L specializes to a theory of stalks if L approximates the structure of a closed interval in the real line. The system V(X) of Vietoris-Rips complexes for a data cloud X becomes both a simplicial fuzzy set and a simplicial sheaf in this general framework. This example is explicitly discussed in this paper, in stages.]]></description>
      <pubDate>Thu, 19 Dec 2019 23:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-1-3</link>
      <guid>https://doi.org/10.32408/compositionality-1-3</guid>
      <author>Jardine, J. F.</author>
      <dc:creator>Jardine, J. F.</dc:creator>
      <content:encoded><![CDATA[This note presents a presheaf theoretic approach to the construction of fuzzy sets, which builds on Barr's description of fuzzy sets as sheaves of monomorphisms on a locale. A presheaf-theoretic method is used to show that the category of fuzzy sets is complete and co-complete, and to present explicit descriptions of classical fuzzy sets that arise as limits and colimits. The Boolean localization construction for sheaves and presheaves on a locale L specializes to a theory of stalks if L approximates the structure of a closed interval in the real line. The system V(X) of Vietoris-Rips complexes for a data cloud X becomes both a simplicial fuzzy set and a simplicial sheaf in this general framework. This example is explicitly discussed in this paper, in stages.]]></content:encoded>
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      <title>An effect-theoretic reconstruction of quantum theory</title>
      <description><![CDATA[An often used model for quantum theory is to associate to every physical system a C*-algebra. From a physical point of view it is unclear why operator algebras would form a good description of nature. In this paper, we find a set of physically meaningful assumptions such that any physical theory satisfying these assumptions must embed into the category of finite-dimensional C*-algebras. These assumptions were originally introduced in the setting of effectus theory, a categorical logical framework generalizing classical and quantum logic. As these assumptions have a physical interpretation, this motivates the usage of operator algebras as a model for quantum theory. In contrast to other reconstructions of quantum theory, we do not start with the framework of generalized probabilistic theories and instead use effect theories where no convex structure and no tensor product needs to be present. The lack of this structure in effectus theory has led to a different notion of pure maps. A map in an effectus is pure when it is a composition of a compression and a filter. These maps satisfy particular universal properties and respectively correspond to `forgetting' and `measuring' the validity of an effect. We define a pure effect theory (PET) to be an effect theory where the pure maps form a dagger-category and filters and compressions are adjoint. We show that any convex finite-dimensional PET must embed into the category of Euclidean Jordan algebras. Moreover, if the PET also has monoidal structure, then we show that it must embed into either the category of real or complex C*-algebras, which completes our reconstruction.]]></description>
      <pubDate>Tue, 17 Dec 2019 23:00:00 +0000</pubDate>
      <link>https://doi.org/10.32408/compositionality-1-1</link>
      <guid>https://doi.org/10.32408/compositionality-1-1</guid>
      <author>van de Wetering, John</author>
      <dc:creator>van de Wetering, John</dc:creator>
      <content:encoded><![CDATA[An often used model for quantum theory is to associate to every physical system a C*-algebra. From a physical point of view it is unclear why operator algebras would form a good description of nature. In this paper, we find a set of physically meaningful assumptions such that any physical theory satisfying these assumptions must embed into the category of finite-dimensional C*-algebras. These assumptions were originally introduced in the setting of effectus theory, a categorical logical framework generalizing classical and quantum logic. As these assumptions have a physical interpretation, this motivates the usage of operator algebras as a model for quantum theory. In contrast to other reconstructions of quantum theory, we do not start with the framework of generalized probabilistic theories and instead use effect theories where no convex structure and no tensor product needs to be present. The lack of this structure in effectus theory has led to a different notion of pure maps. A map in an effectus is pure when it is a composition of a compression and a filter. These maps satisfy particular universal properties and respectively correspond to `forgetting' and `measuring' the validity of an effect. We define a pure effect theory (PET) to be an effect theory where the pure maps form a dagger-category and filters and compressions are adjoint. We show that any convex finite-dimensional PET must embed into the category of Euclidean Jordan algebras. Moreover, if the PET also has monoidal structure, then we show that it must embed into either the category of real or complex C*-algebras, which completes our reconstruction.]]></content:encoded>
      <slash:comments>0</slash:comments>
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