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.
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.
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.
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 […]
