Language Modeling with Reduced Densities

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.

Homotopy theory of Moore flows (I)

A reparametrization category is a small topologically enriched symmetric semimonoidal category such that the semimonoidal structure induces a structure of a commutative 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 closed 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.

Compositionality of Rewriting Rules with Conditions

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 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.

Categorical Stochastic Processes and Likelihood

We take a category-theoretic perspective on the relationship between probabilistic modeling and gradient based optimization. We define two extensions of function composition to stochastic process subordination: one based on a co-Kleisli category and one based on the parameterization of a category with a Lawvere theory. We show how these extensions relate to the category of Markov kernels Stoch through a pushforward procedure.We extend stochastic processes to parametric statistical models and define a way to compose the likelihood functions of these models. We demonstrate how the maximum likelihood estimation procedure defines a family of identity-on-objects functors from categories of statistical models to the category of supervised learning algorithms Learn.Code to accompany this paper can be found on GitHub (https://github.com/dshieble/Categorical_Stochastic_Processes_and_Likelihood).

Closing the category of finitely presented functors under images made constructive

For an additive category P we provide an explicit construction of a category Q(P) whose objects can be thought of as formally representing im(γ)im(ρ)∩im(γ) for given morphisms γ:A→B and ρ:C→B in P, even though P does not need to admit quotients or images. We show how it is possible to calculate effectively within Q(P), provided that a basic problem related to syzygies can be handled algorithmically. We prove an equivalence of Q(P) with the smallest subcategory of the category of contravariant functors from P to the category of abelian groups Ab which contains all finitely presented functors and is closed under the operation of taking images. Moreover, we characterize the abelian case: Q(P) is abelian if and only if it is equivalent to fp(Pop,Ab), the category of all finitely presented functors, which in turn, by a theorem of Freyd, is abelian if and only if P has weak kernels.The category Q(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.

Infinite products and zero-one laws in categorical probability

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 ⨂i∈JXi come in two versions: a weaker but more general one for families of objects (Xi)i∈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.

Assignments to sheaves of pseudometric spaces

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.

Distributive laws for Lawvere theories

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.

Network Models from Petri Nets with Catalysts

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.

Fuzzy sets and presheaves

This paper 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. Presheaves are used to give explicit descriptions of limit and colimit descriptions in fuzzy sets on an interval. The Boolean localization construction for sheaves on a locale specializes to a theory of stalks for sheaves and presheaves on an interval.The system V∗(X) of Vietoris-Rips complexes for a data set X is both a simplicial fuzzy set and a simplicial sheaf in this general framework. This example is explicitly discussed through a series of examples.