We highlight the underlying category-theoretic structure of measures of information flow. We present an axiomatic framework in which communication systems are represented as morphisms, and information flow is characterized by its behavior when communication systems are combined. Our framework includes a variety of discrete, continuous, and, conjecturally, quantum information measures. It also includes some familiar mathematical constructs not normally associated with information, such as vector space dimension. We discuss these examples and prove basic results from the axioms.
A theory of sketches for arithmetic universes (AUs) is developed. A restricted notion of sketch, called here "context", is defined with the property that every non-strict model is uniquely isomorphic to a strict model. This allows us to reconcile the syntactic, dealt with strictly using universal algebra, with the semantic, in which non-strict models must be considered. For any context T, a concrete construction is given of the AU AU<T> freely generated by it. A 2-category Con of contexts is defined, with a full and faithful 2-functor to the 2-category of AUs and strict AU-functors, given by T…
Reasoning in the 2-category Con of contexts, certain sketches for arithmetic universes (i.e. list arithmetic pretoposes; AUs), is shown to give rise to base-independent results of Grothendieck toposes, provided the base elementary topos has a natural numbers object. Categories of strict models of contexts $T$ in AUs are acted on strictly on the left by non-strict AU-functors and strictly on the right by context maps, and the actions combine in a strict action of a Gray tensor product. Any context extension $T_0 \subset T_1$ gives rise to a bundle. For each point of $T_0$ - a model $M$ of $T_0$…
In arithmetic statistics and analytic number theory, the asymptotic growth rate of counting functions giving the number of objects with order below $X$ is studied as $X\to \infty$. We define general counting functions which count epimorphisms out of an object on a category under some ordering. Given a probability measure $μ$ on the isomorphism classes of the category with sufficient respect for a product structure, we prove a version of the Law of Large Numbers to give the asymptotic growth rate as $X$ tends towards $\infty$ of such functions with probability $1$ in terms of the finite moments…
The category-valued trace assigns to a bimodule category over a linear monoidal category a linear category. It generalizes Drinfeld centers of monoidal categories and the relative Deligne product of bimodule categories. In this article, we study bimodule categories that are given as categories of bicomodules over a Hopf algebra. Our main result is a representation-theoretic realization of the category-valued trace as a category of generalized Hopf bimodules.
We develop category-theoretic framework for universal homogeneous objects, with some applications in the theory of Banach spaces, linear orderings, and in topology of compact spaces.
We present a new, category theoretic point of view on finite Ramsey theory. Our aims are as follows: -- to define the category theoretic notions needed for the development of finite Ramsey Theory, -- to state, in terms of these notions, the general fundamental Ramsey results (of which various concrete Ramsey results are special cases), and -- to give self-contained proofs within the category theoretic framework of these general results. We also provide some concrete illustrations of the general method.
Through careful analysis of an argument of Brooke-Taylor and Rosicky, we show that the powerful image of any accessible functor is closed under colimits of $κ$-chains, $κ$ a sufficiently large almost measurable cardinal. This condition on powerful images, by methods resembling those of Lieberman and Rosicky, implies $κ$-locality of Galois types. As this, in turn, implies sufficient measurability of $κ$, via a paper of Boney and Unger, we obtain an equivalence: a purely category-theoretic characterization of almost measurable cardinals.
Gambino and Garner proved that the syntactic category of a dependent type theory with identity types can be endowed with a weak factorization system structure, called identity type weak factorization system. In this paper we consider an enrichment of Joyal's notion of tribe, which we call a tribe with weakly stable path objects, and prove a purely category-theoretic version of the identity type weak factorization system, thus generalizing Gambino and Garner's result. We conclude showing that this structure subsumes also the weak factorization systems coming from the topological and simplicial …
In this paper, we define a generalization of indexed categories and contextual categories which we call contextually indexed (contextual) categories. While contextual categories are models of ordinary type theories, contextually indexed (contextual) categories are models of indexed type theories. We also define type-theoretic semi-fibration categories which generalize type-theoretic fibration categories. Every model category in which cofibrations are stable under pullbacks is a type-theoretic semi-fibration category. We show that type-theoretic semi-fibration category gives rise to a contextua…
Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical constructions are permissible. In this expository paper we summarize and compare a number of such "set-theoretic foundations for category theory," and describe their implications for the everyday use of category theory. We assume the reader has some basic knowledge of category theory, but little or no prior exp…
We study 2-monads and their algebras using a Cat-enriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2-categorical points of view. Every 2-category with finite limits and colimits has a canonical model structure in which the weak equivalences are the equivalences; we use these to construct more interesting model structures on 2-categories, including a model structure on the 2-category of algebras for a 2-monad T, and a model structure on a 2-category of 2-monads on a fixed 2-category K.
We investigate what would be a correct definition of categorical completeness for C*-categories and propose several variants of such a definition that make the category of Hilbert modules over a C*-algebra a free (co)completion. We extend results about generators and comparison theory known for W*-categories with direct sums and splitting of symmetric projections to our "complete C*-categories" and we give an abstract characterization of categories of Hilbert modules over a C*-algebra or a C*-category as "complete C*-category having enough absolutely compact morphisms (and a generator)". We th…
This PhD thesis deals with some new models of intensional type theory and the Univalence Axiom introduced by Vladimir Voevodsky. Our work takes place in the framework of the definitions of type-theoretic fibration categories (the notion of model under consideration in this thesis) and universe in a type-theoretic fibration category, definitions due to Michael Shulman. The goal of this thesis consists mainly in the exploration of the stability of the univalence axiom, in particular in the following sense: being given a type-theoretic fibration category C equipped with a univalent universe U, we…
The ergodic decomposition theorem is a cornerstone result of dynamical systems and ergodic theory. It states that every invariant measure on a dynamical system is a mixture of ergodic ones. Here we formulate and prove the theorem in terms of string diagrams, using the formalism of Markov categories. We recover the usual measure-theoretic statement by instantiating our result in the category of stochastic kernels. Along the way we give a conceptual treatment of several concepts in the theory of deterministic and stochastic dynamical systems. In particular, - ergodic measures appear very natural…
This thesis is about trying to understand various aspects of partial symmetry using ideas from semigroup and category theory. In Chapter 2 it is shown that the left Rees monoids underlying self-similar group actions are precisely monoid HNN-extensions. In particular it is shown that every group HNN-extension arises from a self-similar group action. Examples of these monoids are constructed from fractals. These ideas are generalised in Chapter 3 to a correspondence between left Rees categories, self-similar groupoid actions and category HNN-extensions of groupoids, leading to a deeper relations…
