# Atlantic Category Theory Seminar

Details about the 2021/2022 Atlantic Category Theory (ATCAT) Seminar can be found below. Information about previous seminars can be found here.

### Fall 2021 schedule

• September 14th, 2021 [slides]
Jason Parker, Brandon University
Locally bounded enriched categories
Abstract

Based on ideas of Freyd and Kelly in [1], the notion of locally bounded category was explicitly introduced by Kelly in [2, Chapter 6], where a given locally bounded symmetric monoidal closed category was used as the basis for a general treatment of enriched limit theories. Locally bounded categories subsume locally presentable categories and a significant number of other examples, including many topological categories and (concrete) quasitoposes that are not locally presentable. As such, the notion of locally bounded category is much weaker than the notion of locally presentable category: for example, a locally bounded category (such as the category of topological spaces and continuous maps) need not have a small dense subcategory or even a strong generator. Nevertheless, locally bounded categories still retain some of the convenient features of locally presentable categories, such as reflectivity results for orthogonal subcategories (see [1, 3]) and results on the existence of free monads, colimits of monads, and colimits in categories of algebras (see [3] and further references therein). The enriched limit theories over locally bounded closed categories treated by Kelly in [2, Chapter 6] generalize the enriched finite limit theories that Kelly introduced in [4], where he also introduced the notions of locally presentable symmetric monoidal closed category and locally presentable enriched category. However, notably absent from the literature has been a notion of locally bounded enriched category that would complete the parallel between the locally bounded and locally presentable settings.

In this talk based on joint work with Rory Lucyshyn-Wright, we define and study the notion of a locally bounded enriched category over a (locally bounded) symmetric monoidal closed category, generalizing the locally bounded ordinary categories of Freyd and Kelly. Along with providing many new examples of locally bounded (closed) categories, we demonstrate that locally bounded enriched categories admit fully enriched analogues of many of the convenient results enjoyed by locally bounded ordinary categories. In particular, we prove full enrichments of Freyd and Kelly's reflectivity and local boundedness results for orthogonal subcategories and categories of models for sketches and theories. We also provide characterization results for locally bounded enriched categories in terms of enriched presheaf categories, and show that locally bounded enriched categories satisfy useful adjoint functor theorems. In addition, we prove some general results for constructing examples of locally bounded enriched categories. We also define and study the notion of an alpha-bounded-small limit with respect to a locally alpha-bounded closed category, which parallels Kelly's notion of alpha-small limit with respect to a locally alpha-presentable closed category.

[1] P.J. Freyd and G.M. Kelly. Categories of continuous functors I. Journal of Pure and Applied Algebra Vol. 2, Issue 3, 169-191, 1972.

[2] G.M. Kelly. Basic concepts of enriched category theory. Repr. Theory and Applications of Categories, No. 10, 2005, Reprint of the 1982 original [Cambridge University Press].

[3] G.M. Kelly. A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on. Bulletin of the Australian Mathematical Society Vol. 22, 1-83, 1980.

[4] G.M. Kelly. Structures defined by finite limits in the enriched context I. Cahiers de Topologie et Géométrie Catégoriques Différentielle 23, No. 1, 3-42, 1982.

• September 21st, 2021
JS Lemay, Mount Allison University
The Fundamental Theorems of Calculus and Zinbiel Algebras
Abstract

In commutative algebra, derivations and integrations (also called O-operators) generalize the notions of the derivative and integral from calculus. On the other hand, Zinbiel algebras are a special kind of commutative non-unital associative algebras that capture the notion of riffle shuffle permutations.

Zinbiel algebras and integrations are closely related: every integration induces a Zinbiel algebra, and every Zinbiel algebra induces an integration. However, these constructions are not inverses! Indeed, while every Zinbiel algebra is constructed from an integration, not every integration is constructed from a Zinbiel algebra.

In this talk, I will explain how an integration is constructed from a Zinbiel algebra precisely when said integration comes equipped with a derivation, and that together they satisfy the two fundamental theorems of calculus. Thus we obtain an equivalence of categories, which therefore provides an equivalent characterization of Zinbiel algebras in terms of the fundamental theorems of calculus.

This story provides a new perspective on "O-operators on associative algebras and dendriform algebras" by Bai, Guo and Ni. arXiv:1003.2432

• September 28th, 2021
Frank Fu, Dalhousie University
A description of a reflective functor in the category of presheaves
Abstract

It is well-known that the subcategory of limit preserving functors in the category of presheaves is reflective. But what is the exact definition of the left adjoint? In this talk, we consider the subcategory of product preserving functors (a special case of limit preserving functors) and give a description of the left adjoint using some concepts from multi-sorted term algebra.

[This is a joint work with Peter Selinger, Kohei Kishida and Neil J. Ross]

• October 5th, 2021 [slides]
Jason Parker, Brandon University
Presentations and algebraic colimits of enriched monads for a subcategory of arities
Abstract

We present a general framework for studying signatures, presentations, and algebraic colimits of enriched monads for a subcategory of arities, even in enriched categories that are not locally presentable. Given any small subcategory of arities $j : J \rightarrow C$ in an enriched category C satisfying certain assumptions, we show the existence of free J-ary monads on J-ary endofunctors and the existence of small algebraic colimits of J-ary monads, where a monad or endofunctor is J-ary if it preserves left Kan extensions along j . We then deduce that every signature or presentation generates a J-ary monad, and that every J-ary monad has a presentation; moreover, we show that J-ary monads are monadic over signatures. Our results subsume earlier results of Kelly, Power, and Lack on finitary monads and finitary signatures when C is a locally finitely presentable V-category over a locally finitely presentable closed category V. We conclude by showing that our main results hold for any suitable subcategory of arities in any locally bounded enriched category.

Joint work with Rory Lucyshyn-Wright.

• October 12th, 2021
Martin Szyld, Dalhousie University
Minimal Bicategories of Fractions and Colimits of Bicategories
(Joint work with Pablo Bustillo Vazquez and Dorette Pronk)
Abstract

I will present a new set of fractions axioms for a family W of arrows of a bicategory B. Instead of asking for W to be closed under identities, compositions and invertible 2-cells as in the original set of axioms [1], we give a weaker closure axiom, making the family W minimal such that the localization of B at W can still be computed as a bicategory of fractions "using spans with legs in W". We call such a bicategory a "minimal bicategory of fractions".

We know from SGA that we can compute any pseudo-colimit of categories by localizing the associated fibration at the Cartesian arrows, and that if the indexing category is pseudofiltered we can construct this localization as a category of fractions. We can think of this construction as a "categorification" of (pseudo)filtered colimits of sets; when applied to such a diagram it gives a category with the same objects as this colimit, whose arrows witness the germ equivalence relation.

Using the same strategy, we have computed tricolimits of bicategories as bicategories of fractions, and I will explain how this led to the discovery of minimal bicategories of fractions. Fortunately looking at the SGA case is enough for this: for any fibration, when the base B is pseudofiltered, the family C of all Cartesian arrows satisfies the original fractions axioms, but the "chosen" ones (that is, those given by a cleavage) is a smaller family W that satisfies only the minimal axioms.

For the category of fractions computing the pseudo-colimit, whether your spans have legs in C or in W doesn't really make a difference (you get isomorphic categories), but in a bicategory of fractions, the actual spans (and not their equivalence classes) are the arrows, so different choices of legs for the spans lead to different bicategories. If you want your tricolimit to be a "bicategorification" of the SGA one, then you need your spans to have legs in W.

[1] Pronk, Dorette, Etendues and stacks as bicategories of fractions, Compositio Mathematica, Tome 102 (1996).

• October 26th, 2021
Zachary Murray, Dalhousie University
Proof Assistants in Practice
Abstract

History shows that formal foundations for mathematics are worth studying, but, out of sheer impracticality, one rarely hand writes a formal proof. Proof assistants offer at least a partial solution in automation, but there are still serious questions regarding their practicality, most notably the extent to which automation helps, their ease of use, and why we should trust a proof assistant. We will investigate such concerns practically by exploring the Agda proof assistant and discussing my work on a constructive analysis library in Agda.

• November 2nd, 2021
Colleen Delaney, Indiana University Bloomington
A new diagrammatic construction on fusion categories and applications
Abstract

In this talk I'll explain the "zesting" construction for fusion, braided fusion, and ribbon fusion categories and discuss the connection to topological phases of matter and topological quantum computing in the case when the category is a unitary modular or supermodular tensor category.

• November 16th, 2021
Michael Lambert, Mount Allison University
A Topos View of Blockchain Safety
Abstract

Estimate safety in an abstract approach to blockchain consensus protocols is formulable in the internal language of a topos. In particular, safety is equivalent to a forcing statement. That is, the estimator in the consensus protocol induces a geometric morphism of categories of copresheaves. Associated to this geometric morphism is a forcing relation between statements in the codomain topos and states in the domain topos [Awodey, Kishida, Kotszch 2014]. Estimate safety is precisely the statement that a given proposition about consensus values (in the codomain topos) is forced by all subsequent protocol states (in the domain topos). The key ingredient in making this work is the cosieves given as part of the subobject classifier in the domain topos. This leads to a completely elementary version of estimate safety using a little internal category theory.

• November 23rd, 2021
Peter Huston, Ohio State University
From Lattice Models to Tensor Categories
Abstract

Topological phases of matter are physical systems described at low energy by a TQFT, which can be obtained from an appropriate flavor of monoidal higher category. Lattice models are discrete physical models of topological phases. In this talk, we will first see how localized excitations in a Levin-Wen string-net model are described by a semisimple category. We will then introduce nets of categories, a discrete analogue of a factorization algebra, as a formal means of recovering a braided monoidal structure which is naturally adapted to lattice models. Finally, we will sketch the application of nets of categories in other settings, including the study of fracton phases of matter.

• November 30th, 2021
Marcello Lanfranchi, Dalhousie University