The n-Category Café

Saturday, October 26th

9-9:30 - David Jaz Myers

9:35-10:05 - Julia Morin

10:10-10:40 - Christopher Dean

Break

11:10-11:40 - Giuseppe Leoncini

11:45-12:15 - Aaron Fairbanks

Lunch

2:00-3:00 - John Baez

3:05-3:35 - Chris Grossack

Break

4:05-4:35 - Marcello Lanfranchi

4:40-5:10 - JS Lemay

Sunday, October 27th

9-9:30 - Jacob Neumann

9:35-10:05 - Rose Kudzman-Blais

10:10-10:40 - Jean-Baptiste Vienney

Break

11:10-11:40 - Henri Riihimäki

11:45-12:15 - Andrew Krenz

12:20-12:50 - Robin Cockett

Titles and Abstracts

Speaker: David Jaz Myers

Title: Contextads: Para and Kleisli constructions as wreath products

Abstract: Given a comonad D on a category C, we can produce a double category whose tight maps are those of C and whose loose maps are Kleisli maps for D — this is the Kleisli double category kl(D). Given a monoidal right action & : C x M $\to$ C, we can produce a double category Para(&) whose tight maps are those of C and whose loose maps A -|-> B are pairs (P, f : A & P $\to$ B) of a parameter space P in M and a parameterised map f.

In this talk, we’ll see both these as special cases of a general construction: the Ctx construction which takes a contextad on a (double) category and produces a new double category. We’ll see that this construction is “just” the wreath product of pseudo-monads in Span(Cat). We’ll then exploit this observation to find 2-algebraic structure on the Ctx constructions of suitably structured contextads; vastly generalizing the old observation that a colax monoidal comonad has a monoidal Kleisli category.

Joint work with Matteo Cappucci

Speaker: Julia Morin

Title: Joyal’s representation theorem for Heyting categories

Abstract: An elegant, unpublished result by Andre Joyal, believed to have been for- mulated around 1970, extends the Stone representation theorem for Boolean algebras to Heyting categories. Specifically :

Every small Heyting category H can be embedded into one of the form Func(C,Set) via a Heyting, conservative functor.

Through the construction of syntactic categories built from first-order theories, Joyal’s representation theorem offers completeness results for intuitionistic first-order logic. While a model-theoretic proof was originally presented by Makkai and Reyes in 1977, we introduce in this work a new, purely categorical approach.

Speaker: Christopher Dean

Title: Abstract Grothendieck fibrations and directed path objects

Abstract: A tribe in the sense of Joyal is a convenient framework for reasoning about the path-lifting properties of isofibrations in abstract homotopy theory. I will introduce a directed analogue of a tribe, which I call a ditribe, that axiomatically captures the cartesian-lifting properties of Grothendieck fibrations and opfibrations. I will provide examples based on many different flavours of higher category, and introduce a universal property describing directed path objects in this setting. The overarching goal of this work is to establish that, just as tribes serve as a convenient 1-categorical middle ground between the weak world of (∞,1)-categories and the strict syntax of homotopy type theory, ditribes provide a convenient middle ground between the world of (∞,n)-categories and a future syntax of directed homotopy type theory.

Speaker: Giuseppe Leoncini

Title: Homotopy cocompletions enriched over a general base

Abstract: Starting from a 1-categorical base V which is not assumed endowed with a choice of model structure (or any kind of homotopical structure), we define homotopy colimits enriched in V in such a way that: (i) for V = Set, we retrieve the classical theory of homotopy colimits and (ii) restricting to isomorphisms as weak equivalences, we retrieve ordinary and enriched 1-colimits. We construct the free homotopy V-cocompletion of a small V-category in such a way that it satisfies the expected universal property. For V = Set, we retrieve Dugger’s construction of the universal homotopy theory on a small category C. We define the homotopy theory of internal $\infty$-groupoids in V as the homotopy V-enriched cocompletion of a point, and argue that V-enriched homotopy colimits correspond to weighted colimits in $\infty$-categories enriched in internal ∞-groupoids in V, thus providing a convenient model to perform computations. Again, taking V = Set, this retrieves the classical notions for ordinary (∞, 1)-categories. We compare our approach with some previous definitions of enriched homotopy colimits, and we show that, when the latter are defined and well behaved, they coincide with ours up to Quillen homotopy. As an application, we show that, under our definitions, the so-called genuine (or fine) homotopy theory of G-spaces is the G-equivariant homotopy cocompletion of a point. This is a fact conjectured by Hill that, in the case of a finite group, was recently proven by completely different methods.

Speaker: Aaron Fairbanks

Title: Traces in categories of contractions

Abstract: See processes taking multiple inputs and yielding multiple outputs? You are probably dealing with a monoidal category. Is it possible to feed the outputs of a process back to its own inputs? You are probably dealing with a traced monoidal category. The monoidal category of finite-dimensional vector spaces with tensor product is traced, where feeding the output of a linear operator back to its own input means taking the usual trace from linear algebra. Less famous, but also interesting: the monoidal category of finite-dimensional vector spaces with direct sum is partially traced. (“Partially” means that the trace is only defined for certain maps.) Here the trace would give meaning to feeding the outputs (rows) of a matrix back to its own inputs (columns)! Such a trace could model iteration in quantum computing. For finite-dimensional Hilbert spaces in particular, it is a totally defined trace when restricted to the monoidal subcategory of isometries (Bartha 2014), or more generally contractions (Andrés-Martínez 2022). We prove this result in the most general abstract context where we can see it works: a dagger finite biproduct category with negatives and Moore-Penrose inverses. This is joint work with Peter Selinger.

Speaker: John Baez

Title: 2-Rigs in topology and representation theory

Abstract: A rig is a “ring without negatives”. There are many ways to categorify this concept, but here we explain one specific definition that sheds new light on topology and representation theory. Examples of such 2-rigs include categories of group representations, coherent sheaves and vector bundles. We explain some theorems and conjectures about 2-rigs with simple universal properties. For example, the free 2-rig on one generator is called the 2-rig of “Schur functors” because it acts as endofunctors of every 2-rig. It has as objects all finite direct sums of irreducible representations of symmetric groups. Furthermore, the “splitting principle” for vector bundles has a universal formulation in terms of this 2-rig. This is joint work with Joe Moeller and Todd Trimble.

Speaker: Chris Grossack

Title: Life in Johnstone’s topological topos

Abstract: A topos can be thought of as an alternate universe in which to do mathematics, and Johnstone’s famed Topological Topos is a universe in which every “set” automatically has some topological structure and every “function” is automatically continuous. This topos is very well studied, with many folklore results scattered throughout papers and forums online, but it can be hard to find these results and even harder to find proofs. In a recent blog post (which will become a paper sometime soon), I collected together all the facts I had heard about (and a fair number that I discovered for myself along the way!) along with their proofs. In this talk, we’ll discuss some of these previously folklore results, applications of the topological topos, as well as some important differences between its internal logic and the ordinary logic we use every day.

Speaker: Marcello Lanfranchi

Title: Towards a formal theory of tangent objects

Abstract: In 1972, Ross Street published a paper titled “The formal theory of monads” in which he explored the idea of defining monads in an arbitrary 2-category and finding conditions to construct the Eilenberg-Moore object of such a monad, revealing a new understanding of the notion of algebras of a monad. Can we introduce a similar approach to tangent category theory? In this talk, I would like to discuss some initial work towards a formal theory of tangent objects.

Speaker: JS Lemay

Title: Additive enrichment from coderelictions

Abstract: Differential linear categories provide the categorical semantics of Differential Linear Logic. Briefly, a differential linear category is a symmetric monoidal category that is enriched over commutative monoids, with a monoidal coalgebra modality that also comes equipped with a natural transformation called the codereliction. It turns out that from the codereliction and the induced canonical bialgebra structure of the monoidal coalgebera modality, one can infer the additive structure. As such, by adding more bialgebra axioms to the monoidal coalgebra modality, differential linear categories can be equivalently axiomatized without assuming additive enrichment. Explicitly, a differential linear category is equivalent to a symmetric monoidal category with a monoidal bialgebraic modality equipped with a codereliction.

Speaker: Jacob Neumann

Title: A type theory for synthetic 1-category theory

Abstract: When performing category-theoretic constructions, one often incurs a significant amount of bureaucratic work: the category laws must be checked whenever one defines a category, functoriality must be checked when defining a functor, naturality when defining a natural transformation, and so on. While it’s easy enough (if not too principled) to “hand-wave” away many of these conditions when working informally, formal developments of category theory (such as those done in a computer proof assistant) cannot avoid them, and consequently require significant amounts of time and effort.

An exciting solution to this issue is the development of directed type theory. In a directed type theory, every type is a synthetic category, that is, the terms of that type come equipped with a hom-type structure between them which automatically admits an associative composition operation and appropriate identities. Every function one can express in such a theory automatically comes with a morphism part operating on the hom-types, which is automatically functorial. And so on. In synthetic category theory, a huge portion of the busywork of category theory is taken care of in a formal, principled way, allowing theorists to focus on the important details of their work.

In this talk, I’ll detail the syntax of a directed type theory currently in development (j.w.w. Thorsten Altenkirch), and show how it achieves basic synthetic 1-category theory. In particular, I’ll discuss how composition, associativity, functoriality, and naturality are all consequences of our principle of directed path induction. Time permitting, I’ll discuss some of the semantics of this directed type theory, and compare it to other approaches.

Speaker: Rose Kudzman-Blais

Title: Linearly distributive Fox theorem

Abstract: Linearly distributive categories (LDCs), introduced by Cockett and Seely to model multiplicative linear logic, are categories equipped with two monoidal structures that interact via linear distributivities. A key result in monoidal category theory is the Fox theorem, which characterizes cartesian categories. The aim of this work was to extend the Fox theorem to linearly distributive categories. To establish an adjunction with the 2-category of cartesian LDCs, we introduced the concepts of medial linearly distributive categories, medial linear functors, and medial linear transformations. This adjunction between cartesian LDCs and symmetric strong medial LDCs was successfully proven, and several new examples of medial LDCs were presented. Notably, medial LDCs induced by distributive symmetric monoidal categories with zero objects, such as REL and VEC, highlight the utility of this new definition.

Speaker: Jean-Baptiste Vienney

Title: What’s wrong with higher-order derivatives?

Abstract: Some interesting problems happen in mathematics when we differentiate a function at least two times. And some machinery has been developed to deal with them. A first problem happens when we differentiate a polynomial over a field of positive characteristic. Some derivatives are equal to zero even if they shouldn’t. The notion of Hasse–Schmidt derivative has been invented to deal with this. Another question arises in differential geometry. The tangent bundle approximates linearly a manifold around a point. The derivative of a function between two manifolds approximates the function by sending a linear approximation in the first manifold to a linear approximation in the second manifold. Can we approximate a manifold around a point by a polynomial curve of order two, three etc… instead of a line? The notion of higher-order tangent bundle has been invented to deal with this. As for the usual tangent bundle, it is given by a functor which can also be applied to a function between two manifolds, producing approximations of order two, three etc. It seems that both of these notions could be integrated into the framework of differential and tangent categories.

In this talk I’ll explain the Hasse–Schmidt derivative and the higher-order tangent bundles, how we could maybe define Hasse–Schmidt differential categories and higher-order tangent categories and how they should be related. Interestingly, there does not seem to be work in the mathematics literature relating the Hasse–Schmidt derivative to higher-order tangent bundles. But we hope that they would be closely related in the differential categories framework. It will be an appetizer to the full-fledged story I would like to tell in my PhD thesis.

Speaker: Henri Riihimäki

Title: Hochschild homology of directed graphs: from connectivity functors to reachability category

Abstract: There is currently an active interest in homotopy and homology theories in the world of graphs and directed graphs; discrete homotopy theory, magnitude homology and path homology are few prominent themes. In a joint work with Luigi Caputi we extended the use of Hochschild homology of directed graphs via so called connectivity structures, of which I will present functorial examples. Given a filtration of directed graphs we can then introduce persistent Hochschild homology. Unfortunately, to get an actually computable pipeline we have to resort to non-functorial constructions. To remedy this leads to defining the reachability category of a directed graph. In addition to application to persistence, basic equivalence of categories allows us to give a very direct proof that recently introduced commuting algebras are Morita equivalent to incidence algebras.

Speaker: Andrew Krenz

Title: Categorical median algebra

Abstract: The two-element Boolean algebra 2 induces a Stone-type duality between poc-sets and median algebras. In this talk, I will introduce these categories and show how this duality arises from a two-variable tensor-hom adjunction involving the categories of poc-sets (Poc), median algebras (Med), and Boolean algebras (Bool), and consequently, that every Boolean algebra gives rise to a contravariant adjunction between Poc and Med. In fact, the category of contravariant adjunctions between Poc and Med is equivalent to Bool. I plan to give a detailed account of an intermediate result used in our proof of this assertion. In particular, I will construct a pair of normal limit sketches P, M such that Poc $\simeq$ Mod(P), Med $\simeq$ Mod(M), with the added property that the underlying category of each is contained in the other’s category of models. With slightly more work, it is possible to argue that FinPoc is a finite limits theory for Med and vice versa. Time permitting, I will demonstrate that the double dualization monads associated to the 2-induced adjunction are the codensity monads of the full inclusions FinPoc $\hookrightarrow$ Poc, FinMed $\hookrightarrow$ Med.

Speaker: Robin Cockett

Title: Localizations of restriction categories

Abstract: There are at least three ways to factorize a restriction functor, there is the usual full ( bijective on objects)/faithful factorization, then there is the localic/hyperconnected factorization and in between these two is the localizing/separating factorization. This factorization is induced by inverting certain restrictions and is a special sort of Grothendiek topology albeit seen at a less structured level.

The talk will be mostly concerned about what happens to the localizing/separating factorization when one considers the category of join restriction categories. The main problem is simply that localizing a join restriction category does not necessarily yield a join restriction category. This forces one into considering a join completion process which shall be described.

An example of this machinery in use is in the construction of schemes for commutative rigs (semi-rings) which I will sketch.