Johns Hopkins Category Theory Seminar

Intermittent Tuesdays
pretalk 4pm, talk 5:30pm

Gilman 377

Spring 2019 Schedule (intermittent Tuesdays; pretalk 4pm, talk 5:30pm): Title: Higher categories from higher-dimensional manifolds
     While higher groupoids have a natural model in spaces, higher categories have no such well-accepted model. This makes the question of correctness of a given definition of higher categories difficult to answer. We argue that the question has a simple answer “locally”, namely, categories are locally modelled on so-called manifold diagrams. The corresponding “local model" for spaces/groupoids can be formulated in classical terms by a generalised Thom-Pontryagin construction. The idea of locally modelling higher categories by manifold diagrams (most prominently in the case of Gray-categories) is not new and has been proposed by multiple authors. However, the niceness of this manifold-based perspective on higher categories has been somewhat obfuscated by the complexity of manifold geometry in higher dimensions in the past. We will discuss a fully algebraic formulation of this manifold perspective. Interestingly, the model of higher categories that is based on this algebraic formulation is not fully weak: It is a generalisation of (unbiased) Gray-categories to higher dimensions. This is the starting point of a wealth of further research, which reaches from a (version of) Simpson’s conjecture to presentations of the extended cobordism n-category and the homotopy and cobordism hypotheses.
Title: Polynomial functors, a degree of generality
     The talk will begin by recalling our partial answers to the question “What is an LCC(C)?”. With that understanding reached, we will categorify the concept of a polynomial into that of a categorical polynomial — a structure determined by three maps. These maps induce a functor and we will see how, in the internal language of an LCC, such functors induced from polynomials directly capture our intuitions — developing the necessary results about LCC's as we go.
     We will then explore some results characterising those functors arising from such polynomials, the polynomial functors, before assembling them into a variety of categorical structures so as to better understand their totality. Time permitting we will turn to examine a specific class of monads whose underlying functor is polynomial and relate them to the notion of inductive types in the underlying category.
     The theory and its connections are both wide and deep; we will not aim to exhaust the subject or the audience.
Title: Bar-cobar and Koszul duality for algebraic operads, continued
     The talk last time was actually about &lqduo;several equivalent definitions of operads and cooperads.” This time I will get to the bar-cobar and Koszul duality of algebraic operads. We will quickly recap what I talked last time, and then get to the conclusion. After that I will talk about (not-so-detailed) details as long as time permits.
Title: Monoids, Ultrafilters, and Monads
     We will introduce monads through motivating examples, namely monoids and ultrafilters. We will first generalize the notion of a monoid to obtain the classical definition of a (1-categorical) monad. Through this introduction, we’ll see that all monads are induced by adjunctions; the monad is the data of the adjunction “visible” from the codomain of the right adjoint. Viewing monoids as “free monoid algebras” will motivate the Eilenberg-Moore construction. We’ll go on to define monadic functors and display their desirable properties, namely the preservation and reflection of certain limits and colimits. The main reference for this section of the talk is chapter 5 of Category Theory in Context.
     Once we’re comfortable by monads induced by adjunctions, we’ll move on to ultrafilters and codensity monads. Similarly to the first section, we’ll start with a motivating example (ultrafilters) before tackling the general theory. Codensity monads will broaden our perspective, as it allows us to induce monads from non-monadic functors. The main reference for this section will be Codensity and the ultrafilter monad by Tom Leinster.
     Time permitting, we’ll also briefly touch on free monads, algebras of endofunctors, and polynomial monads.
Title: Does knowledge of 1-category theory provide morally sufficient grounds upon which to fake knowledge of ∞-category theory?
     This talk will offer an ethical tactic for engaging with ∞-categories as a non-expert. It will start by explaining exactly what an ∞-category is from the point of view of much of the literature that works with them. Along the way, it will also illustrate the similarities and differences between 1-categories and ∞-categories by giving an in-depth discussion of one of the equivalences between ∞-categories that is used without comment in the Gepner-Haugseng-Kock paper.
Title: Introduction: Infinity Operads as Analytic Monads by Gepner, Haugseng and Kock
      The notion of an Operad has been around for decades now, going back to May in the 70s. They are used to capture the computational combinatorics of algebraic structures in various situations. Classically, Operads are defined using symmetric sequences of sets, these sequences give rise to “Analytic Endofunctors” which are Monads when the symmetric sequence is an operad. For sets, there is an equivalence between the notion of an algebra on an operad and the algebra on it's associated monad. Lifting to operads defined as sequences of spaces, this algebraic equivalence is lost in general. The Authors Gepner, Haugseng, and Kock notice this problem has to do with higher structures and so devise a definition of an infinity operad as an analytic monad to recover an analog of the algebraic equivalence in the setting of higher category theory. Several other equivalent models of infinity operads exist, for example as dendroidal segal spaces by Cisinski and Moerdijk. GHK's final result proves their analytic model of infinity operads is equivalent to the dendroidal segal spaces.
      This introduction will focus on the main constructions and results of the paper that this seminar aims to study in detail during the remainder of the semester.
Title: Polynomials & polynomial functors over the ∞-category of spaces
     Polynomial functors are functors between slices of the ∞-category of spaces that can be built as a composite of pullbacks, dependent sums and dependent products, so the information of a polynomial functor is recorded by a certain diagram of spaces called a “polynomial”. After giving an intrinsic description of polynomial functors, we discuss how to express several constructions involving polynomial functors (such as the composite of polynomial functors, the cartesian morphisms between polynomial functors, the colimit of a diagram of polynomial functors) in terms of the corresponding representing polynomials.
Title: Generatingfunctorology
      “A generating function is a clothesline on which we hang up a sequence of numbers for display.” So begins the much beloved generatingfunctionology by Herbert Wilf. A generating function is a (hopefully) analytic function whose power series expansion encodes a sequence of combinatorial data. Using them gives elegant and joyfully quirky proofs of combinatorial truths.
     One might wonder whether those elegant and joyfully quirky proofs of combinatorial truths can be given in a combinatorial manner -- that is, by actually exhibiting a bijection which shows that two things being counted will have the same count. What we want, then, is a generating functor, an "analytic" functor whose actions on sets (or something else) encodes the combinatorial data.
     In this talk, we will hang our homotopy types out to dry and watch as they spin in the wind with an action of the symmetric groups. We will find that any such clothesline determines an analytic functor on homotopy types (which, just a reminder, includes the more familiar case of sets), and will characterize the analytic functors in intrinsic terms. Finally, we will see that trees can be described in the same language, giving an interpretation of generating functors as strange forests of twisting trees, ripe for an operadic harvest.
Title: Initial Algebras and Free Monads
      Our job here is to prepare some generalities on monads and algebras, before specializing them to analytic monads in the following section.​
​ As we saw (in my previous talk), the free operad construction on symmetric sequences plays an important role in the theory of operads. We will see that an analogous explicit construction of free monads on endofunctors is available in the infinity-categorical setting. On the way to it, we will construct the initial Lambek algebra (which is a kind of a “pre-algebra” on a “pre-monad”=endofunctor) via the bar-cobar-adjunction between algebras and coalgebras with the space of twisting morphisms in between. We will also formulate the parametrized (over varying (co)domains of endofunctors) version of this construction. I will emphasize the analogy to the classical constructions we saw in my talk.
​ ​ Since the content is totally independent of the previous sections, anyone with no sense of what was going on in those talks will be able to join and understand this talk.
Title: A Combinatorial Model for Infinity Operads
     Category theory embeds into simplicial sets via the nerve functor. Simplicial sets are controlled by a nice combinatorial category Δ. By axiomatizing how composition of categories is captured by the nerve and weakening this axiom suitably we get a nice combinatorial model for (∞, 1)-category theory via quasicategories. An operad (multicategory) is like a category, but we are allowed to map from many object to one, and we have multi-linear composition. This composition is captured by the tree category Ω, and operads likewise embed as presheaves on Ω with a strict multicomposition. By weakening this axiomatization we get a nice combinatorial model of ∞-operads completely analogous to quasicategories. In my first talk I'll discuss this “dendroidal set” model for ∞-operads so that we have a better understanding of what GHK are trying to accomplish in Section 5. I'll also discuss the closely related complete dendroidal segal space model that is used in GHK. Notably this talk is independent from the paper (but not randomly so).

Title: Analytic Monads as Infinity Operads
     In section four we learned that analytic endofunctor are identified with preshaves on elementary trees, or equivalently as preshaves on the category of trees with inert morphisms. This is almost dendroidal spaces, but we are missing an extension to the full tree category Ω. This amounts to adding the “active” maps, which control the multicomposition of dendroidal sets. It's not too far fetched then to believe that this extension relates to adding a multiplicative structure to our analytic endofunctors. In my second talk I will sketch the proof of this equivalence between dendroidal segal spaces and analytic monads.

Fall 2018 Schedule (random Thursdays; talk 4pm, open discussion 5:30pm): Title: Separating the operations from the algebras: an introduction to topological operads
     Operads were introduced by May in order to study operations on k-fold loop spaces. Since then they have been employed throughout algebra and Homotopy theory. In this talk we will introduce (topological) operads in an example based manner by following this historical development.
Title: An Introduction to Topoi
      A topos is a category that behaves like the category of sheaves of sets on a topological space. In that sense, it is a kind of generalization of a geometric space. It was introduced by Grothendieck in order to study the category of étale sheaves on a scheme. It also turns out that a topos is the right kind of category to model theories in higher-order typed logic. In this talk, we will define Grothendieck and elementary topoi, work through several examples, and try to demonstrate the usefulness of the concept. Title: The logic is coming from inside the category
      Usually when our objects form a category, we talk about them by drawing diagrams. The arrows in these diagrams represent morphisms, which can feel very complicated depending on the category we're in. But there is another way to talk about the objects of the category, a more native language in which the various universal constructions of category theory become familiar constructions from naive set theory. In this talk, we will introduce this internal logic of a category. We'll keep adding to it until we can talk about the logic of the category inside the category itself.

     Why should you care? If you think that quasi-coherent sheaves of modules are complicated, but modules are simple, then the internal logic may be right for you; in the internal logic of a category of sheaves, a quasi-coherent sheaf of modules is just a module, and therefore all theorems* about modules proved normally are proved for quasi-coherent sheaves.

*terms and conditions may (and will) apply.

Title: What is an LCC?
      What is an LCC? This question may be difficult to answer in that really the object of study is arguably more properly an LCCC. We will attempt to side-step this difficulty, together, in our journey to understanding what an LCC is from at least one distinct perspective in this talk. Through team-based intuition building, and internal language usage, we will aim to understand the how LCCs are a natural setting for type theory, the general theory of these objects, and polynomials. Title: Products in a Category with One Object
      A category with one object is of course a monoid; the requirement that its object is a product with itself imposes an extra binary operation on the monoid satisfying interesting identities. I will discuss joint work with Aaron Gray on a universal monoid U of this type and show that any finite monoid has a non-canonical injection into U.
      I learned last month that some parts of our work were anticipated by Rick Statman with applications to theoretical computer science. I will also discuss some of Statman's work. Title: Bar-Cobar and Koszul duality theory of (algebraic) operads
      Operad theory is a language to talk about operations themselves, separated from algebras they act on (as explained by Daniel in the first talk of the semester). This is a crucial viewpoint first introduced to discuss on the relations between loop spaces and algebraic structure on them; the main theorem of this theory states that having a homotopy type of a loop spaces amounts to admitting a “homotopically relaxed” monoid (called A-space) structure on it. Through an analogy between space and chain complex, we can define the dg-analog called A-algebras.
      As suggested from the loop-space description, this kind of homotopically relaxed algebraic structure is homotopy-invariant, in the sense that the algebraic structure transfers not only to isomorphic objects but also to homotopy equivalent ones. This result is very useful, for example, in recovering the lost information when taking cohomology ring of a dg-algebra by transferring the A-structure on homology (n-ary product here is known as Massey products).
      This observation leads us to consider the following problems: How can we systematically define those relaxed algebraic structures starting from usual ones? How can we prove the homotopy invariance? Can we find a handy description of them? The goal of this talk is to give an explicit recipe as an answer to these questions. In this theory, a certain duality between operads and cooperads, which is a generalization of the bar-cobar construction between algebras and coalgebras, plays an important role. Construction using bar-cobar duality gives an operad which describe the relaxed algebraic structure, though it is usually huge. When we start with “quadratic” operads this bar-cobar duality can be refined (called Koszul duality) to give handier (actually “minimal”) model for the relaxed structure.
      Although I will quickly review the definition(s) of operads, the audience who have not seen any of them is advised to look up what operads are.

Spring 2018 Schedule (sporadic Thursdays; pretalk 4pm, talk 5:30pm; Shaffer 2): Title: Points in Spaces as Sheaves
     An introduction to sheaves by way of a natural model for 'points' in 'spaces'. With topologies as a proxy for categories, I'll present a motivating example using sheaves to define points with respect to consistent local observations. Then I'll relate this example back to classical ideas of convergence in point set topology. Background material will be given special attention. Specifically, we'll spend time defining sheaves, coverages, and the relevant concepts from point set before applying these definitions in the special case of the category of closed sets under inclusion.
Title: Introduction to applied category theory
     In this talk, we give a basic introduction to the nascent field of applied category theory, which primarily refers to applications of category theory outside of pure mathematics, computer science and quantum physics. We'll mention a few of the main themes and categorical constructions found in applied category theory, using an application in natural language processing as our main example.

Fall 2017 Schedule (approximately alternate Thursdays; pretalk 4pm, talk 5:30pm; Krieger 413): Title: Basic concepts of enriched category theory
     The plan is to give a leisurely introduction to the basic concepts of enriched category, in which the collection of morphisms between each fixed pair of objects is itself an object of another category, which will also function as a review of unenriched category theory, in which this collection of morphisms is a mere set. In particular, we will define enriched categories, functors, natural transformations, adjunctions, and at least state the enriched Yoneda lemma.
     At the break we will hold the organizational meeting for the seminar to assign speakers and topics for the rest of the semester. Title:Yoneda, rich and poor
     Our goal is to discuss the Yoneda lemma, first in the setting of enriched category theory, and then in the specific setting of preorders. The Yoneda lemma talks about the embedding of a (V-)category in the category of (V-valued) presheaves on it. We first recall how any closed symmetric monoidal category is enriched over itself, and we will discuss enriched notions of natural transformation. This will allow us to state the enriched Yoneda lemma, and to understand why it is true. In the poor (but, I will argue, not quite bankrupt) setting where V is the preorder 2 = {0 → 1}, a V-category is of course just a preorder, and Yoneda’s embedding is an example of an ideal completion of a preorder. Research into completions of preorders has developed mostly separately from enriched category theory. We will mention a few constructions of other preorder completions, and discuss if, and how, they relate to the general enriched setting.
Title:All good things must come to an end
     In this talk we will begin by examining the calculus of (co)wedges through the careful distillation of select properties of the usual adjunction Set(AxB,C) = Set(A,[B,C]); the further motivating example of identity arrows in a category will lead us to consider the general notion of (co)ends. From here we will move to discuss several exciting and important theorems about ends, such as (but not limited to): (co)limits as (co)ends as (co)limits, Fubini's theorem, the (co)Yoneda lemma(s), and Kan extensions as ends. As we go we will examine as many examples as time and interest allow for, all with an eye to the connections with enriched category theory.
Title: Weighted (co)limits in the unenriched setting
     We'll start by reviewing the classical notion of (co)limit and reinterpreting it in terms of cones. We'll then generalize these notions by adding weights. We'll see the Grothendieck construction of a weighted (co)limit. Finally we'll rewrite these in terms of (co)ends so that they can be generalized to an enriched setting. The reference is Ch. 7.1 and 7.2 from Categorical Homotopy Theory by Emily Riehl.
Title: Enriched weighted colimits
     This week we will enrich what we've learned about ends and colimits over the last two weeks. The first half of the lecture will develop the theory of enriched weighted limits, conical limits, and enriched ends. Special emphasis will be placed on the power of representability, and how these constructions can actually be computed in a V-(co)complete setting. The second half will focus on specific examples and applications of weighted colimits primarily in topology, though participants are encouraged to bring their favorite examples, time permitted. Title: Yoga of Four Operations
     Push, pull, tensor, and hom; these are the four operations of the calculus of bimodules. In this talk, we'll take an "equipment-theoretic" approach to enriched category theory, and express some of the techniques we have learned so far in terms of bimodules between enriched categories, their tensors, and their homs. In particular, we will define Kan extensions and weighted (co)limits and prove a few of their elementary properties. Finally, we will see how these notions play out in the (relatively) simple equipment of sets, functions, and relations. Hopefully, everyone will leave this talk well equipped for the wonderful world of enriched categories. Title:Basics on 2-categories
     We will give an elementary description of double category and 2-category. We will learn how to paste 2-cells. And we will enrich our definitions in 1-categories to 2-categories. Title:Introduction to model categories
     Model categories are good places to do homotopy theory. In such a category, one can indeed define a good notion of homotopy between maps. For example, the category of topological spaces can be endowed with a model structure in which the notion of homotopy between continuous maps is the usual one. In this talk, we will first see how model categories and (right) homotopies are defined, and how their associated homotopy category is constructed. Then we will introduce the injective and projective model structures on a category of diagrams in a “good” model category, which allows us for example to define homotopy colimits (or limits) as the left (or right) adjoint of the homotopy diagonal functor. Finally, we will see that these model structures also apply on enriched diagrams in some closed symmetric monoidal model categories. Title:2-cats with PIEs
     We will specialise the theory of weighted (co)limits to the case V=Cat. By means of several examples, we will learn to spell out the universal property of weighted (co)limits (in both its 1- and 2-dimensional aspect), and construct weighted (co)limits in the 2-category of small categories. We will also give conditions for a 2-category to admit certain classes of weighted (co)limits.
Fall 2016 Schedule (Occasional Mondays; pretalk 3pm, talk 4:30pm; Maryland 114): Title: Factorisation systems in category theory
     An (orthogonal) factorisation system on a category C consists of two classes (E,M) of morphisms in C, subject to an "orthogonality" axiom and both closed under composition and containing the isomorphisms, such that every morphism f in C factorises as f = me, with e in E and m in M. The axioms imply that the (E,M)-factorisations of a given morphism are unique up to unique isomorphism. As in the example of the image factorisation system (surjective, injective) on the category of sets, one can generally see a factorisation system on a category as providing a notion of image to its morphisms.
     Following an introduction to the basic properties and examples of factorisation systems, this talk will survey the role of factorisation systems in such diverse topics of category theory as reflective subcategories, notions of epimorphism, regular categories and (bi)categories of relations, and constructions of associated sheaves, as dictated by time and interest.
Title: Simplicial objects
     Simplicial objects provide a combinatorial model for doing homotopy theory, that generalize topological spaces and chain complexes simultaneously. In this talk we'll try and understand a multitude of examples of simplicial objects. We'll describe the homotopy groups of Kan complexes (fibrant objects in Quillen's model category). We'll state the connection between simplicial categories and other categories via dold-kan and geometric realization functors.
Title: An introduction to the Galois theory of Grothendieck
     In SGA1, Grothendieck re-imagines Galois theory in terms of an axiomatic characterisation of categories of group actions. This approach leads to generalisations of Galois theory which applies to infinite field extensions and to categories of commutative algebras. Maybe more surprisingly he is able to demonstrate that, in this framework, Galois theory and the theory of covering spaces / the fundamental group become examples of the same categorical formalism. Ultimately, Grothendieck's insight gains it most abstract and all encompassing form in the work of Joyal and Tierney, in which they demonstrate that any Grothendieck topos, that is to say any category of of generalised sheaves, is equivalent to a category of continuous actions of some localic (spatial) groupoid. We might call this the Galois groupoid of the topos.
     In this talk I intend to provide an elementary introduction to Grothendieck's Galois theory, proving as many of his core results as I can in the time available. We will examine both finite and pro-finite variants, which will enable us to discuss the Galois theory of (infinite) algebraic closures and the theory of covering spaces of spaces that lack a universal covering space. I also hope to briefly touch upon the primary themes and motivations that arise in Joyal and Tierney's work. My plan is to rely on few categorical preliminaries, and to explain those that I do need as we go along. Title: A monad is just ... (with an eye to universal algebra)
      In this talk we aim to explore some of the elementary results in the theory of monads. We will begin by examining how monads might naturally arise from notions in monoidal categories and attempt to understand monads and their modules (specifically algebras) in this light — finding, as we will, the generation of monads from monoids to be unsatisfactory in that the inverse problem does not admit an immediately natural and adequate resolution. With that source of monads exhausted we will turn to the next and perhaps most fruitful approach, viz., the generation of monads from adjunctions.
     Once some basic results have been established we will address the inverse problem again and find here the famous results of Kleisli and Eilenberg-Moore: every monad arises from an adjunction in two canonical and somehow universal ways. We will explore the Kleisli category and Eielenberg-Moore categories for some common monads in order to impress the niceties of the latter, the category of algebras. In particular we will interest ourselves with the observation that the construction of Eilenberg-Moore inherits all limits and suitable colimits from the base category and so prompt the answering of a very practical question: how might we tell, for a given “algebraic” system (groups, monoids, categories ...), whether and indeed which limits and colimits exist? This will lead us to the notion of monadicity and the theorem of Beck.
     Time permitting we will consider the idea of a distributive law so that we may more clearly describe the situation of one monad acting over another — the generalisation of the case of unital rings in which the multiplicative monoid and additive group interact suitably.
     Only a knowledge of categories, functors, natural transformations, limits and adjoints will be assumed.

Contact Emily Riehl.