Johns Hopkins Category Theory Seminar

Spring 2024 Schedule: (Tuesdays excluding the first of the month, 4:15-5:15pm; Krieger 413)

The condensed mathematics seminar will continue in Spring 2024, organized by Rok Gregoric and Anish Chedalavada. The new seminar website is here. The category theory seminar will resume in parallel on the following schedule.

Title: Fibered Yoneda Lemma in Lean 4

Abstract: In this talk, I would like to give you a sense of what it’s like to do category theory in a computer. I use an interactive theorem prover called Lean4, a complex piece of software based on dependent type theory. In the first part, I will give a quick overview of category theory in Lean 4. In the second part of the talk, I will present two interactive proofs in Lean 4: the proof of the Yoneda Lemma and the proof of Fibered Yoneda Lemma.

Title: Calculus of fractions for quasicategories

Abstract: A central object of interest in abstract homotopy theory is the localization of marked categories. One case of interest is when the marked category admits a “calculus of fractions”. This condition was introduced by P. Gabriel and M. Zisman to construct better-behaved models for the localization. For instance, in their model of the localization, morphisms are given by spans rather than arbitrary zig-zags, and the localization functor preserves any finite colimits that exist in the original category.

In this talk, I will present a generalization (joint with C. Kapulkin and Z. Lindsey) of calculus of fractions to the setting of infinity categories (specifically quasicategories) and show how a workable model for the localization can be constructed using a marked variant of Kan’s Ex-functor. No prior familiarity with calculus of fractions or quasicategories is required!

Title: Synthetic ∞-category theory

Abstract: ∞-categories are ubiquitous in modern algebraic geometry and homotopy theory. They can be understood as a homotopical version of ordinary categories in that composition is not given as a strict operation but up to a contractible choice of higher-dimensional data. This observation opens up a new way to reason about them: using the language of homotopy type theory, one can natively express such homotopy-invariant conditions leading to simpler definitions of proofs that traditionally would take a lot of time to establish. I’ll give a non-technical introduction to this synthetic approach to ∞-category theory, relying on a directed extension of homotopy type theory by Riehl and Shulman. This includes a recent project, jointly with Kudasov and Riehl, where we developed a computer-checked proof of the Yoneda lemma for ∞-categories. No prior knowledge of ∞-categories or type theory is required.

Title: On the Construction of a Realizability Topos

Abstract: Partial combinatory algebras come out of a program started by Kleene to give a semantics for intuitionistic logic. As such any partial combinatory algebra gives a model of intuitionistic logic. Given a partial combinatory algebra A, one can derive from its model of intuitionistic logic an elementary topos called the realizability topos on A. What makes this topos interesting is that it is a large category but it is not a Grothendieck topos nor does it generally have countable limits or colimits and is not equivalent to a small category like FinSet. Realizability toposes also possess the existence of internal categories that are weakly complete (i.e. has all limits with respect to internal diagrams) and yet are not internal preorders. The effective topos is of particular note since this topos is built out of partial recursive mathematics and serves as a model of computable mathematics.

I’ll will give an overview of partial combinatory algebras as well as their connection to lambda calculus which will play a role in giving semantics for intuitionistic logic. We will also go over triposes and the tripos-to-topos construction needed to construct the realizability topos as well as the construction of the products, exponents and the subobject classifier of the realizability topos. Time permitting, I might talk about the internal weakly complete category of modest sets.

Title: The simply typed 2λ-calculus

Abstract: 2λ-calculus is a language with three levels: types, terms, and rewrites. I will explore their properties and give categorical description of the equational theory on the rewrites of the 2λ-calculus

Title: A formal language for formal category theory

Abstract: Equipments have shown to be a powerful environment to express formal category theory. We build a model structure on the category of double categories and double functors whose fibrant objects are the equipments, and combine this together with Makkai’s early approach to equivalence invariant statements in higher category theory via FOLDS (First Order Logic with Dependent Sorts) and Henry’s recent connection between model structures and formal languages, to show a result on the equivalence invariance of formal category theory.

Fall 2023 Schedule: (Tuesdays excluding the first of the month, 3-4pm; Krieger 413)

In Fall 2023 the category theory seminar will aim to introduce condensed mathematics.

References include:

A contemporaneous seminar:

Past workshops and mini-courses:

Applications and related topics:


Title: Pyknotic sets

Abstract: In this talk, I will introduce the idea of Pyknotic/Condensed sets and how they can potentially be used to solve the problem of the category of topological spaces not being abelian. I will also go over the prerequisite concepts in category theory such as the notion of Grothendieck topology, pro-object in a category etc.

Title: Pyknotic/condensed sets II

Abstract: We will continue the discussion of pyknotic sets, and introduce the notion of condensed sets.

Title: Condensed abelian groups I

Abstract: We will talk about the categories of κ-condensed abelian groups with Grothendieck axioms and why they are the abelian category of the nicest possible sort.

Title: Coherence Theorems in the style of Mac Lane

Abstract: The aim is to discuss different equational structures defined on categories like monoidal, symmetric monoidal and cartesian monoidal structures. The aim is to shed light on the associated coherence theorems and to show how these are related to the construction of simple type theories. This suggests a more general approach regarding coherence theorems. We will revisit the coherence theorem of monoidal categories proved by Mac Lane.

Title: Condensed abelian groups II

Abstract: In the previous talk, we learned about how k-Condensed Abelian Groups are abelian categories of an especially nice sort. In this talk, I will discuss how this fixes issues in some of our motivating problems and endows additional structure on the category that we will use later. Time permitting, I will also discuss some cardinality issues when considering different cardinals k.

Title: Cohomology

Abstract: Our goal in this talk is to recover the classical sheaf cohomology on topological spaces from related condensed set and condensed abelian groups. I will also discuss some infinity category theory, because from some point of view, that is where the actual story of derived category and cohomology happens.

Title: Locally compact abelian groups

Title: Solid locally analytic representations

Abstract: I will discuss common work with Joaquin Rodrigues Jacinto about new foundations for representation theory of p-adic Lie groups using condensed mathematics. I will explain how different categories of representations fit in the new theory of analytic geometry of Clausen and Scholze, and also how some group cohomological comparisons can be understood in the language of six functor formalisms.

Title: The (∞,2)-category of internal (∞,1)-categories

Abstract: Results which concern the classification of parametrized structures over a given base by means of internal constructions within that base are fairly ubiquitous in logic, algebra and topology. These results are by and large formal consequences of reflection properties of an associated externalization functor. In this talk, we define the formal externalization of ∞-categories internal to an ambient ∞-category C and study the resulting (∞,2)-categorical enhancement thus obtained. We then construct a lift of this theory to the context of model categorical presentations, and relate it to homotopy theoretical constructions of Hovey, Dugger, and Riehl and Verity.

Title: How to invent pyknotic/condensed mathematics

Spring 2022 Schedule: (almost always on Tuesday, pretalk 4-5pm, talk 5:30-6:30pm)

Title: Codescent objects and lax coherence

Abstract: A 2-categorical colimit called a codescent object has several applications in 2-dimensional algebra — for instance in the proof of Steve Lack’s Coherence theorem and in a statement of a 2-dimensional analogue of Beck’s monadicity theorem. I will go through ways to compute various codescent objects and also give exposition on a lesser known version of Steve Lack’s Coherence theorem for lax algebras. The talk will feature numerous examples.

Title: On the ∞-topos semantics of homotopy type theory I

Abstract: In Fall 2021 I taught a semester-long graduate topics course in homotopy type theory without telling the class what any of it means. What I mean by this is we neglected entirely to discuss the semantics of homotopy type theory in traditional set-based mathematics. In the first part of a two part series, I plan to sketch the construction, due to Voevodsky, that interprets theorems in homotopy type theory as theorems about simplicial sets. In the second part, I’ll then sketch the proof due to Shulman et al that extends this result to an arbitrary ∞-topos.

Title: On the ∞-topos semantics of homotopy type theory II

Abstract: In Fall 2021 I taught a semester-long graduate topics course in homotopy type theory without telling the class what any of it means. What I mean by this is we neglected entirely to discuss the semantics of homotopy type theory in traditional set-based mathematics. In the first part of a two part series, I plan to sketch the construction, due to Voevodsky, that interprets theorems in homotopy type theory as theorems about simplicial sets. In the second part, I’ll then sketch the proof due to Shulman et al that extends this result to an arbitrary ∞-topos.

Title: Making internal logics to order with geometric definability

Abstract: We’ve seen how homotopy type theory can be interpreted in any ∞-topos. But what if we have a specific ∞-topos E in mind. What axioms can we add to HoTT to make it so that we are working in E specifically (or a suitably similar topos)?

In this talk, we’ll answer this question for 1-toposes using a theorem due to Ingo Blechschmidt: geometric definability. Geometric definability gives a unification between syntax and semantics in the internal logic that we can use to give practical axioms for working in the internal logic of any topos. Along the way, we will learn about the yoga of geometric logic and classifying toposes.

Time permitting, I’ll put forward a speculative conjecture which I’ll call geometric constructability, which would extend this method for working in the internal logic to ∞-toposes and homotopy type theory. This opens up the possibility of a type theory which would let us work in all toposes over a given base topos simultaneously, moving between them via geometric morphisms.

Title: Sheaves of t-categories

Abstract: The notion of a t-category, or a triangulated category equipped with a t-structure, was introduced by Beilinson, Bernstein, and Deligne in their study of perverse sheaves. To avoid certain pathologies inherent in the category of triangulated categories, it is more convenient to upgrade this notion to that of a stable infinity-category equipped with a t-structure. Since the infinity-category of stable infinity-categories admits limits, this allows us to consider sheaves of t-categories. A main example is the derived infinity-category of a scheme. We will begin with a quick introduction to stable infinity-categories and t-structures, attempt to motivate why one might be interested in such things, and conclude with some more recent results on existence of bounded t-structures and applications to algebraic K-theory.

Title: Toposic view on deep neural networks

Abstract: First of all this lecture will be an attempt to understand and to learn from the article Topos and stacks of deep neural networks of J.C.Belfiore and D.Bennequin. All the original research work is due to them. Deep Neural Networks are very useful in machine learning and thus have many modern applications. After recalling some basic facts about those networks, I will expose the approach of Belfiore and Bennequin consists in associating a topos to a Deep Neural Network and then interpret some features of it (which are useful for machine learning perspective such as learning dynamics, invariant structures in the layers, semantical functioning, …) in the topos associated to the Deep Neural Network. One could hope with such an interpretation, one could in the long term understand better the network or design better networks by listening to the suggestions of the topos.

Title: Toposic view on deep neural networks

Abstract: First of all this lecture will be an attempt to understand and to learn from the article Topos and stacks of deep neural networks of J.C.Belfiore and D.Bennequin. All the original research work is due to them. Deep Neural Networks are very useful in machine learning and thus have many modern applications. After recalling some basic facts about those networks, I will expose the approach of Belfiore and Bennequin consists in associating a topos to a Deep Neural Network and then interpret some features of it (which are useful for machine learning perspective such as learning dynamics, invariant structures in the layers, semantical functioning, …) in the topos associated to the Deep Neural Network. One could hope with such an interpretation, one could in the long term understand better the network or design better networks by listening to the suggestions of the topos.

Title: Homotopy theory of Virtual Double Simplicial Sets

Abstract: The first part of the talk will be a review of virtual double categories (vdcs), the co-Yoneda lemma, and generalized T-multicategories. I will work with the examples Span(C), and V-prof for vdcs and detail the formal construction of a vdc as a fc-multicategory. If we remove the assumption of pullback in C, or we allow for varying V in “V-prof”, we are forced us to compose weakly. This warrants a simplicial version of a virtual double category.

The second part is under construction, so will act as more of a discussion after tea. I will talk about the development of a homotopy theory of virtual double simplicial sets (vdsSets). This development involves defining a notion of representability that will allows us to define (1) coskeletal induction, (2) horn conditions and (3) slice categories. (1) constructs examples of vdsSets (2) defines virtual double oo-categories and (3) defines op/cartesian cells. These are all important for understanding the vdsSet “Ont” whose objects are (object:category) pairs c:C, d:D, and whose pro-morphisms are comma objects F(c)-f->G(d):M, and for describing a yoneda lemma in Ont. Ont is a toy model for a meta-theory of ontology.

Title: The étale fibration over topoi

Abstract: The purpose of the talk is to show how ideas of dependent type theory (dependent sums and dependent products) can be used to revisit features of topology (locally contractible and compact spaces). The theory of ∞-topoi is based on the universe of ∞-groupoids (of the universe of sets in the 1-categorical setting). This universe is encoded by the fibration of étale maps over the category of ∞-topoi. We shall see that this universe has dependent sums, but not dependent products. We shall then see that the morphisms of topoi along which this universe has dependent sums are the locally contractible morphisms, and the morphisms along which dependent products exists are proper morphisms. This is jww Jonathan Weinberger.

Title: A 2-Categorical Proof of Frobenius in HoTT

Abstract: In the context of models of HoTT, the Frobenius condition says that the pushforward along fibrations preserve fibrations. This is used for the interpretation of dependent product (Π) types. Coquand gave a simple constructive proof of Frobenius in Cubical Type Theory, taking advantage of the fact that fibration structures can be reduced to composition structures. We give a 2-categorical proof of the Frobenius condition guided by the category-theoretic proofs by Gambino & Sattler and by Awodey. This is based on joint work with Emily Riehl.

Title: The Objective Metatheory of Simply Typed Lambda Calculus

Abstract: Normalisation by evaluation was initially discovered by accident, with the idea of leveraging a Lisp interpreter for reducing lambda calculus terms. Since this discovery, it was clear that NbE had some underlying categorical content, and, in 1995, Altenkirch et al published the first categorical normalisation proof. Discovering this content requires first asking the question “What is STLC?”, perhaps preceded by the question “What is a type theory?”. Clarifying these notions yields the concept of evaluation, and, from there, one can inductively construct normalisation proofs by assembling proof objects in an appropriately structured category. This talk will lay out all of these ideas with care, following along our recent formalisation of Altenkirch’s proof in Cubical Agda. As a bonus, a modification of our formalisation employing setoids of computational traces yields the implementation of a NbE powered stepper (Stepping by Evaluation).

Title: A tale of two model structures

Abstract: In this talk I will present joint work with Lyne Moser and Maru Sarazola, in which we construct two model structures on the category of double categories. The novel feature of these with respect to existent ones in the literature is that both of our model structures “contain” the homotopy theory of 2-categories. Moreover, they induce Lack’s model structure on 2-categories via different versions of the horizontal embedding of 2-categories into double categories. We will introduce these two model structures, discuss their main properties, and show how they relate to each other.

Fall 2021 Schedule: (essentially every Wednesday; pretalk 4:30-5:30pm, talk 6-7pm)

Title: Formally adjoining duals to symmetric monoidal (∞-)categories

Abstract: I will review the familiar notion of dual objects in a symmetric monoidal category C. I will then consider the less-familiar construction which freely adjoins duals to the objects of C to obtain a new symmetric monoidal category D(C) in which every object is dualizable. In order for this construction to be well-defined, it is convenient to work in an ∞-categorical setting (but, as I hope to convey, the essential ideas I will discuss are generally motivated from basic 1-categorical considerations). I will discuss some results from my thesis concerning the functor D and some close variants thereof where care is taken to respect additional structure on C such as certain colimits. In addition to several abstract motivations for considering such constructions, I will discuss an application, giving a new universal property for genuine equivariant stable homotopy theory in terms of equivariant Spanier-Whitehead duality.

Title: Classifying spaces of categories of structures

Abstract: Probably the most common two categories whose objects are used to model homotopy types are topological spaces and simplicial sets (or variations on these). I will survey a few alternatives to these, including other presheaf categories, and most fundamentally, the category Cat of (small) categories itself. Here a category is identified with the homotopy type of its classifying space. Much of this theory is systematized in the theory of localizers laid out by Grothendieck, and studied by Thomason, Maltsiniotis, Cisinski, and others.

As we will see, every homotopy type is modeled by some category. In the second half of the talk, I will turn to the perplexing fact that nevertheless, most naturally occurring categories are weakly contractible. This raises a challenge: given an interesting homotopy type X, find an interesting category C with that homotopy type. I will discuss joint work with Jinhe Ye and Gregory Cousins addressing one version of this question, with applications to logic (and specifically to model theory).

Title: Kripke-Joyal Semantics for Dependent Type Theory

Abstract: Every topos has an internal higher-order intuitionistic logic. In the first part of my talk I will give an expository account of this internal logic. I will also overview the so-called Kripke–Joyal semantics of a topos which has proved to be a powerful technique in topos theory. In its most basic formulation, Kripke-Joyal semantics provides a way to test the validity a formula of the internal logic of a topos.

In the second part of my talk, I will explain how to replace the internal logic of a topos by a more expressive internal type theory. I will also introduce a generlization of Kripke-Joyal semantics which provides a convenient way of moving back and forth between the internal type theory and the diagrams in a topos. This talk is based on a joint work with Steve Awodey and Nicola Gambino.

Title: Kripke-Joyal Semantics for Homotopy Type Theory

Abstract: We will use the Kripke-Joyal Semantics of the internal dependent type theory of presheaf toposes to work with various algebraic structures in presheaf toposes. I shall demonstrate how our extended Kripke-Joyal Semantics can be used to relate some of the recent type-theoretic and category-theoretic developments in finding constructive models of Homotopy Type Theory.

This talk is based on a joint work with Steve Awodey and Nicola Gambino.

Title: Double categories and algebraic K-theory

Abstract: Algebraic K-theory is a subject that touches upon a variety of fields, techniques, and applications. Starting with the work of Grothendieck in 1957, it was quickly taken up by Bass and Milnor, who defined the lower K-groups of a ring, and bloomed under Quillen—and later, Waldhausen—who introduced the rich higher invariants we know today. At its heart, it consists of a machinery that takes some flavor of algebraic structure as input, and produces a space, or a spectrum, whose homotopical structure records key data about the original object.

This talk hopes to be an introduction to algebraic K-theory from a category theorist’s point of view. After briefly talking about the historical motivation, we will delve into some of the different types of categories that a classical algebraic K-theory machinery may take as an input, and how (bi)simplicial sets are constructed from these categories to obtain their K-theory spaces. We will also talk about how K-theory is “universally additive” in some precise sense, and mention other classical foundational results along the way.

No previous background of K-theory required!

Title: Double categories and algebraic K-theory

Abstract: In recent work, Campbell and Zakharevich introduced a new type of structure, called ACGW-category. These are double categories satisfying a list of axioms that seek to extract the properties of abelian categories which make them so particularly well-suited for algebraic K-theory. The main appeal of these double categories is that they generalize the structure of exact sequences in abelian categories to non-additive settings such as finite sets and reduced schemes, thus showing how finite sets and schemes behave like the objects of an abelian category for the purpose of algebraic K-theory.

In this talk, we will explore the key features of ACGW categories and the intuition behind them. Then, we will move on to recent work with Brandon Shapiro where we further develop this program by defining “ACGW categories with weak equivalences”, which allow us to get a notion of chain complexes and quasi-isomorphisms for finite sets. These satisfy an analogue of the classical Gillet–Waldhausen Theorem, providing an alternate model for the K-theory of finite sets.

Title: S-algebras and the Field with One Element

Abstract: Absolute geometry, or geometry over the oxymoronic “field with one element” F1, seeks to extend the classical algebraic geometry of rings so that arithmetic schemes (such as Spec Z) can be treated on a similar footing to schemes defined over fields. The program’s loftiest ambitions include proving the Riemann hypothesis, whose analogue over finite fields has been known since Deligne in the 1970s. Since the initial conjecture of F1-geometry by Smirnov and Manin in the late 1980’s and early 1990’s, numerous different objects have been proposed as foundations for it, from monoids and semirings to more exotic objects like hyperrings, blueprints, and Lambda-rings.

In this talk, we will briefly go over the motivations for F1-geometry and a few notable past attempts at constructing it. Then, for the bulk of the talk, we will focus on Alain Connes and Katia Consani’s recent attempts to construct an absolute geometry over the category of Gamma-sets. We will lay out the main features of Connes and Consani’s project so far, which include proposals for a compactified scheme Spec Z-bar, the spectrum of a general S-algebra, and an analogue of singular cohomology with coefficients in a Gamma-space; as well as discussing its connections with other versions of absolute geometry. Finally, we will note what questions about this geometry over S remain open.

Primary references for this talk are Connes and Consani’s papers “Absolute algebra and Segal’s Gamma-rings: Au dessous de Spec Z-bar,” “On Absolute Algebraic Geometry: The Affine Case,” and “Spec Z-bar and the Gromov Norm,” along with their sequels; the speaker may also reference his own recent scribblings based on these papers. Descriptions of the classical setup are largely based on Hartshorne’s Algebraic Geometry (Sec. I.6, IV.1) and Dino Lorenzini’s Invitation to Arithmetic Geometry (Ch. 8-10), and background on F1-geometry is primarily drawn from Oliver Lorscheid’s overview “F1 for everyone,” together with the papers it references for specific constructions.

Title: An example in cohesive topos theory

Abstract: I will try to explain parts of Urs Schrieber’s work on cohesive topos, in particular, the example of smooth Deligne complex.

Title: An introduction to A1-homotopy theory

Abstract: There are intricate links between algebraic varieties and topological spaces. For instance algebraic varieties over the real or complex numbers give rise to topological spaces, so with that in mind, how much of homotopy theory can be developed for algebraic varieties or schemes? A1-homotopy theory is one approach which “uses the affine line as a substitute for the unit interval in topology”. The rigidity of algebra (only polynomial functions are allowed!) makes it unclear how exactly to set up the theory correctly, so I will attempt to give a motivated introduction to how one proceeds. I will not assume the audience is familiar with schemes or algebraic varieties, and instead will give a user-friendly crash course for anything we need. See you there!

Title: In Homotopy Type Theory, the moduli stack of somethings is the type of those things

Abstract: The title is already the abstract, but in particular: I’ll introduce homotopy type theory, a logical framework which, on the one hand: - is all about the idea that every mathematical object is a certain type of mathematical objects, and what it means to identify two objects is determined by what type of thing they are; and, somehow also, - is a way of working directly with higher stacks (sheaves of infinity-groupoids) as if they were defined by what elements they have, just like sets are. After that single sentence, we’ll look at the moduli stack of elliptic curves over C as a type, and compute its homotopy type.

Title: stable category theory(??) and absolute algebra(???)

Abstract: I will explain how to make sense of “(ω-)categories with negative dimensional cells,” which categorifies “spaces with negative dimensional cells,” aka spectra, following chapter 13 of German Stefanich’s thesis. After that, time/preparation/my math ability/vibe permitting, I’ll explain how this might potentiall relates to the idea of absolute algebras (in the sense Sean explained in a previous talk).

Title: Schur Functors and Categorified Plethysm

Abstract: It is known that the Grothendieck group of the category of Schur functors is the ring of symmetric functions. This ring has a rich structure, much of which is encapsulated in the fact that it is a “plethory”: a monoid in the category of birings with its substitution monoidal structure. We show that similarly the category of Schur functors is a “2-plethory”, which descends to give the plethory structure on symmetric functions. Thus, much of the structure of symmetric functions exists at a higher level in the category of Schur functors.

Fall 2020 Schedule (various Wednesdays; gather 5pm, talk 5:15pm Baltimore time):

Title: BI DOUBLing categories we’ll see // MULTIple morphisms acting weakly // Two out of the four // Have laws rather poor // But the last is coherent VIRTUALLY!

Abstract: Although 1-dimensional categories of various flavours are sufficient as a universe of discourse for many applications, the theory of categories themselves only truly reveals itself in dimension 2. For this reason, and for other naturally occurring motivating examples, it is of use to make rigorous and study various notions of 2-dimensional category.

Of particular interest is the ability of notions of 2-dimensional categories to introduce “weakness”. Lifted from the confines of a single dimension we are now free to ask for, and study examples where, composition of 1-dimensional morphisms in a 2-dimensional category is no longer strictly associative. Moreover, morphisms between 2-dimensional categories themselves are now free to obey functoriality to varying degrees.

The talk will take the form of a high-level overview of the definitions, examples, motivations for, and early theory of bi-categories, multicategories, pseudo-double categories and virtual double categories. These definitions will be compared, and each will be paired with an accompanying notions of morphisms of varying strictness.

The goal of the talk is to advocate for the view that, when composition is defined by a universal property, the complex laws and theorems of non-strict composition are automatic and functoriality is the natural result of the graded preservation of universality.

Prerequisites: limits in a category, especially products and pullbacks;monoidal categories, and strict/strong/lax monoidal functors; the 2-category Cat of categories, functors, and natural transformations

Suggested background: We will find motivation in theory of pro-functors and their composites, as well as the theory of bi-modules over rings and their tensors.

Title: Getting equipped for formal category theory

Abstract: In his 1973 paper Metric Spaces, Generalized Logic, and Closed Categories, Lawvere notes that not only are the objects of interest in mathematics organized into categories, they often are categories (of some flavor) in their own right. There are many different flavors of category — enriched, internal, and stranger — each of which have their own category theory. But just as there are many concepts throughout mathematics which are unified by their expression in the algebra of composition of maps, there are many concepts in category theory which are unified by their expression in the algebra of composition of natural transformations between bimodules. This algebra of natural transformations between bimodules is described by a virtual equipment. In this talk, we’ll see a bit of what category theory looks like in a general virtual equipment.

Title: Partial maps and PCAs in categories

Abstract: This talk will give a brief introduction to restriction categories and Turing categories, including their motivations which relate to partial maps and partial combinatory algebras (PCAs). First we will talk about restriction categories where we’ll introduce notions such as restriction idempotents and ways they can be split. We’ll then talk about how to handle products within restriction categories. After this, we’ll introduce Turing categories and describe in depth their relationship with partial combinatory algebras.

Title: 2-lessons from Australian Category Theory: mates (and doctrinal adjunction)

Abstract: Part of the lesson of category theory is that adjunctions permeate mathematics at almost all levels, from logic and topology to algebra and language grammar. The categorical method, however, teaches us that understanding is obtained by collecting our objects of interest into a category and studying their structure — as given by their morphisms — en masse. What then, on this view, is the structure supported by the category whose /objects/ are adjunctions? What is an appropriate notion of morphism here?

In this talk we will begin to answer this question in greater (but not greatest) generality by extracting, from any 2-category, at first a category and then later double categories whose objects (and later pro-arrows) are adjunctions. This will allow us to see how morphisms between left adjoints correspond to morphisms between right adjoints, the “mate correspondence”. Then, in the double-categorical context, we will realise mates as a /very/ natural isomorphism between certain double functors. We will exploit this naturality to give a theorem about the transfer of all propositions and structures on left adjoints expressible in a certain language, to corresponding propositions and structures on their right adjoints. That is, we will aim to make rigorous Leinster’s view that “all imaginable statements about mates are true.”

We will find applications of our theory of mates in re-proving some of the early theorems of adjunctions and, if time and interest permits, to the celebrated “Doctrinal Adjunction” result of Kelly. The central result of this paper relates adjunctions of lax- and colax-morphisms of 2-dimensional algebras for 2-dimensional monads to adjunctions of the underlying objects. As a particular consequence, in an adjunction of monoidal categories the left adjoint is oplax-monoidal iff. the right adjoint is lax-monoidal.

Prerequisites: Adjunctions (equational definition in Cat); 2-categories, 2-functors (definitions and Cat as an example); double categories, double functors (definitions)

Inessential but suggested: horizontal and vertical double-natural transformations

Title: Quillen model structures in 2-category theory

Abstract: The goal of this talk is to introduce some of the basic concepts of model category theory from the point of view of a 2-category theorist. All motivation and examples will be drawn from 2-category theory.

Title: Unstraightened Algebra

Abstract: Naive attempts to define homotopy coherent algebraic structures in topology lead to issues of book-keeping and the problem of data versus structure as one must specify an infinite amount of coherence data to ensure homotopical associativity. We study monoids via simplicial objects known as their bar constructions, and show how weakening some of the assumptions in the bar construction leads to the appropriate homotopy coherent structure. This way of thinking applies readily to category theory, where a strict monoidal category is a strict monoid object, and a monoidal category is a homotopy coherent monoid. In the case of categories we can go a step further and associate a cocartesian fibration to a monoidal category’s bar construction via the Grothendieck construction. This allows us to recast the theory of monoidal categories, lax monoidal functions, and algebras over operads in terms of cocartesian fibrations over the simplex category. While packaging coherence data this way might seem like overkill, since normal category theory is naturally only 2-dimensional, it gives a way to extend the theory of monoidal structures to the ∞-category world. Indeed this is the approach taken by Lurie in Higher Algebra to define and study ∞-operads and monoidal ∞-categories.

Title: Comparing left and right derived functors with double categories

Abstract: For functors between abelian categories which fail to be exact, there are left and right derived functors which encode obstructions to exactness on a homological level. These derived functors are quite readily described using the language of model categories applied to categories of chain complexes. In this talk, we will explore the double functoriality of taking left and right derived functors, and use this to give a derived version of the projection formula for base change along a proper map between locally compact Hausdorff spaces.

Title: A cup of HoTT cocoa

Abstract: Join me around the fire with a cup of HoTT cocoa for orbifold storytime. In homotopy type theory, we may define a group as the type of self-identifications — or symmetries — of an object. But HoTT offers a radical change in perspective on groups: instead of working with the symmetries directly, we work with the images of an object — those things which are identifiable with it (though not in any canonical way). We’ll play around with this viewpoint and see how to represent groups and actions via images. Then we’ll see how to define some simple orbifolds in HoTT, and learn that orbifolds have points just like manifolds do, and that it can be very fun to work with them. Along the way I will drop some puzzles to chew on, which we will share our favorite solutions to at the end.

Spring 2020 Schedule (single talk; 2:30pm-4pm UTC):

Title: ∞-category theory for undergraduates; slides

Abstract: At its current state of the art, ∞-category theory is challenging to explain even to specialists in closely related mathematical areas. Nevertheless, historical experience suggests that in, say, a century’s time, we will routinely teach this material to undergraduates. This talk describes one dream about how this might come about — under the assumption that 22nd century undergraduates have absorbed the background intuitions of homotopy type theory/univalent foundations.

Fall 2019 Schedule (most Tuesdays; pretalk 4pm, talk 5:30pm):

Title: Sketches of an Elephant: an Introduction to Topos Theory

Abstract: We briefly outline the history of topos theory, from its origins in sheaf theory which lead to the notion of a Grothendieck topos, through its unification with categorical logic which lead to the notion of an elementary topos, to a glimpse of the modern topos-theoretic outlook. PTJ describes this point of view as “the rejection of the idea that there is a fixed universe of “constant” sets within which mathematics can and should be developed, and the recognition that the notion of “variable structure” may be more conveniently handled within a universe of continuously variable sets.” Time permitting, we’ll sketch an application of the universal language of the topos of sheaves on the spectrum of a commutative ring that allows one to regard the ring as a local ring, at least locally.

Title: Induction and construction: the pointless theory of localic topox

Abstract: In this talk we will explore a half-way notion between the concrete description of the topos of sheaves of continuous functions on a topological space, and the fully general description of topos on a site. Specifically, we will concern ourselves with the replacement of topological spaces by categories of a particular nature. These categories are the locales of the title, and we will sketch some of their theory that is of independent interest so as to gain an intuition for the weirdness to come.

To mention some highlights, we will encounter spaces without points, and examine both extrinsic and intrinsic descriptions of when a topos is equivalent to a topos of sheaves on a locale.

The connection with topological spaces will then motivate the definition of geometric morphisms, whose properties and utility will be of central concern in the coming talks.

For those playing along at home, references include, as ever, Mac Lane - Moerdijk (the chapter bearing the same name as the title of this talk), the nLab (articles: locale, localic topos, and geometric morphism) but also the book of Picado - Pultr, “Frames and Locales”.

Title: Introduction to Grothendieck Topologies and Sheaves

Abstract: We will start by motivating Grothendieck Topologies as a means of studying geometric objects whose local structures can be probed by maps more general than an embedding of an open neighborhood. We will briefly consider the case of orbifolds, where it is natural to consider covers which locally express an orbifold as a quotient of a Euclidean space by a finite group of diffeomorphisms. We will then give the definition of a site and the category of sheaves on it, giving examples as we go along.

We will then proceed to generalize what we saw for the categories of sheaves on a topological space to the categories of sheaves on a site. In particular, that there is a left exact left adjoint of the inclusion of sheaves into presheaves (the sheafification functor) and that a category of sheaves on a site is an elementary topos.

Title: A Model of Type Theory with Directed Univalence in Bicubical Sets

Abstract: Directed type theory is an analogue of homotopy type theory where types represent infinity-categories, generalizing groupoids. A bisimplicial approach to directed type theory, developed by Riehl and Shulman, is based on equipping each type with both a notion of path and a separate notion of directed morphism. In this work, we investigate the extent to which the cubical techniques used to give computational models of homotopy type theory can be extended to a variant of the Riehl-Shulman directed type theory. Using Agda as the internal language of the presheaf topos of bicubical sets, we formalize the definitions of directed type theory, reformulate them to better correspond to the filling problems of cubical type theory, and implement them for various type constructors. We show that there is a universe of covariant discrete fibrations (covariant functors valued in types that themselves have trivial morphism structure), where functions are a retraction of morphisms in this universe. This retraction is “three quarters” of the desired directed univalence equivalence between functions and morphisms. We also describe some work in progress towards the final quarter, inspired by Cavallo, Sattler and Riehl’s proof of directed univalence in the bisimplicial model. Adapting their method to the bicubical setting is a significant extension, because simplices have a special property of being Reedy, which cubes lack. To ameliorate this, we define a Reedy category of prismatic sets, showing that cubical sets are equivalent to sheaves on prisms, and showing that the type theory’s fibrations can be interpreted in the model structure transferred to bicubical sets from prismatic cubical sets. The final quarter of the proof is not obviously non-constructive, though it will take further work to determine whether it has computational content. This is joint work with Dan Licata.

Title: Modalities and fibrations for synthetic (∞,1)-categories

Abstract: Higher-dimensional categories play an increasing role in many areas of Mathematics, such as topology, geometry, number theory, and logic.

Higher categories are often defined analytically, i.e. entirely based on sets and ordinary categories (endowed with some additional structure), even though these objects themselves do not carry any higher-dimensional information.

Riehl and Shulman have introduced a synthetic framework for (∞,1)-category theory where, in contrast, the basic objects intrinsically are of homotopical flavor. One motivation in developing such a synthetic theory is to find a language that is closer to ordinary 1-category theory, and hide much of the (mathematical) implementation of higher-dimensional and homotopical objects.

We present a variation of this synthetic theory that allows for additional constructions such as opposite categories and twisted arrow spaces. Our approach is crucially based on Licata-Riley-Shulman’s fibrational framework for modal type theories.

Furthermore, we suggest a notion of synthetic co-/cartesian fibrations, based on Riehl-Shulman’s work on synthetic discrete co-/contravariant fibrations.

The topics presented are joint work in progress with Ulrik Buchholtz.

Title: More on Geometric Morphisms (and Points)

Abstract: Now we have the category of topoi (or topox if you want to be neutral), namely, the category whose objects are topoi and morphisms are geometric morphisms (=pairs of adjoint functors whose left adjoint is left exact). We should spend some time to understand what these morphisms look like. First, we will study examples of geometric morphisms and define several properties of geometric morphisms with suggestive names (which meant to generalize the counterpart for (nice) topological spaces). Among those, we will define the notion of a point. To study points of Grothendieck topoi, we will exploit the “presentation” by a site, using Kan extensions and the resulting “tensor-hom” adjunctions. This will lead us to the notion of flat functors, and a point of a sheaf topos on a site turns out to correspond a “continuous flat functors” from the site to the category of sets. Elaborating this we will see that, under some conditions, a functor between sites defines a geometric morphism between their sheaf topoi in explicit ways. Title: Why (algebraic/arithmetic) geometers care about topos theory? Abstract: This talk is meant to be a rather informal discussion on the topic in the title. I will bring some materials and scatter around on what motivated Grothendieck to find the notion of topoi, how is it effective as a vastly generalized notion of spaces. More specifically, I will mention several topologies on the categories of (affine) schemes and its abelian sheaf cohomology (such as etale cohomology), locally ringed spaces (or more generally structured spaces) and algebraic spaces, and some other things, and try to elaborate accordingly (with help of participants).

Title: An Introduction to Classifying Toposes

Abstract: In mathematics we often run into structures on objects that can be obtained by pulling back along some universal example of that structure. For example, the cohomology classes of a space X with coefficients in an abelian group G, Hn(X,G), can be obtained by pulling back a universal class ηn in Hn(K(n,G),G) along maps into the Elienberg-Maclane spaces X→ K(n,G). In the first half of this talk we will introduce the philosophy of classifying topoi as a way to classify certain “theories” of topostopos, for example the objects of a topos and the ring objects of a topos.

After the break we will return to singular cohomology and show how we can abstract H1(-, G) into a theory of G-torsor objects in a topos. We will then show that there exists a classifying topos for G-torsors.

Title: Cohomology of statistical systems

Abstract: In this talk, I will introduce the category of information structures and some of its properties. These structures serve as models of systems of measurements in physics and analogue concepts in computer science and logic. (No background in probability or statistics will be assumed.) Then I will introduce a cohomology theory naturally attached to them, called “information cohomology”. Several functions appear as cocycles (for adapted modules of coefficients): Shannon entropy and some of its generalizations, the multinomial coefficients and their generalizations, the log of the determinant, the dimension of affine subspaces… The cocycle conditions encode remarkable recurrence relations. These are first steps towards a general “topology of statistical systems” in the vein of topos theory.

Title: Introduction to homotopy type theory with a view towards higher group theory

Abstract: Type theory was originally perceived to address Russell’s famous paradox in set theory. In the second half of the 20th century it was developed as a constructive foundation for mathematics, with a strong view towards computability. Computer implementations of type theory have been used to develop libraries of formalized mathematics, which have been used in the verification of major mathematical theorems as well as more practically in the verification of complex pieces of software and hardware. Voevodsky’s simplicial model of type theory, in which the beautiful univalence axiom was shown to hold, then came as somewhat of a shock, but since its initial conception it has become clear that the homotopy interpretation of type theory is here to stay. By now, constructive models of the univalence axiom have been developed, and it has been shown that homotopy type theory is the internal language of an arbitrary infinity topos. In this introductory talk we look at some of the fundamental concepts of homotopy type theory, and we explore some of the ways of thinking that are typical for this field.

Title: The long n-exact sequence of homotopy n-groups

Abstract: An n-group in homotopy type theory is nothing but a pointed connected n-type. Its underlying type is the loop space, which comes equipped with a higher group-like structure. The fundamental homotopy n-group of a pointed type is then just the n-truncation of the connected component of the base point. We formulate a notion of n-exactness, and show that any fiber sequence F → E → B of pointed types induces a long n-exact sequence of homotopy n-groups.

Title: Elementary topoi

Abstract: This talk will primarily be an introduction to elementary topos theory as well as an introduction to some interesting facts, examples, and constructions in an elementary topos. There will be two parts to the talk.

In the first part, I will go over the definition of an elementary topos as well as some important facts. In particular, I will define the subobject classifier and show that it is an internal Heyting algebra. I will also discuss how higher intuitionistic logic relates to elementary topoi.

In the second part I will discuss an interesting example of an elementary topos which is not a Grothendieck topos namely the effective topos. I will also explain what a realizability tripos is and describe the tripos-to-topos construction which result in topoi similar to the effective topos called realizability topoi. I will discuss Grothendieck topologies and their relationship to nuclei on the subobject classifier as well as mention geometric modalities. If time permits, I might briefly go over the lifting of Grothendieck universes in a sheaf topos.

Title: Within and not without, an apology for internal languages

Abstract: In this talk we will learn how (part of) classical set theory may be reinterpreted within an elementary topos. Our interest in this will be twofold: - First, the nature of this internalisation is such that it makes both expedient and clear arguments which would otherwise be phrased diagrammatically, and it does so without incurring any loss in the generality of the theorems. That is, the internal language is of utility to the mathematician working in a topos. - Second, this reinterpretation bears interesting and non-trivial consequences when the target topos carries with it a geometric character. We will examine the notion of local truth, and time allowing, the specialisation of this interpretation to the setting of a Grothendieck topos — the so-called “sheaf semantics”.

Overall we will chart a course through understanding: the role of the subobject classifier in lending structure to the topos; the Mitchell-Bènabou language of a topos; the Kripke-Joyal semantics which proves a soundness and completeness result for this language; and the construction of a real numbers object internal to the topos.

Understanding will be emphasised over details wherever possible.

Title: A sheaf of rings is a ring of sheaves, and other tales from algebra on the inside

Abstract: In this talk, we’ll begin exploring algebra (rings, modules, etc) through the looking glass. We’ll see what these notions mean when interpreted in the internal logic of a topos, and discuss some particular examples from algebraic and differential geometry.

Title: “Generally…”, “At a point…”, and other ways of being true

Abstract: We will revisit Lawvere-Tierney topologies on a topos from the internal point of view. Here they take the form of modal operators, ways that a proposition may be true. For each modal operator, there is a “modal translation” of a proposition (generalizing Godel’s double negation translation), and that modal translation holds if and only if the proposition holds in the associated sheaf topos.

To use this machinery, we introduce the notion of a geometric formula — a formula that behaves well under change of topos. In particular, for a geometric formula, the modal translation of a proposition is equivalent to its modal statement. This abstruse logical lemma has a geometrical purchase: if a geometrical formula holds at a point, it holds on an open neighborhood. We use this fact to quickly solve a few exercises from Vakil’s Rising Sea.

Title: The Big Picture

Abstract: We peek in at the internal logic of the big Zariski topos of a scheme. Here, the affine line A behaves as one would naively hope for algebraic geometry: the ring of functions (any functions) on the locus of zeros of polynomials f_1 … f_n is the polynomial ring A[x_1,…,x_m]/(f_1,…,f_n). We explore some consequences of this fact: A is an algebraically closed* field* with nilpotent infinitesimals*.

We will end by wondering aloud about how to work with monoidal structures in an internal way. *Come and see what these words mean internally!

Title: The arithmetic site

Abstract: Since 1998, the adele class space is at the core of the approach of A.Connes and then A.Connes and C.Consani of the Riemann zeta function. In 2014, A.Connes and C.Consani introduced a semiringed topos, the arithmetic site, which provides a geometrical understanding of the adele class space. In this lecture, I will first give some historical and heuristical reasons that led A.Connes and C.Consani to introduce the adele class space and the arithmetic site (and properly define those objects). I will then try in the remaining time to explain as precisely as possible the relation between those two objects, ie that the set of points of the arithmetic site over R+max is the quotient of the adele class space by Ž*.

Title: The Extended Cobordism Hypothesis: A survey of an overview

Abstract: Over the course of 2 talks, spanning 2 hours, I will attempt to distill about 70 pages of content from Lurie’s “On the Classification of Topological Field Theories”, which itself is just an overview. Although from this description it may seem like a rollercoaster of a talk, we will opt for a more relaxed approach. We will survey the relevant geometric* ideas used in Lurie’s Paper and avoid too many technical details (except for the cute ones).

The first talk will cover the necessary ideas to state the Extended Cobordism Hypothesis (ECH). We will discuss relevant ideas in SMCs and 2-categories, including adjoints and duals, and how the (1-)category of n-cobordisms has these duals. The duals will allow us to reduce the calculation of a SM-functor to its value on certain generating cobordisms. This will lead us to versions of the classical Cobordism Hypothesis. We will then mention (∞,n)-categories as segal spaces, and the construction of the (∞,n)-category of cobordisms, Bordn. Hopefully we will arrive at a statement of the Extended Cobordism Hypthosis.

In the second talk, I will reveal that the ECH was a ruse, and that the real cobordism hypthosis has a lot more structure. Namely, Bordn must be restricted to framed manifolds. This may seem like a huge restriction, but it will allow us to carry over an O(n)-structure via the (framed) ECH. The framed ECH is rather easy to prove, and from it we can extend the cobordism hypothesis to versions of Bordn with weaker restrictions. From these various levels of cobordism hypotheses, we will show how to deduce the general cobordism hypothesis.

Spring 2019 Schedule (intermittent Tuesdays; pretalk 4pm, talk 5:30pm):

Title: Higher categories from higher-dimensional manifolds

Abstract: 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

Abstract: 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

Abstract: The talk last time was actually about “ 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

Abstract: 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?

Abstract: 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

Abstract: The notion of an Operad has been round 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

Abstract: 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

Abstract: “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

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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.<p>

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.<p>

*terms and conditions may (and will) apply.

Title: What is an LCC?

Abstract: 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.

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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

Abstract: 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)

Abstract: 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 Eilenberg-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.