Johns Hopkins Category Theory (Virtual) Seminar

Various Wednesdays
gather 5pm, talk 5:15-6:15pm, social til 7pm

Fall 2020 Schedule (various Wednesdays; gather 5pm, talk 5:15pm Baltimore time; zoom connection information and time zone conversation will be available at 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!
     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.
     - 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
     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
     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.
     - 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
      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
      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
      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
      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; zoom connection information available at Title: ∞-category theory for undergraduates; slides
     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
     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
     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
     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
     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
     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)
     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?
     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
     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
     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
     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
     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
      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
     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
     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
     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
      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
     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
     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
     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 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
     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.