talks

[16/08/23] Basic Category Theory I - Categories & Functors

Speaker - Arghan Dutta
Abstract - We understand the basic language of categories and functors through a plethora of examples illustrating their ubiquitous nature all through mathematics. We start with a bit of history on the subject and then quickly move onto defining categories. The instances of this concept are abundant, since the category axioms don't demand much - indeed any kind of mathematical structure which has an associated notion of mapping compatible with the structure gives rise to a category. We see examples of categories, both "concrete" and "non-concrete". We give methods to construct new categories out of existing ones. We then stress on the point that categories are mathematical objects in their own right so that it is natural to talk about a notion of map between them AKA a Functor. We define functors and give examples. We end the talk with the definition of a comma category. These ideas will be used all throughout the seminar series. The main theme is : Understand the relationships to know the objects !

[23/08/23] Basic Category Theory II - Natural transformations, Universal properties & (co)Limits

Speaker - Raghuram Sundararajan
Abstract - We continue with the discussion of the basic language of Category Theory. Categories are mathematical objects and functors are maps of categories. Perhaps surprisingly, we can consider functors as mathematical objects and ask about maps between them. Such maps are called natural transformations and they capture the concept of "naturality" in mathematics. Of course, defining natural transformations at last is sort of an anachronistic approach since Maclane and Eilenberg had come up with category theory to formalize the idea of "naturality" in mathematics. Our motivating example is that for a finite dimensional vector space V, V is isomorphic to both its dual and double dual. While there is a canonical isomorphism of V onto its double dual, an attempt to construct one between V and its dual fails since we have to "choose an arbitrary basis" to construct it, depending on V. For a fixed pair of categories C and D, we show that functors from C to D and their natural transformations form the functor category [C, D]. We then quickly move onto representable functors which encode "Universal properties" in mathematics. We end with a description of certain special universal constructions : (co)limits of diagrams (functors) and their abundant examples.

[30/08/23] Categorification and Monoidal Categories

Speaker - Subhajit Das

Abstract - A good way to think about Category Theory is that it is a refinement of Set Theory. Categorification is the process of turning set theoretic concepts into category theoretic ones. We classify Categorification broadly into two types : (1) Horizontal categorification or "Oidification" which attempts to understand elements of a mathematical structure as arrows in a single object category and (2) perhaps the most important one in our study of Higher Categories, Vertical Categorification. We stress on the idea of Vertical Categorification as a section of Decategorification, something which we have been doing for a long time in our mathematical venture. Decategorification forgets category theoretic structure : Turns categories to sets, sets to cardinals etc. Vertical Categorification on the other hand, reverses the procedure : Sets become categories, categories become 2-categories, 2-categories become 3-categories and so on. At each step, more structure gets added and the notion of equality gradually weakens. We apply the idea of Vertical Categorification to a monoid to obtain the notion of a monoidal category. We verify that this is indeed taken by Decategorification to the notion of a monoid. We study some basic properties of monoidal categories and categorify monoid maps to monoidal functors. The importance of categorification in this context is readily exhibited : We have a 2-category structure on the category of Monoidal categories, in contrast to the 1-category of monoids. We present some examples and then a schematic diagram of horizontal and vertical categorifications, indicating in particular that a monoid is categorified vertically by a monoidal category which is further horizontally categorified by a Bicategory (2-category), to be defined later. We finally end with a discussion of a coherence theorem for Monoidal Categories.

[13/09/23] Structures on Monoidal Categories

Speaker - Arghan Dutta

Abstract - We continue with the discussion of the previous week. Starting with a category with a monoidal structure, we produce further, three different monoidal structures on the underlying category. We recall the definition of monoidal functors and define monoidal natural transformations between them. A significant part of the talk is spent on structures on monoidal categories, starting with braidings and symmetries. We then introduce the vital concept of an internal monoid in a monoidal category. We present examples to illustrate its abundance all through out mathematics. We then spend some time on Monads, which are monoids in the category of endofunctors on some category. Monads are crucial objects which capture algebraic theories and will be particularly useful in our study of Higher categories. We define the category of algebras over a monad which roughly capture the "models" of an algebraic theory. We then show that every adjunction defines a monad on the domain of the left adjoint. As an example, the free-forgetful adjunction between Graphs and Categories induces a monad on the category of graphs which sends a graph to the underlying graph of the free category generated by it. We demonstrate that the converse of this statement is true by constructing the infamous Eilenberg-Moore adjunction associated to a monad. We then move on to (non-degenerate) pairings and duality on monoidal categories. We end with the notions of rigid and pivotal monoidal categories.

[04/10/23] Ideas from Enriched Category Theory - Towards Higher Categories

Speaker - Subhajit Das

Abstract - The concept of a (locally small) category is firmly grounded on the category Set : The hom-sets are merely sets. However, many categories enjoy the property that their hom sets have additional structure that "behave well" with composition : the hom sets of R-Mod have an abelian group structure, the hom sets of Cat have a category structure, the hom-sets of n-Cat have an n-Category structure and the hom sets of sSet themselves have a simplicial structure. While Ordinary Category Theory fails to capture this, Enriched Category Theory formalizes this idea. In this talk, we shall begin with the observation that the (Cartesian) monoidal structure on the category Set is what allows it to support the definition of a category "over" it.

We then move on to defining categories enriched in arbitrary monoidal categories. A monoidal category V behaves like a dial which when set to different values gives a familiar type of category : When V = Set, Ab, R-Mod, Cat, etc, a category enriched in V is precisely an ordinary category, a preadditive category, a linear category, a (strict) 2-category respectively. We understand the definition of a strict 2-category in full detail. We spend some time on Lawvere's beautiful (and astonishing) idea that metric spaces or posets are enriched categories : Metric spaces are ([0, ∞)*, +, 0)-enriched categories while posets are (false → true, ∧, true)-enriched categories. Most of these concepts are spinoffs of Lawvere's ideas, back when Category Theory was in stages of development. We follow this up with the appropriate notion of enriched functors and natural transformations which would define the 2-category V-Cat of categories enriched in V. We then define the "underlying category" functor V-Cat → Cat, and show that it is a strict 2-functor. We show how putting a symmetry structure on the monoidal category V endows V-Cat with the structure of a monoidal 2-category. Further, we see that that the monoidal category V being closed (that is, the monoidal product functor being divisible on one side) endows V with an internal hom bifunctor V* x V → V, making V into a self-enriched category. This then paves the way for an enriched version of category theory, allowing us in particular to define representable V-functors, talk about universal properties etc. We end with an inductive definition of strict n-categories, as (n-1)-Cat enriched categories, with 0-Cat = Set. This definition, although elegant, needs more attention. Accordingly, we give a rough idea of n-dimensional globular sets, which are the higher dimensional analogs of graphs and which are the right context to put forth an explicit definition of strict n-categories. This will be discussed in detail, in the upcoming lectures.

[11/10/2023] The Theory of Bicategories

Speaker - Arghan Dutta

Abstract - In the previous lecture, we ended with an iterative (local) definition of strict higher categories. As it turns out, these are very rare. What are abundant are their weak cousins, weak higher categories. We embark on with the 2-dimensional notion of a weak higher category, namely, a bicategory. We begin with the definition, noting that bicategories are obtained as a categorification of the notion of an ordinary category. We also note that the definition bears a resemblace with that of a monoidal category, the reason being that a bicategory with one object is precisely, a monoidal category. We then define maps of bicategories and lax transformations and present a plethora of examples. We then observe that bicategories are well structured to support a definition of adjunctions between 1-cells and accordingly develop a theory of the same. We end with a coherence theorem for bicategories, analogous with Maclane's coherence theorem for monoidal categories.

[22/11/2023] n-Globular sets and Strict n-categories

Speaker - Arghan Dutta

Abstract - While the definition stating that a strict (n+1)-category is Str-n-Cat enriched is concise, it lacks explicitness. This talk introduces the concept of globular sets, which serve as an n-dimensional analogue to directed graphs. Subsequently, we give a precise formulation of strict n-categories using globular sets dues to Leinster. We give several examples and then define maps between these objects. We hint at a proof of the equivalence between the local and global definitions next. Finally, we move to strict infinity categories. We start with a global definition in terms of infinite dimensional globular sets, more precisely, as presheaves on a quotient category of the free category generated by the graph with objects being the natural numbers and with a pair of arrows from n to n+1, for each n ≥ 0. We use this to define strict infinity categories in full detail. We conclude with a local definition of such objects, namely by defining the category of strict infinity categories as the limit of the diagram
... → Str-n-Cat → Str-(n-1)-Cat → Str-(n-2)-Cat → .... → Cat → Str-0-Cat = Set in Cat, where every functor strips the top dimensional cells.

[29/11/23] Kan Extensions, (co)ends and weighted (co)limits

Speaker - Subhajit Das

Abstract - We introduce the notion of a Kan Extension in the general setup of a bicategory. Kan Extensions are universal objects and hence defined as a representation of a Set-valued functor. We formulate a characterisation in terms of adjoints to precomposition with a fixed functor. We then specialise to the case of Kan Extensions in the 2-category CAT. This is followed by a brief digression towards (co)ends and weighted (co)limits. We then note that under certain conditions on the categories involved, Kan Extensions admit a pointwise formula, in terms of a (co)end or a weighted (co)limit. Finally, we justify Mac Lane's statement " All concepts are Kan Extensions" : (co)limits, adjunctions, (co)Yoneda Lemma, etc. We end with a short discussion on the colimit-density of representables in the presheaf category on a small category.