Tag - Category theory

Paolo Perrone: Kan extensions are partial colimits

11th February, 2021

One way of interpreting a left Kan extension is as taking a kind of 'partial colimit', where one replaces parts of a diagram by their colimits. We make this intuition precise by means of the 'partial evaluations' sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the category CAT of locally small categories. We also define a morphism of monads between them, which we call 'image', and which takes the 'free colimit' of a diagram. This morphism allows us in particular to generalize the idea of 'cofinal functors', i.e. of functors which leave colimits invariant in an absolute way. This generalization includes the concept of absolute colimit as a special case. The main result of this work says that a pointwise left Kan extension of a diagram corresponds precisely to a partial evaluation of its colimit. This categorical result is analogous to what happens in the case of probability monads, where a conditional expectation of a random variable corresponds to a partial evaluation of its centre of mass.

Jiří Rosický: Metric monads

4th February, 2021

We develop universal algebra over an enriched category and relate it to finitary enriched monads. Using it, we deduce recent results about ordered universal algebra where inequations are used instead of equations. Then we apply it to metric universal algebra where quantitative equations are used instead of equations. This contributes to understanding of finitary monads on the category of metric spaces.

Alistair Savage: Affinization of monoidal categories

2nd February, 2021

We define the affinization of an arbitrary monoidal category, corresponding to the category of string diagrams on the cylinder. We also give an alternative characterization in terms of adjoining dot generators to the category. The affinization formalizes and unifies many constructions appearing in the literature. We describe a large number of examples coming from Hecke-type algebras, braids, tangles, and knot invariants.

John Bourke: Accessible ∞-cosmoi

10th December, 2020

Riehl and Verity introduced ∞-cosmoi - certain simplicially enriched categories - as a framework in which to give a model-independent approach to ∞-categories. For instance, there is an infinity cosmos of ∞-categories with finite limits or colimits, or of cartesian fibrations. In this talk, I will introduce the notion
of an accessible ∞-cosmos and explain that most, if not all, ∞-cosmoi arising in practice are accessible. Applying results of earlier work, it follows that accessible ∞-cosmoi have homotopy weighted colimits and admit a broadly applicable homotopical adjoint functor theorem.

Martin Bidlingmaier: Model categories of lcc categories and the gros model of dependent type theory.

26th November, 2020

In this talk we discuss various model categories of locally cartesian closed (lcc) categories and their relevance to coherence problems, in particular the coherence problem of categorical semantics of dependent type theory. We begin with Lcc, the model category of locally cartesian closed (lcc) sketches. Its fibrant objects are precisely the lcc categories, though without assigned choices of universal objects. We then obtain a Quillen equivalent model category sLcc of strict lcc categories as the category of algebraically fibrant objects of Lcc. Strict lcc categories are categories with assigned choice of lcc structure, and their morphisms preserve these choices on the nose. Conjecturally, sLcc is precisely Lack’s model category of algebras for a 2-monad T , where T is instantiated with the free lcc category functor on Cat. We then discuss the category of algebraically cofibrant objects of sLcc and show how it can serve as a "gros" model of dependent type theory.

Maru Sarazola: The stable homotopy hypothesis

19th November, 2020

The homotopy hypothesis is a well-known bridge between topology and category theory. Its most general formulation, due to Grothendieck, asserts that topological spaces should be "the same" as infinity-groupoids. In the stable version of the homotopy hypothesis, topological spaces are replaced with spectra.

In this talk we will review the classical homotopy hypothesis, and then focus on the stable version. After discussing what the stable homotopy hypothesis should look like on the categorical side, we will use the Tamsamani model of higher categories to provide a proof.

Ivan Di Liberti: Enriched Locally Generated Categories

5th November, 2020

We introduce the notion of M-locally generated category for a factorization system (E,M) and study its properties. We offer a Gabriel-Ulmer duality for these categories, introducing the notion of nest. We develop this theory also from an enriched point of view. We apply this technology to Banach spaces showing that it is equivalent to the category of models of the nest of finite-dimensional Banach spaces.

Fiona Torzewska: Mapping class groupoids and motion groupoids

4th November, 2020

An LMS online course in mapping class groupoids.

A topological phase of matter is a physical system whose behaviour may be effectively described via a topological quantum field theory i.e. functor from cob to vect. The study of topological quantum field theories has applications in quantum computing but also involves a lot of beautiful mathematics which is interesting in its own right. A central role in the description of topological phases of matter in 2 (spatial) dimensions is played by the representations of braid groups. A natural generalisation to study the statistics of higher (spatial) dimensional phases of matter is then to look for generalisations of the braid group. Braid groups can be equivalently defined as the mapping class groups or as the motions groups of points in a disk, as well as in several other equivalent ways. In these lectures we will introduce generalisations of these two definitions. In each case we will show first that these give us groupoids and then that we can get back to the classical definitions by considering the endomorphisms of a single object. The mapping class groupoid is a simpler construction but is not in general the right notion to take when considering particles moving through space. We will construct a functor from the motion groupoid to the mapping class groupoid and hence see which cases we can study only the mapping class groupoid. We will use lots of examples to aid intuition and intend this talk to be accessible to those with minimal knowledge of topology.