In this talk I will show that the simplicial Moore path functor, first defined by Van den Berg and Garner, is a polynomial functor. This result, which surprised us a bit at first, has helped a great deal in developing effective Kan fibrations for simplicial sets.
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.
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.
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.
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.
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.
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.
Among Banach spaces approximate injectivity is more important than injectivity. We will treat it from the point of view of enriched category theory - as enriched injectivity over complete metric spaces.
We give a definition of the Gray tensor product in the setting of scaled simplicial sets which is associative and forms a left Quillen bi-functor with respect to the bicategorical model structure of Lurie. We then introduce a notion of oplax functor in this setting, and use it in order to characterize the Gray tensor product by means of a universal property. A similar characterization was used by Gaitsgory and Rozenblyum in their definition of the Gray product, thus giving a promising lead for comparing the two settings.
The classical Pontryagin-Thom isomorphism equates manifold bordism groups with corresponding stable homotopy groups. This construction moreover generalizes to the equivariant context. I will discuss work which establishes a Pontryagin-Thom isomorphism for orbispaces (an orbispace is a 'space' that is locally modelled on Y/G for Y a space and G a finite group; examples of orbispaces include orbifolds and moduli spaces of pseudo-holomorphic curves). This involves defining a category of orbispectra and an involution of this category extending Spanier-Whitehead duality. Global homotopy theory also plays a key role.
In this talk we will study the lax orthogonal factorization systems (LOFSs) of Clementino and Franco, with a particular focus on finding equivalent definitions of them.
In particular, we wish to define them as a pair of classes ℰ and ℳ subject to some conditions. To achieve this, we will reduce the definition of a LOFS in terms of algebraic weak factorization systems (defined as a KZ 2-comonad L and KZ 2-monad R on the 2-category of arrows [2, 𝒞] with a 2-distributive law LR ⇒ RL) to a more property-like definition (meaning a definition with less data but more conditions).
To do this, we replace strict KZ 2-monads with the property-like definition of KZ pseudomonads in terms of Kan-extensions due to Marmolejo and Wood. In addition, pseudo-distributive laws involving KZ pseudomonads have a property-like description which will be used. Thus one can deduce the conditions the classes ℰ and ℳ must satisfy.
We will also consider some similarities and differences between LOFSs and (pseudo-)orthogonal factorization systems, and will extend their definitions to include universal fillers for squares which only commute up to a comparison 2-cell.
We present explicit constructions of infinite families of CW-complexes of arbitrary dimension with buildings as the universal covers. These complexes give rise to new families of C*-algebras, classifiable by their K-theory.
The underlying building structure allows explicit computation of the K-theory. We will also present new higher-dimensional generalizations of the Thompson groups, which are usually difficult to distinguish, but the K-theory of C*-algebras gives new invariants to recognize non-isomophic groups.
An elementary homomorphism from a free group to the pure braid group yields interesting connections between braid groups, homotopy theory, and low dimensional topology. This map induces a map on the Lie algebra obtained from the descending central series. Further, this map induces a morphism of simplicial groups. All of these maps are shown to be injective.
Brunnian braids are discussed. The analogous maps of Lie algebras induced on the filtration quotients of the mod-p descending central series is again an injection. Using these facts it turns out that the homotopy groups of this simplicial group, those of the 2-sphere, are isomorphic to natural subquotients of the pure braid group. In addition, the mod-p analogues give a connection between the classical unstable Adams spectral sequence, and the mod-p analogues of Vassiliev invariants of pure braids.
