I will discuss a construction of a new model structure on simplicial objects in a countably lextensive category (i.e., a category with well-behaved finite limits and countable coproducts). This builds on previous work on a constructive model structure on simplicial sets, originally motivated by modelling Homotopy Type Theory, but now applicable in a much wider context.
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.
An LMS online lecture course in model categories.
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.
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