Gabriel-Ulmer duality for topoi and its relation with site presentations

Let $\kappa$ be a regular cardinal. We study Gabriel-Ulmer duality when one restricts the 2-category of locally $\kappa$-presentable categories with $\kappa$-accessible right adjoints to its locally full sub-2-category of $\kappa$-presentable Grothendieck topoi with geometric $\kappa$-accessible morphisms. In particular, we provide a full understanding of the locally full sub-2-category of the 2-category of $\kappa$-small cocomplete categories with $\kappa$-colimit preserving functors arising as the corresponding 2-category of presentations via the restriction. We analyse the relation of these presentations of Grothendieck topoi with site presentations and we show that the 2-category of locally $\kappa$-presentable Grothendieck topoi with geometric $\kappa$-accessible morphisms is a reflective sub-bicategory of the full sub-2-category of the 2-category of sites with morphisms of sites genearated by the weakly $\kappa$-ary sites in the sense of Shulman [37].Comment: 25 page

The \'etale symmetric K\"unneth theorem

Let $k$ be an algebraically closed field, $l\neq\operatorname{char} k$ a prime number, and $X$ a quasi-projective scheme over $k$. We show that the \'etale homotopy type of the $d$th symmetric power of $X$ is $\mathbb Z/l$-homologically equivalent to the $d$th strict symmetric power of the \'etale homotopy type of $X$. We deduce that the $\mathbb Z/l$-local \'etale homotopy type of a motivic Eilenberg-Mac Lane space is an ordinary Eilenberg-Mac Lane space.Comment: revised version, comments welcome

Derived ∞\infty-categories as exact completions

We develop the theory of exact completions of regular $\infty$-categories, and show that the $\infty$-categorical exact completion (resp. hypercompletion) of an abelian category recovers the connective half of its bounded (resp. unbounded) derived $\infty$-category. Along the way, we prove that a finitely complete $\infty$-category is exact and additive if and only if it is prestable, extending a classical characterization of abelian categories. We also establish $\infty$-categorical versions of Barr's embedding theorem and Makkai's image theorem

The effective model structure and ∞\infty-groupoid objects

For a category $\mathcal E$ with finite limits and well-behaved countable coproducts, we construct a model structure, called the effective model structure, on the category of simplicial objects in $\mathcal E$, generalising the Kan--Quillen model structure on simplicial sets. We then prove that the effective model structure is left and right proper and satisfies descent in the sense of Rezk. As a consequence, we obtain that the associated $\infty$-category has finite limits, colimits satisfying descent, and is locally Cartesian closed when $\mathcal E$ is, but is not a higher topos in general. We also characterise the $\infty$-category presented by the effective model structure, showing that it is the full sub-category of presheaves on $\mathcal E$ spanned by Kan complexes in $\mathcal E$, a result that suggests a close analogy with the theory of exact completions

The category of equilogical spaces and the effective topos as homotopical quotients

We show that the two models of extensional type theory, those given by the category of equilogical spaces and by the effective topos, are homotopical quotients of categories of 2-groupoids