Related papers: Accessible $\infty$-cosmoi

A topological space $X$ is cometrizable if it admits a weaker metrizable topology such that each point $x\in X$ has a (not necessarily open) neighborhood base consisting of metrically closed sets. We study the relation of cometrizable…

Given an algebraic theory $\ct$, a homotopy $\ct$-algebra is a simplicial set where all equations from $\ct$ hold up to homotopy. All homotopy $\ct$-algebras form a homotopy variety. We give a characterization of homotopy varieties…

We discuss some finite homogeneous structures, addressing the question of universality of their automorphism groups. We also study the existence of so-called Kat\v{e}tov functors in finite categories of embeddings or homomorphisms.

We define the notion of an indexed profunctor over a 2-category, and use it to develop an abstract theory of limits. The theory subsumes (conical) limits, weighted limits, ends and Kan extensions. Results include an abstract version of the…

We show how several useful properties of Ind-constructions in $\infty$-categories extend to arbitrary free colimit completion constructions.

We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal's…

We prove a refinement of Quillen's Theorem A, providing necessary and sufficient conditions for a functor to be cofinal with respect to diagrams valued in a fixed $\infty$-category. We deduce this from a general duality phenomenon for…

We give a sharper version of a theorem of Rosicky, Trnkova and Adamek, and a new proof of a theorem of Rosicky, both about colimit preservation between categories of structures. Unlike the original proofs, which use category-theoretic…

This article provides some basic results on weight structures, weight complex functors and homotopy categories. We prove that the full subcategories K(A)^{w < n}, K(A)^{w > n}, K(A)^- and K(A)^+ (of objects isomorphic to suitably bounded…

We give sufficient conditions for the existence of a Quillen model structure on small categories enriched in a given monoidal model category. This yields a unified treatment for the known model structures on simplicial, topological, dg- and…

It is known that the existence of localization with respect to an arbitrary (possibly proper) class of maps in the category of simplicial sets is implied by a large-cardinal axiom called Vopenka's principle.In this article we extend the…

For a given compact Hausdorff space $X$, we construct the space $OS_{f}(X)$ of normed, order-preserving, weakly additive, positively homogeneous and semi-additive functionals (for brevity, semi-additive functionals) and it is proved that…

In this note we prove the following useful fact that seems to be missing from the literature: the $\infty$-category of coherent ordinary topoi (in the sense of SGA4) is equivalent to the $\infty$-category of coherent $1$-localic…

Let K be a comonad on a model category M. We provide conditions under which the associated category of K-coalgebras admits a model category structure such that the forgetful functor to M creates both cofibrations and weak equivalences. We…

We show that the Cantor-Schr\"oder-Bernstein Theorem for homotopy types, or $\infty$-groupoids holds in the following form: For any two types, if each one is embedded into the other, then they are equivalent. The argument is developed in…

We define $A_{\infty}$-structures -- algebras, coalgebras, modules, and comodules -- in an arbitrary monoidal DG category or bicategory by rewriting their definitions in terms of unbounded twisted complexes. We develop new notions of strong…

Internal categories feature notions of limit and completeness, as originally proposed in the context of the effective topos. This paper sets out the theory of internal completeness in a general context, spelling out the details of the…

We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of…

In this paper, we develop the notion of presentability in the parametrised homotopy theory framework of Barwick-Dotto-Glasman-Nardin-Shah over orbital categories. We formulate and prove a characterisation of parametrised presentable…

We present a way of constructing a Quillen model structure on a full subcategory of an elementary topos, starting with an interval object with connections and a certain dominance. The advantage of this method is that it does not require the…