Related papers: Free colimit completion in $\infty$-categories

We develop a number of basic concepts in the theory of categories internal to an $\infty$-topos. We discuss adjunctions, limits and colimits as well as Kan extensions for internal categories, and we use these results to prove the universal…

We construct a compact closed category out of any symmetric monoidal category by freely adding adjoints to its objects. The morphisms of the completion are defined as string diagrams annotated by objects and morphisms from the original…

We construct free monoids in a monoidal category with finite limits and countable colimits, in which tensoring on either side preserves reflexive coequalizers and colimits of countable chains.

We demonstrate equivalence between two definitions of lower finite highest weight categories. We also show that, in the presence of a duality, a lower finite highest weight structure on a category is unique. Finally, we give a new proof for…

We study the relationship between presheaf constructions and free cocompletions in the context of formal category theory, elucidating the coincidence between the two concepts in familiar settings. We show that, in a virtual equipment…

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…

We show that the notion of $(\infty,n)$-limit defined using the enriched approach and the one defined using the internal approach coincide. We also give explicit constructions of various double $(\infty,n-1)$-categories implementing various…

We show that the free construction from multicategories to permutative categories is a categorically-enriched non-symmetric multifunctor. Our main result then shows that the induced functor between categories of algebras is an equivalence…

We show that a profinite completion functor for (simplicial or topological) operads with good homotopical properties can be constructed as a left Quillen functor from an appropriate model category of infinity-operads to a certain model…

We apply some tools developed in categorical logic to give an abstract description of constructions used to formalize constructive mathematics in foundations based on intensional type theory. The key concept we employ is that of a Lawvere…

We extend the notion of exact completion on a weakly lex category to elementary doctrines. We show how any such doctrine admits an elementary quotient completion, which freely adds effective quotients and extensional equality. We note that…

Every category $\mathcal K$ has a free completion $\mathcal P \mathcal K$ under colimits and a free completion $\Sigma\mathcal K$ under coproducts. A number of properties of $\mathcal K$ transfer to $\mathcal P \mathcal K$ and…

We give a description of the Tripos To Topos construction in terms of four free constructions. We prove that these compose up to give a free construction from the category of triposes and logical morphisms to the category of toposes and…

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)…

In the first part of this paper we study fibrations of $(\infty,2)$-categories. We give a simple characterization of such fibrations in terms of a certain square being a pullback, and apply this to show that in some cases…

For a differential graded k-quiver Q we define the free A-infinity-category FQ generated by Q. The main result is that for an arbitrary A-infinity-category A the restriction A-infinity-functor A_\infty(FQ,A) -> A_1(Q,A) is an equivalence,…

We use a category-theoretic formulation of Aczel's Fullness Axiom from Constructive Set Theory to derive the local cartesian closure of an exact completion. As an application, we prove that such a formulation is valid in the homotopy…

We construct a left semi-model structure on the category of intensional type theories (precisely, on $\mathrm{CxlCat_{Id,1,\Sigma(,\Pi_{ext})}}$). This presents an $\infty$-category of such type theories; we show moreover that there is an…

We give an elementary construction of the exact completion of a weakly lex category for categories enriched in the cartesian closed category $\mathsf{Pos}$ of partially ordered sets. Paralleling the ordinary case, we characterize categories…

In this expository paper we explain in detail how to construct bicategorical colimits of several kinds of tensor categories, for example essentially small finitely cocomplete K-linear tensor categories. The constructions are direct and…