相关论文: On the straightening of every functor

In this expository article, we will give an efficient functorial proof of the equivalence of various characterisations of purity in a finitely accessible additive category $\mathcal C$. The complications of the proofs for specific choices…

We demonstrate that any full and faithful $*$-functor between approximable categories of locally finite coarse spaces induces a coarse embedding between the underlying spaces. Furthermore, we establish a general characterisation of such…

We distinguish between faint, weak, strong and strict localizations of categories at morphism families and show that this framework captures the different types of derived functors that are considered in the literature. More precisely, we…

In this paper, we state and prove precise theorems on the classification of the category of (braided) categorical groups and their (braided) monoidal functors, and some applications obtained from the basic studies on monoidal functors…

We consider two categories of C*-algebras; in the first, the isomorphisms are ordinary isomorphisms, and in the second, the isomorphisms are Morita equivalences. We show how these two categories, and categories of dynamical systems based on…

For a small category A, we prove that the homotopy colimit functor from the category of simplicial diagrams on A to the category of simplicial sets over the nerve of A establishes a left Quillen equivalence between the projective (or Reedy)…

We prove a generalization of the Arone-Ching chain rule for Goodwillie derivatives by showing that for any pair of reduced finitary functors $F \colon \mathcal{D} \to \mathcal{E}$ and $G \colon \mathcal{C} \to \mathcal{D}$ between…

Riehl and Verity have introduced an "$\infty$-cosmic" framework in which they redevelop the category theory of $\infty$-categories using 2-categorical arguments. In this paper, we begin with a self-contained review of the parts of their…

We prove that any spectrum is equivalent to the nonconnective K-theory of a stable $\infty$-category. We use these results to construct a stable $\infty$-category $\mathcal{C}$ with a bounded t-structure such that…

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 construct various multiple categories, based on generalised Ehresmann quintets. The main construction is a multiple category whose objects are all the `lax' multiple categories; the transversal arrows are their strict multiple functors…

We verify a confluence result for the rewriting calculus of the linear category introduced in our previous paper. Together with the termination result proved therein, the generalized coherence theorem for linear category is established.…

We prove an ambidexterity result for $\infty$-categories of $\infty$-categories admitting a collection of colimits. This unifies and extends two known phenomena: the identification of limits and colimits of presentable $\infty$-categories…

We prove a result of equivalence invariance of formal category theory for statements that can be expressed within an equipment. To do this, we exploit Henry and Bardomiano Mart\'inez's link between Makkai's FOLDS (first order logic with…

We prove a number of results of the following common flavor: for a category $\mathcal{C}$ of topological or uniform spaces with all manner of other properties of common interest (separation / completeness / compactness axioms), a group (or…

From every pair of adjoint functors it is possible to produce a (possibly trivial) equivalence of categories by restricting to the subcategories where the unit and counit are isomorphisms. If we do this for the adjunction between effect…

This note is a contribution written for the second volume of the Encyclopedia of mathematical physics. We give an informal introduction to the notions of an $(\infty,n)$-category and $(\infty,n)$-functor, discussing some of the different…

We prove that a lax $\mathbb{E}_{n+1}$-monoidal functor from $\mathcal V$ to $\mathcal W$ induces a lax $\mathbb{E}_n$-monoidal functor from $\mathcal V$-enriched $\infty$-categories to $\mathcal W$-enriched $\infty$-categories in the sense…

Let $R$ be a commutative ring If $\mathcal{C}_1$ and $\mathcal{C}_2$ are $R$-linear triangulated categories then we can give an obvious triangulated structure on $\mathcal{C} = \mathcal{C}_1 \oplus \mathcal{C}_2$ where $Hom_\mathcal{C}(U,…

We examine various categorical structures that can and cannot be constructed. We show that total computable functions can be mimicked by constructible functors. More generally, whatever can be done by a Turing machine can be constructed by…