Related papers: Codescent theory II: Cofibrant approximations

We show that the quasicategory of frames of a cofibration category, introduced by the second-named author, is equivalent to its simplicial localization.

We study the accessibility properties of trivial cofibrations and weak equivalences in a combinatorial model category and prove an estimate for the accessibility rank of weak equivalences. In particular, we show that the class of weak…

The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive $2$-representations with a fixed apex $\mathcal{J}$ of a fiat $2$-category $\cC$ and the set of equivalence classes of…

We further develop Weber's notion of elementary 2-topos by proposing certain new axioms. We show that in a 2-category C satisfying these axioms, the "discrete opfibration (DOF) classifier" S is always an internal elementary 1-topos, in an…

We produce a cofibrantly generated simplicial symmetric monoidal model structure for the category of (small unital) C*-categories, whose weak equivalences are the unitary equivalences. The closed monoidal structure consists of the maximal…

An n-truncated model structure on simplicial (pre-)sheaves is described having as weak equivalences maps that induce isomorphisms on certain homotopy sheaves only up to degree n. Starting from one of Jardine's intermediate model structures…

Let $\mathcal C$ be a $\mathcal V$-enriched model category. We say that an object $x$ of $\mathcal C$ is homotopy tiny if the total right derived functor of $\mathcal C(x, -) : \mathcal{C} \rightarrow {\mathcal V}$ preserves homotopy…

Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previously known results. We describe the close…

We build model structures on the category of equivariant simplicial operads with a fixed set of colors, with weak equivalences determined by families of subgroups. In particular, by specifying to the family of graph subgroups (or, more…

One goal of applied category theory is to better understand networks appearing throughout science and engineering. Here we introduce "structured cospans" as a way to study networks with inputs and outputs. Given a functor $L \colon…

We construct a functor from the category of graphs to the category of groups which is faithful and "almost" full, in the sense that it induces bijections of the Hom sets up to trivial homomorphisms and conjugation in the category of groups.…

Diffeological spaces are generalizations of smooth manifolds. In this paper, we study the homotopy theory of diffeological spaces. We begin by proving basic properties of the smooth homotopy groups that we will need later. Then we introduce…

In a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and L^2-Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility of those notions with homotopy colimits of…

Given subsets $\mathcal{C},\mathcal{F}$ of a preorder $\mathcal{A}$, we give necessary and sufficient conditions for $\mathcal{A}$ to admit the structure of a model category whose cofibrant objects are $\mathcal{C}$ and whose fibrant…

We consider two categories related to symplectic manifolds: 1. Objects are symplectic manifolds and morphisms are symplectic embeddings. 2. Objects are symplectic manifolds endowed with compatible almost complex structure and morphisms are…

Coherent strings of composable morphisms play an important role in various important constructions in abstract stable homotopy theory (for example algebraic K-theory or higher Toda brackets) and in the representation theory of finite…

For a recollement $(\mathcal{D}B,\mathcal{D}A,\mathcal{D}C)$ of derived categories of algebras, we investigate when the functor $j^*:\mathcal{D}A\rightarrow\mathcal{D}C$ is an eventually homological isomorphism. In this context, we compare…

We construct an abelian category C and exact functors in C which on the Grothendieck group descend to the action of a simply-laced quantum group in its adjoint representation. The braid group action in the adjoint representation lifts to an…

Coalgebras for an endofunctor provide a category-theoretic framework for modeling a wide range of state-based systems of various types. We provide an iterative construction of the reachable part of a given pointed coalgebra that is inspired…

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…