Related papers: Six model categories for directed homotopy
One of the primary methods of studying the topology of configurations of points in a graph and configurations of disks in a planar region has been to examine discrete combinatorial models arising from the underlying spaces. Despite the…
We identify Grandis' directed spaces as a full reflective subcategory of the category of multipointed $d$-spaces. When the multipointed $d$-space realizes a precubical set, its reflection coincides with the standard realization of the…
We construct a real combinatorial model for the configuration spaces of points of compact smooth oriented manifolds without boundary. We use these models to show that the real homotopy type of configuration spaces of a simply connected such…
Diagrammatic sets admit a notion of internal equivalence in the sense of coinductive weak invertibility, with similar properties to its analogue in strict $\omega$-categories. We construct a model structure whose fibrant objects are…
In this article, we develop a notion of Quillen bifibration which combines the two notions of Grothendieck bifibration and of Quillen model structure. In particular, given a bifibration $p:\mathcal E\to\mathcal B$, we describe when a family…
We construct a zig-zag of Quillen adjunctions between the homotopy theories of differential graded and simplicial categories. In an intermediate step we generalize Shipley-Schwede's work on connective DG algebras by extending the Dold-Kan…
We compare the structure of a mapping cone in the category Top^D of spaces under a space D with differentials in algebraic models like crossed complexes and quadratic complexes. Several subcategories of Top^D are identified with algebraic…
In this note, we construct a closed model structure on the category of $\mathbb{Z}/2\mathbb{Z}$-graded complexes of projective systems of ind-Banach spaces. When the base field is the fraction field $F$ of a complete discrete valuation ring…
We study Quillen's model category structure for homotopy of simplicial objects in the context of Janelidze, Marki and Tholen's semi-abelian categories. This model structure exists as soon as the base category A is regular Mal'tsev and has…
Our subject is that of categories, functors and distributors enriched in a base quantaloid Q. We show how cocomplete Q-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored,…
We define model structures on exact categories which we call exact model structures. We look at the relationship between these model structures and cotorsion pairs on the exact category. In particular, when the underlying category is weakly…
The purpose of this paper is to develop a homotopical algebra for graphs, relevant to zeta series and spectra of finite graphs. More precisely, we define a Quillen model structure in a category of graphs (directed and possibly infinite,…
The aim of this paper is to construct exact model structures from so called extendable cotorsion pairs. Given a hereditary Hovey triple $(\mathcal{C}, \mathcal{W}, \mathcal{F})$ in a weakly idempotent complete exact category with enough…
In this paper we study a 2-dimensional version of Quillen's homotopy category construction. Given a category $\mathscr{A}$ and a class of morphisms $\Sigma \subset \mathscr{A}$ containing the identities, we construct a 2-category…
In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections on principal bundles, and complex…
In this paper, we try to realize the unbounded derived category of an abelian category as the homotopy category of a Quillen model structure on the category of unbounded chain complexes. We construct such a model structure based on…
Building on work by Fiore-Pronk-Paoli, we construct four model structures on the category of double categories, each modeling one of the following: simplicial spaces, Segal spaces, $(\infty,1)$-categories, and $\infty$-groupoids.…
The definition of the homotopy limit of a diagram of left Quillen functors of model categories has been useful in a number of applications. In this paper we review its definition and summarize some of these applications. We conclude with a…
The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on "higher props," we show that the category of all small colored…
We give a homotopy classification of the global defects in ordered media, and explain it via the example of biaxial nematic liquid crystals, i.e., systems where the order parameter space is the quotient of the $3$-sphere $S^3$ by the…