Related papers: A semi-model structure for Grothendieck weak 3-gro…

The goal of this paper is to address the problem of building a path object for the category of Grothendieck (weak) $\infty$-groupoids. This is the missing piece for a proof of Grothendieck's homotopy hypothesis. We show how to endow the…

We introduce and study a notion of cylinder coherator similar to the notion of Grothendieck coherator which define more flexible notion of weak infinity groupoids. We show that each such cylinder coherator produces a combinatorial…

We show that if the canonical left semi-model structure on the category of Grothendieck $n$-groupoids exists, then it satisfies the homotopy hypothesis, i.e. the associated $(\infty,1)$-category is equivalent to that of homotopy $n$-types,…

We present a slight variation on a notion of weak \infty-groupoid introduced by Grothendieck in Pursuing Stacks and we study the homotopy theory of these \infty-groupoids. We prove that the obvious definition for homotopy groups of…

A short introduction to Grothendieck weak omega-groupoids is given. Our aim is to give evidence that, in certain contexts, this simple language is a convenient one for constructing globular weak omega-groupoids. To this end, we give a short…

We define the Grothendieck group of an n-angulated category and show that for odd n its properties are as in the special case of n=3, i.e. the triangulated case. In particular, its subgroups classify the dense and complete n-angulated…

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy types of their classifying spaces. Double categories (Ehresmann, 1963) have well-understood geometric realizations, and…

We show that there exists a canonical functor from the category of strict \infty-groupoids to the category of Grothendieck \infty-groupoids and that this functor is fully faithful. As a main ingredient, we prove that free strict…

We define a linear structure on Grothendieck's arithmetic fundamental group $\pi_1(X, x)$ of a scheme $X$ defined over a field $k$ of characteristic 0. It allows us to link the existence of sections of the Galois group ${\rm Gal}(\bar k/k)$…

We construct a "diagonal" cofibrantly generated model structre on the category of simplicial objects in the category of topological categories sCat_{Top}, which is the category of diagrams [\Delta^{op}, Cat_{Top}]. Moreover, we prove that…

We show that fundamental groups of compact, orientable, irreducible 3-manifolds with toroidal boundary are Grothendieck rigid.

We introduce a new higher categorical structure called a weakly globular n-fold category. This structure is based on iterated internal categories and on the notion of weak globularity. We identify a suitable class of pseudo-functors whose…

We introduce a new quasi-isometry invariant for finitely generated groups and show that every group with this property admits a subshift which is effectively closed by patterns and that cannot be realized as the topological factor of any…

In this paper we treat Grothendieck Duality for noetherian rings via rigid dualizing complexes. In particular, we prove that every ring, essentially finite type over a regular base ring, has a unique rigid dualizing complex. The rigid…

Grothendieck Duality -- the theory of the twisted inverse image pseudofunctor (-)^! over a suitable category of scheme-maps -- can be developed concretely, with emphasis on explicit constructions, or abstractly, with emphasis on…

This paper studies the homotopy theory of the Grothendieck construction using model categories and semi-model categories, provides a unifying framework for the homotopy theory of operads and their algebras and modules, and uses this…

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…

Let $\Lambda$ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of mod$\Lambda$. We show that $\mathcal{M}$ has an additive generator if and only if the n-almost split sequences form a basis for the relations for the…

In this work we shall introduce a new model structure on the category of pro-simplicial sheaves, which is very convenient for the study of \'etale homotopy. Using this model structure we define a pro-space associated to a topos, as a result…

The Grothendieck construction is a fundamental link between indexed categories and opfibrations. This work is a detailed study of the Grothendieck construction over a small tight bipermutative category in the context of Cat-enriched…