15,621 research outputs found
2-nerves for bicategories
We describe a Cat-valued nerve of bicategories, which associates to every
bicategory a simplicial object in Cat, called the 2-nerve. We define a
2-category NHom whose objects are bicategories and whose 1-cells are normal
homomorphisms of bicategories, in such a way that the 2-nerve construction
becomes a full embedding of NHom in the 2-category of simplicial objects in
Cat. This embedding has a left biadjoint, and we characterize its image. The
2-nerve of a bicategory is always a weak 2-category in the sense of Tamsamani,
and we show that NHom is biequivalent to a certain 2-category whose objects are
Tamsamani weak 2-categories.Comment: 23 page
Monads with arities and their associated theories
After a review of the concept of "monad with arities" we show that the
category of algebras for such a monad has a canonical dense generator. This is
used to extend the correspondence between finitary monads on sets and Lawvere's
algebraic theories to a general correspondence between monads and theories for
a given category with arities. As application we determine arities for the free
groupoid monad on involutive graphs and recover the symmetric simplicial nerve
characterisation of groupoids.Comment: New introduction; Section 1 shortened and redispatched with Section
2; Subsections on symmetric operads (3.14) and symmetric simplicial sets
(4.17) added; Bibliography complete
Rigidification of higher categorical structures
Given a limit sketch in which the cones have a finite connected base, we show
that a model structure of "up to homotopy" models for this limit sketch in a
suitable model category can be transferred to a Quillen equivalent model
structure on the category of strict models. As a corollary of our general
result, we obtain a rigidification theorem which asserts in particular that any
-space in the sense of Rezk is levelwise equivalent to one that
satisfies the Segal conditions on the nose. There are similar results for
dendroidal spaces and -fold Segal spaces.Comment: 30 pages, new introductio
A Quillen model structure for Gray-categories
A Quillen model structure on the category Gray-Cat of Gray-categories is
described, for which the weak equivalences are the triequivalences. It is shown
to restrict to the full subcategory Gray-Gpd of Gray-groupoids. This is used to
provide a functorial and model-theoretic proof of the unpublished theorem of
Joyal and Tierney that Gray-groupoids model homotopy 3-types. The model
structure on Gray-Cat is conjectured to be Quillen equivalent to a model
structure on the category Tricat of tricategories and strict homomorphisms of
tricategories.Comment: v2: fuller discussion of relationship with work of Berger;
localizations are done directly with simplicial set
The homotopy theory of dg-categories and derived Morita theory
The main purpose of this work is the study of the homotopy theory of
dg-categories up to quasi-equivalences. Our main result provides a natural
description of the mapping spaces between two dg-categories and in
terms of the nerve of a certain category of -bimodules. We also prove
that the homotopy category is cartesian closed (i.e. possesses
internal Hom's relative to the tensor product). We use these two results in
order to prove a derived version of Morita theory, describing the morphisms
between dg-categories of modules over two dg-categories and as the
dg-category of -bi-modules. Finally, we give three applications of our
results. The first one expresses Hochschild cohomology as endomorphisms of the
identity functor, as well as higher homotopy groups of the \emph{classifying
space of dg-categories} (i.e. the nerve of the category of dg-categories and
quasi-equivalences between them). The second application is the existence of a
good theory of localization for dg-categories, defined in terms of a natural
universal property. Our last application states that the dg-category of
(continuous) morphisms between the dg-categories of quasi-coherent (resp.
perfect) complexes on two schemes (resp. smooth and proper schemes) is
quasi-equivalent to the dg-category of quasi-coherent complexes (resp. perfect)
on their product.Comment: 50 pages. Few mistakes corrected, and some references added. Thm.
8.15 is new. Minor corrections. Final version, to appear in Inventione
Pro-categories in homotopy theory
The goal of this paper is to prove an equivalence between the model
categorical approach to pro-categories, as studied by Isaksen, Schlank and the
first author, and the -categorical approach, as developed by Lurie.
Three applications of our main result are described. In the first application
we use (a dual version of) our main result to give sufficient conditions on an
-combinatorial model category, which insure that its underlying
-category is -presentable. In the second application we
consider the pro-category of simplicial \'etale sheaves and use it to show that
the topological realization of any Grothendieck topos coincides with the shape
of the hyper-completion of the associated -topos. In the third
application we show that several model categories arising in profinite homotopy
theory are indeed models for the -category of profinite spaces. As a
byproduct we obtain new Quillen equivalences between these models, and also
obtain an example which settles negatively a question raised by Raptis
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