1,509 research outputs found
Homotopy Fibre Sequences Induced by 2-Functors
This paper contains some contributions to the study of the relationship
between 2-categories and the homotopy types of their classifying spaces.
Mainly, generalizations are given of both Quillen's Theorem B and Thomason's
Homotopy Colimit Theorem to 2-functors.Comment: The paper has been improved following referee's sugestions, to which
the author would like to express his gratitude
Diagonal fibrations are pointwise fibrations
On the category of bisimplicial sets there are different Quillen closed model
structures associated to various definitions of fibrations. In one of them,
which is due to Bousfield and Kan and that consists of seeing a bisimplicial
set as a simplicial object in the category of simplicial sets, fibrations are
those bisimplicial set maps such that each of the induced simplicial set maps
is a Kan fibration, that is, the pointwise fibrations. In another of them,
introduced by Moerdijk, a bisimplicial map is a fibration if it induces a Kan
fibration of associated diagonal simplicial sets, that is, the diagonal
fibrations. In this note, we prove that every diagonal fibration is a pointwise
fibration.Comment: to be published in "Journal of Homotopy and Related Structures
Computability of the (co)homology of cyclic monoids
Leech's (co)homology groups of finite cyclic monoids are computed.Comment: 14 page
Higher cohomologies of commutative monoids
Extending Eilenberg-Mac Lane's methods, higher level cohomologies for
commutative monoids are introduced and studied. Relationships with pre-existing
theories (Leech, Grillet, ...) are stated. The paper includes a cohomological
classification for symmetric monoidal groupoids and several computations for
cyclic monoids.Comment: 37 page
Homotopy colimits of 2-functors
Like categories, small 2-categories have well-understood classifying spaces.
In this paper, we deal with homotopy types represented by 2-diagrams of
2-categories. Our results extend to homotopy colimits of 2-functors lower
categorical analogues that have been classically used in algebraic topology and
algebraic K-theory, such as the Homotopy Invariance Theorem (by Bousfield and
Kan), the Homotopy Colimit Theorem (Thomason), Theorems A and B (Quillen), or
the Homotopy Cofinality Theorem (Hirschhorn).Comment: 32 page
Geometric Realizations of Tricategories
Any tricategory characteristically has associated various simplicial or
pseudo-simplicial objects. This paper explores the relationship amongst three
of them: the pseudo-simplicial bicategory so-called Grothendieck nerve of the
tricategory, the simplicial bicategory termed its Segal nerve, and the
simplicial set called its Street geometric nerve, and it proves the fact that
the geometric realizations of all of these possible candidate 'nerves of the
tricategory' are homotopy equivalent. Our results provide coherence for all
reasonable extensions to tricategories of Quillen's definition of the
'classifying space' of a category as the geometric realization of the
category's Grothendieck nerve. Many properties of the classifying space
construction for tricategories may be easier to establish depending on the
nerve used for realizations. For instance, by using Grothendieck nerves we
state and prove the precise form in which the process of taking classifying
spaces transports tricategorical coherence to homotopy coherence. Segal nerves
allow us to obtain an extension to bicategories of the results by Mac Lane,
Stasheff, and Fiedorowicz about the relation between loop spaces and monoidal
or braided monoidal categories by showing that the group completion of the
classifying space of a bicategory enriched with a monoidal structure is, in a
precise way, a loop space. With the use of geometric nerves, we obtain genuine
simplicial sets whose simplices have a pleasing geometrical description in
terms of the cells of the tricategory and, furthermore, we obtain an extension
of the results by Joyal, Street, and Tierney about the classification of
braided categorical groups and their relationship with connected, simply
connected homotopy 3-types, by showing that, via the classifying space
construction, bicategorical groups are a convenient algebraic model for
connected homotopy 3-types.Comment: 48 pages; modified thank
Bicategorical homotopy fiber sequences
Small B\'{e}nabou's bicategories and, in particular, Mac Lane's monoidal
categories, have well-understood classifying spaces, which give geometric
meaning to their cells. This paper contains some contributions to the study of
the relationship between bicategories and the homotopy types of their
classifying spaces. Mainly, generalizations are given of Quillen's Theorems A
and B to lax functors between bicategories.Comment: 40 pages, corrected some typos, final versio
Bicategorical homotopy pullbacks
The homotopy theory of higher categorical structures has become a relevant
part of the machinery of algebraic topology and algebraic K-theory, and this
paper contains contributions to the study of the relationship between
B\'enabou's bicategories and the homotopy types of their classifying spaces.
Mainly, we state and prove an extension of Quillen's Theorem B by showing,
under reasonable necessary conditions, a bicategory-theoretical interpretation
of the homotopy-fibre product of the continuous maps induced on classifying
spaces by a diagram of bicategories . Applications are given for the study of homotopy pullbacks of
monoidal categories and of crossed modules.Comment: 50 page
Structure and classification of monoidal groupoids
The structure of monoidal categories in which every arrow is invertible is
analyzed in this paper, where we develop a 3-dimensional Schreier-Grothendieck
theory of non-abelian factor sets for their classification. In particular, we
state and prove precise classification theorems for those monoidal groupoids
whose isotropy groups are all abelian, as well as for their homomorphisms, by
means of Leech's cohomology groups of monoids.Comment: 38 pages; corrected typos and made modifications suggested by the
refere
Double groupoids and homotopy 2-types
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 here we deal with homotopy types represented by double
groupoids satisfying a natural `filling condition'. Any such double groupoid
characteristically has associated to it `homotopy groups', which are defined
using only its algebraic structure. Thus arises the notion of `weak
equivalence' between such double groupoids, and a corresponding `homotopy
category' is defined. Our main result in the paper states that the geometric
realization functor induces an equivalence between the homotopy category of
double groupoids with filling condition and the category of homotopy 2-types
(that is, the homotopy category of all topological spaces with the property
that the homotopy group at any base point vanishes for ). A quasi-inverse functor is explicitly given by means of a new `homotopy
double groupoid' construction for topological spaces.Comment: keywords: Double groupoid, classifying space, bisimplicial set, Kan
complex, geometric realization, homotopy typ
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