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

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

Computability of the (co)homology of cyclic monoids

Leech's (co)homology groups of finite cyclic monoids are computed.Comment: 14 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 $\mathcal{A}\to\mathcal{B}\leftarrow \mathcal{A}'$. 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 $n^{\text{th}}$ homotopy group at any base point vanishes for $n\geq 3$). 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