# Category Theory & Homotopy

### Multiple categories: the equivalence of a globular and a cubical approach

#### Abstract:

We show the equivalence of two kinds of strict multiple category, namely:
• the well known globular $\omega$-categories,
• the cubical $\omega$-categories with connections.

#### ftp access:

xxx archive: math.CT/0007009

#### Published in:

Advances in Math. 170 (2002) 71-118.

### Homotopies and automorphisms of crossed modules of groupoids

#### Abstract:

We give a detailed description of the structure of the actor 2-crossed module related to the automorphisms of a crossed module of groupoids. This generalises work of Brown and Gilbert for the case of crossed modules of groups, and part of this is needed for work on 2-dimensional holonomy to be developed elsewhere.

#### ftp access:

xxx archive: math.CT/0008117

#### Published in:

Applied Categorical Structures 11 (2003) 185-206.

### Towards a 2-dimensional notion of holonomy

#### Abstract:

Previous work (Pradines, C.R.Acad.Sci.Paris 263 (1966) 907; Aof and Brown, Topology Appl. 47 (1992) 97) has given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a groupoid by a local smooth coadmissible homotopy (or free derivation) for the crossed module case. The development also has to use corresponding notions for certain types of double groupoids. This leads to a holonomy Lie groupoid rather than double groupoid, but one which involves the 2-dimensional information.

#### Published in:

Advances in Math. 178 (2003) 141-175.

### Local subgroupoids II: Examples and properties

#### Abstract:

The notion of local subgroupoid as a generalisation of a local equivalence relation was defined in a previous paper by the first two authors. Here we use the notion of star path connectivity for a Lie groupoid to give an important new class of examples, generalising the local equivalence relation of a foliation, and develop in this new context basic properties of coherence, due earlier to Rosenthal in the special case. These results are required for further applications to holonomy and monodromy.

#### ftp access:

xxx archive: math.CT/0008165

#### Published in:

Top. Appl. 127 (2003) 393-408.

### Sheaves and Local Subgroupoids

#### Abstract:

This is an introduction to the notion of local subgroupoid introduced by the author and R. Brown. It can also serve as an introduction to an application of sheaf theory, and so could be useful to beginners in that theory.
The main results are the construction of the holonomy groupoid and the notion of s-sheaf for certain local subgroupoids s.

ftp access: 00_16.ps.gz

email: iicen@inonu.edu.tr

### A 2-groupoid enrichment of chain complexes

#### Abstract:

The detailed structure of the category of chain complexes corresponding to an enrichment over the monoidal category of 2-groupids with the Gray tensor product, is given.