Subject: categories: Cocompleteness of infinity categories and groupoids

From: Ronnie Brown <[email-masked]>
Date: Thu, 22 Oct 1998 12:11:50 +0100 (BST)

The existence of colimits can be proved by adjoint functor type 
arguments: Lew Hardy and I wrote out details for the not too 
dissimilar situation of topological groupoids in 

``Topological groupoids I: universal constructions'', {\em Math. Nachr.} 
71 (1976) 273-286. 

Identification of vertices of a groupoid (or category) gives what Philip 
Higgins called a universal groupoid or category, and of course more 
compositions are allowed. Philip constructed this explicitly, but it also 
follows from the general construction. There is a mention of the 
cocompleteness of the multiple situation in 
 (with P.J. HIGGINS), ``On the algebra of cubes'', {\em J. Pure Appl. 
Algebra} 21 (1981) 233-260. (see p. 238). 

Analysis and computation of colimits of various forms of multiple 
groupoids is necessary for applying Generalised Van Kampen Theorems - 
for the groupoid case this is most conveniently done in the `small' 
model of crossed complexes. 

For cat^n-groups (= n-fold categories in 
groups) it is probably most convenient to work in Ellis-Steiner's 
crossed n-cubes of groups (generalising Guin-Walery/Loday's crossed 
squares). Analysing pushouts of crossed squares led Loday and me to the 
non-abelian tensor product of groups (which act on each other).  

The identification of p-cells in an n-category (n>p) to give a new
n-category is discussed in relation to homotopy theory in my survey
``Homotopy theory, and change of base for groupoids and multiple
groupoids'', {\em Applied categorical structures}, 4 (1996) 175-193.    
That is, it shows how complicated and interesting are even simple cases of 
this general idea. We show how the n-adic Hurewicz theorem can be seen as 
an example of this in 
 (with J.-L. LODAY), ``Homotopical excision, and Hurewicz theorems, for 
$n$-cubes of spaces'', {\em Proc. London Math. Soc.} (3) 54 (1987) 176-192. 
(This was the first proof of even a triadic Hurewicz theorem. The 
relative case goes back to early homotopy theory.) 

The general idea is of universally constructing an n-fold category 
from lower dimensional information, or lower dimensional 
identifications.                               

The computational aspect (how to compute the answers) seems really 
interesting. One expects to be able to be more explicit in the groupoid 
case. On the other hand, even general double groupoids are a bit 
mysterious.  

Ronnie Brown