categories: Re: colimits of categories

["Ronald Brown" <ronnie@ll319dg.fsnet.co.uk>]

reply to r.brown@bangor.ac.uk

Another paper to look up is by Philip Higgins in Mathematische  Nachrichten
27 (1963) 115-132 which generalised his earlier work on algebras with a
scheme of operators to partially defined operators.

One way of looking at this is that a morphism f: C \to D of categories (or
groupoids) may identify objects. So one thing to do is factorize it through
through a universal morphism which does just that identification (compare
Steve Vickers' comments). Philip's
book (in fact his 1964 paper on presentations of groups) shows how you
thereby get free groupoids on graphs and free products of groups from one
construction. An advantage of this is you need only one normal form theorem,
whereas the group theory books give two proofs.

There is a paper by M.Zisman looking at the effect of this particular
construction on classifying spaces of categories and groupoids.

This is seen as a `change of base' construction (a favourite notion of
Grothendieck) in my paper

``Homotopy theory, and change of base for groupoids and
multiple groupoids'', {\em Applied categorical structures}, 4
(1996) 175-193.

This also relates change of base  to results in algebraic topology such as
n-adic  excision and  Hurewicz theorems. The n-adic theorems were found via
this route (n=2 is the well known relative case).

In homotopy theory one would like to know what happens to a homotopy type if
you change its lower dimensional part. An algebraic  solution to this would
also give the homotopy types of spheres, since S^n is obtained from a disc
E^n by shrinking the boundary S^{n-1} to a point. So we can't expect a
calculable solution overnight!  In the strict multiple groupoid case some
explicit calculations have been done (using crossed n-cubes of groups,
Ellis-Steiner) , but all this  suggests the interesting difficulty of
calculating with multiple categories. On the other hand, calculating with
groups is not a walk-over either, and the expectation is that some group
theory results are best understood from a higher dimensional viewpoint.

For colimits of topological categories and groupoids see also

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

which gets it from an adjoint functor type construction: this probably
overlaps work of C. Ehresmann.

Ronnie Brown

----- Original Message -----
From: <S.J.Vickers@open.ac.uk>
To: <categories@mta.ca>
Sent: Tuesday, January 29, 2002 2:08 PM
Subject: categories: Re: colimits of categories


> David Carlton asks -
> > Is there a good reference for the construction of colimits of
> > categories?
>
> If I remember correctly, Philip Higgins's little book "Notes on categories
> and groupoids" (Van Nostrand Reinhold 1971) is good on that kind of thing.
>
> You're right that the non-filtered colimits are distinctly messier than
the
> limits. There are two reasons.
>
> The first is that that is the way of algebra anyway - think of colimits of
> monoids or groups, for instance. Universal algebra says that colimits
exist
> for every algebraic theory, but the construction is intricate. You first
> make an algebra of all possible terms (expressions) and then factor out a
> congruence to enforce the equational laws and the cocone commutativities.
>
> The second reason is that categories are models not of an algebraic
theory,
> but of an essentially algebraic theory (some operations - specifically
here
> composition - are only partial, with domain of definition stipulated
> equationally). The techniques of universal algebra still work, by and
large,
> but the proof is even more intricate than the 2-step process in algebra.
> This is because imposing equations can cause new terms to spring into
> existence.
>
> Steve Vickers.
>
>
>
>