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
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
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 the 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.
30-Jan-2002 09:06:16 -0400,3084;000000000000-00000000