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. 30-Jan-2002 08:58:25 -0400,2955;000000000001-00000000

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

David Carlton writes about colimits of categories. Other people have answered many of his questions, but not yet:

Also, while I'm asking, does the decategorification operation (from nCat into (n-1)Cat) commute with colimits? I was somewhat surprised to see that decategorification from 1Cat into 0Cat does commute with filtered colimits; so I'm wondering to what extent that statement generalizes. I.e. can I replace 1Cat by nCat, can I remove the word 'filtered', and for that matter can I replace colimits by limits? (I know decategorification doesn't commute with arbitrary limits of 1Cats - indeed, that's arguably where much of the fun of higher category theory comes into play - though I haven't thought too much about whether or not it commutes with filtered limits.)

If by n-Cat you mean strict n-categories and strict n-functors, then here is a response to the ``colimit'' part of the question. First of all, you can't generalize from filtered colimits to arbitrary colimits, even in the case 1Cat-->0Cat. Write Cat for 1Cat and Set for 0Cat, and P:Cat-->Set for the decategorification functor, which sends a category to its set of isomorphism classes of objects. This functor P preserves filtered colimits, as you said (or see below), and clearly also preserves coproducts, but doesn't preserve the coequalizer f q A ----> B ----> C ----> g where C is the free-living isomorphism, with objects 0 and 1, and two mutually inverse non-identity arrows 0-->1 and 1-->0; where B has objects 0 and 1, and arrows freely generated by 0-->1 and 1-->0, and where q is the evident quotient; where A has two objects 0 and 1 and arrows freely generated by 0-->0 and 1-->1, and where f and g are the evident functors. Alternatively, observe that Cat is locally finitely presentable (lfp), so that if it preserved all colimits it would have a right adjoint, and prove that it does not have one. On the other hand, decategorification P:n-Cat-->(n-1)-Cat does preserve filtered colimits for any n. To see this, write n-Catg for the full subcategory of n-Cat consisting of those n-categories in which all n-cells are invertible. The inclusion I:n-Catg --> n-Cat has a right adjoint R which forgets all non-invertible n-cells. (It also has a left adjoint.) Now RI=1 and PIR=P, so that P will preserve filtered colimits if PI and R do so. But PI:n-Catg ---> (n-1)-Cat has a right adjoint D, which regards an (n-1)-category as an n-category with no non-identity n-cells. Thus PI preserves all colimits, and P will preserve filtered colimits provided R does so. If V is locally finitely presentable, then so is V-Cat [G.M.Kelly and Stephen Lack, V-Cat is locally presentable or locally bounded if V is so, Theory Appl. Cat. 8:555-575, 2001]. From the equation (n+1)-Cat=(n-Cat)-Cat and the fact that 0-Cat(=Set) is lfp, it follows by induction that n-Cat is lfp for every n. Similarly from the equation (n+1)-Catg=(n-Catg)-Cat and that fact that 1-Catg(= the category Gpd of groupoids) is lfp, it follows that n-Catg is lfp for every n. Now R:n-Cat-->n-Catg is a right adjoint functor between lfp categories, so will preserve filtered colimits if and only if its left adjoint I:n-Catg-->n-Cat preserves finitely presentable objects. For an object G of V, write 2_G for the V-category with objects 0 and 1, and homs 2_G(0,0)=2_G(1,1)=I, 2_G(1,0)=0, and 2_G(0,1)=G. By the Kelly-Lack paper, the finitely presentable objects of V-Cat are the closure under finite colimits of the V-categories of the form 2_G for G a finitely presentable object of V. It follows that I:(n+1)-Catg --> (n+1)-Cat will preserve finitely presentable objects if I:n-Catg ---> n-Cat does so. Thus it remains only to show that I:Gpd-->Cat preserves finitely presentable objects, or equivalently that R:Cat-->Gpd preserves filtered colimits. There are various ways to do this. One could use the description of filtered colimits in Cat given in the Kelly-Lack paper to show that IR preserves filtered colimits, and deduce that R does so. Alternatively one could show that the ``free-living isomorphism'' (called C above) is finitely presentable in both Gpd and Cat, and constitutes a strong generator of Gpd, and deduce that I preserves finitely presentable objects. Similarly, P:n-Cat-->(n-1)-Cat will preserve whatever limits PI preserves, and once again an inductive argument shows that PI will preserve whatever limits PI:Gpd-->Set preserves (products, for instance). Steve Lack. 30-Jan-2002 09:02:37 -0400,2558;000000000001-00000000

on 31/1/02 6:27 AM, david carlton at carlton@math.stanford.edu wrote:

For what it's worth, I can show that Decat can't be a left adjoint (in the relevant sense). If it were, its right adjoint would be a functor F from Set (a 1-category trivially extended to a 2-category) to 1Cat such that, for all categories C and sets S, the category Hom_1Cat(C, FS) is equivalent to the set Hom_Set(Decat(C),S) (thought of as a discrete category).

Sorry, when I wrote about Decat as a left adjoint I thought about n-groupoids rather than n-categories. Steve Lack has already clarified the situation. But now I understand you are asking about weak (or pseudo) colimits. They exist in 1-Cat and 2-Cat and can be expressed in terms of appropriate weighted colimits. I never saw a paper about pseudocolimits in 3-Cat (here we can use Gray-categories instead of general tricatgories). They must be expressed as weighted colimits as well, or a codescent object of a simplicial Gray-category. I'd like to have a reference if such a paper already exists. Michael Batanin.