Date: Wed, 26 Nov 1997 13:42:59 -0400 (AST) Subject: Cocompletions of Categories Date: Wed, 26 Nov 97 10:19:37 +0100 From: Jiri Velebil Dear Colleagues, I wonder whether the following description of a free F-conservative completion of any category under C-colimits (where F and C are classes of small categories such that F is a subclass of C and "F-conservative" means "preserving existing F-colimits"). The description of the cocompletion is as follows: Suppose C is a class of small categories and F is a subclass of C. Let X be any category. Denote by [X^op,Set] the quasicategory of all functors and all natural transformations between them. Denote by F^op-[X^op,Set] the quasicategory of all functors which preserve F^op-limits, i.e. limits of functors d : D -> X^op with D^op in F. Claim 1. F^op-[X^op,Set] is reflective in [X^op,Set] (The proof uses the fact that the above Claim holds for the case when X is small - Korollar 8.14 in Gabriel, Ulmer: Lokal pr"asentierbare Kategorien.) By Claim 1., F^op-[X^op,Set] has all small colimits. Denote by D(X) the closure of X (embedded by Yoneda) in F^op-[X^op,Set] under C-colimits. Then one can prove that D(X) is a legitimate category. The codomain-restriction I: X -> D(X) of the Yoneda embedding fulfills the following: 1. D(X) has C-colimits. 2. I preserves F-colimits. 3. D(X) has the following universal property: for any functor H : X -> Y which preserves F-colimits and the category Y has C-colimits there is a unique (up to an isomorphism) functor H* : D(X) -> Y such that H* preserves C-colimits and H*.I = H. In fact, this gives a 2-adjunction between C-CAT_C : the 2-quasicategory of all categories having C-colimits, all functors preserving C-colimits and all natural transformations and CAT_F : the 2-quasicategory of all categories, all functors preserving F-colimits and all natural transformations. The result also holds for V-categories, instead of a class C of small categories one has to work with a class of small indexing types. Thank you, Jiri Velebil velebil@math.feld.cvut.cz Department of Mathematics FEL CVUT Technicka 2 Praha 6 Czech Republic Date: Thu, 22 Oct 1998 12:11:50 +0100 (BST) From: Ronnie Brown Subject: categories: Cocompleteness of infinity categories and groupoids 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