This kind of condition occurs in topology as a fibrant square - but where all the maps are fibrations as is the map A ---> B x_D C . This can be generalised to cubes. See a paper by R. Steiner on `Resolutions of spaces by n-cubes of fibrations', J. London Math. Soc.(2), 34, 169-176, 1986 used to build a complete (strict) algebraic model of homotopy n-types which allows some computations. This raises the spectre in algebra of Resolutions of A by free crossed n-cubes of A. to give a more `nonabelian' homological algebra. Of course crossed n-cubes of A should be equivalent to n-fold groupoids in A. This would presumably bring in higher versions of nonabelian tensor products in A; a bibliography of such a tensor, mainly for n=2, has 90 items. This probably does not help to answer Mike's question on the name! Ronnie www.bangor.ac.uk/r.brown/nonabtens.html ----- Original Message ----- From: "Michael Barr" <mbarr@math.mcgill.ca> To: "Categories list" <categories@mta.ca> Sent: Thursday, December 01, 2005 1:48 AM Subject: categories: Name for a concept
Is there a standard name for a square A ----> B | | | | | | v v C ----> D in which the canonical map A ---> B x_D C is epic? I had always called it a weak pullback, but Peter Freyd claims that that phrase is reserved for the case that it satisfies the existence, but not necessarily the uniqueness of the definition of pullback. In fact, he claims it means that Hom(E,-) converts it to the kind of square I am talking about. What is interesting is that in an abelian category, it satisfies this condition iff it satisfies the dual condition iff the evident sequence A ---> B x C ---> D is exact. Putting a zero at the left end characterizes a genuine pullback and at the other end a pushout.
Michael