Dear Ross, concerning the internal hom of strict omega-categories:
Given two strict omega-categories C and D, how do you define the strict omega-category of omega-functors between C and D?
There is the Crans-Gray tensor product on StrOmegaCat that makes it biclosed monoidal.
I think David was asking about the simpler cartesian closed structure on omega-Cat. This is constructed in
The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987) 283-335
You know all this, but for the record I say the following: The cartesian closed structure has an internal hom that is a restriction of the internal hom wrt the Gray structure. Usually the one of the Gray structure is the one of interest. It is the one closer to the full oo-category theoretic notion (the one with no strictness constraints whatsoever). The Crans-Gray tensor product with its property that G^k otimes G^l is k+l-dimensional is the fix in the globular model for what in the simplicial model is automatic, namely that Delta^k x Delta^l is k+l-dimensional. That this is automatic for the cartesian product in simplicial sets but requires more work for globular sets is one of the reasons why simplicial models for oo-categrories are more highly-developed than globular ones: they are easier. A good brief introduction to this is on the first few pages of Sjoerd Crans' A tensor product for Gray-categories http://www.emis.de/journals/TAC/volumes/1999/n2/5-02abs.html (After that introduction the article goes on to refine the globular Gray tensor product to the case of _weak_ (or rather: semi-strict) 3-categories.) But of course for the purposes of David's application (which I don't know about) the strict version of the internal hom might be sufficient. Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear all, Thank you for your replies. All the constructions I can find rely on strict omega-categories defined as graphs with structure. If instead we define recursively a strict omega-category as a category enriched over a strict omega-category, is there a recursive way to define the omega-category of omega-functors (between two fixed omega-categories)? David [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 26/09/2010, at 12:00 PM, David Leduc wrote:
All the constructions I can find rely on strict omega-categories defined as graphs with structure. If instead we define recursively a strict omega-category as a category enriched over a strict omega-category, is there a recursive way to define the omega-category of omega-functors (between two fixed omega-categories)?
Dear David Let V be a finitely complete cartesian closed category. Then both the category Cat(V) of categories in V and the category V-Cat of small categories enriched in V are also cartesian closed. If V = Set then Cat(V) = V-Cat = Cat. If V = Cat then Cat(V) = Dbl is the category of double categories and double functors while V-Cat = 2Cat is the category of 2-categories and 2-functors. If V = 2Cat then V-Cat = 3Cat . . . and on it goes. Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
-
David Leduc -
Ross Street -
Urs Schreiber