Date: Tue, 19 Nov 1996 21:44:33 -0400 (AST)
Subject: What kind of functor do I have here?

Date: Tue, 19 Nov 1996 19:39:26 -0600
From: Joshua Caplan <caplan@cs.uiuc.edu>

Suppose I have an endofunctor F : C -> C on a category with finite
products and exponents, and a natural transformation s : Exp => Exp o
(Fop x F) such that, for all f in C(A,B),

1
|  \
|   \
|    \
|[f]  \[Ff]
v      v
B^A--->FB^FA
   s_AB

where [f] is (f o pr_2)^, the name of f.

Is this called a "strong functor", similar to the "strong monad" idea?

---------------------------------+---------------------------------------------caplan@cs.uiuc.edu               |Joshua Caplan, Dept. of Computer Sci.
http://acsl.cs.uiuc.edu/~caplan/ |Univ. of Illinois at Urbana-Champaign
---------------------------------+---------------------------------------------


Date: Wed, 20 Nov 1996 11:58:39 -0400 (AST)
Subject: Re: What kind of functor do I have here?

Date: Wed, 20 Nov 96 10:16:50 GMT
From: Roy L. Crole <rlc3@mcs.le.ac.uk>



Re: Joshua Caplan's question about strong functors.

If you regard C as enriched over itself, with C(A,B) = B^A in obC, and the
functor F is C-enriched, and also s_A,B : C(A,B) ---> C(FA,FB) is precisely
the action of F on hom-objects, then F is often called a strong functor,
and s a functorial strength.

Connections between these ideas and tensorial strengths can be found in

A. Kock, Strong Functors and Monoidal Monads, Archiv der Mathematik, 23
1972.

Basically the short answer to your question is "yes". Hope this helps.

Roy Crole