Date: Thu, 23 Sep 93 12:45:19 +0200 From: ldup@alcbel.be (Luc Duponcheel) To: barr@Math.McGill.CA

Michael,

I have a simple question. I work in the following `framework' :

A category which is such that all Hom(A,B) are themselves categories and having the following properties :

first some notation -------------------

morphisms are denoted as F : A -> B, G : C -> D, ... and their composition is denoted as GF

morphisms in Hom(A,B) (called transformations) are denoted as alpha : F -> G, beta : H -> K and their composition is denoted as beta . alpha

here come the actual properties -------------------------------

1) for all transformations alpha : F -> G where F : A -> B and G : A -> B, and all morphisms H : B -> Y there exists a transformation alpha H : FH -> GH.

1a) GF alpha = G (F alpha) 1b) F (beta . alpha) = F beta . F alpha

2) for all transformations alpha : F -> G where F : A -> B and G : A -> B, and all morphisms H : X -> A there exists a transformation H alpha : HF -> HG.

2a) alpha GF = (alpha G) F 2b) (beta . alpha) F = beta F . alpha F

3) G (alpha F) = (G alpha) F

BTW ---

a transformation beta : H -> K is natural if for all transformations alpha : F -> G one has K alpha . beta F = beta G . H alpha

These axioms are a subset of the ones used for 2-categories. The transformations do not need to be (but may, of course, be) natural. I do not, a priory, need any `horizontal' composition "*" of

Can someone help this guy out with this? transformations.

One of the results which I want to prove in this framework is just the fact that certain transformations (who do not a priory need to be natural at all) are nevertheless, under certain conditions (of a different nature) natural.

Is there any *name* for this `framework'?

I could call it *categories with transformations* but if there is any other name which is commonly used, then I would appreciate if you can inform me about it.

Thanks!

Luc.

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