A Reference on Multicategories with “Internal Hom”

The multicategory of Waldhausen categories is “enriched over itself”: the Hom-set of k-exact functors can be given a Waldhausen category structure by letting the morphisms be natural transformations, and cofibrations/weak equivalences be levelwise. With this construction, k-fold composition is a k-exact functor
Hom(Ek1,Ek)××Hom(E0,E1)Hom(E0,Ek).
(It also works for general composition, not just on the 1-level, but I thought that writing that out here would not be worthwhile.)

This does not seem to be (a) new or (b) an example of a new construction, but I am having trouble finding references for it. Can anyone help?

Answer

It might be helpful to look up the notion of closed multicategory. I have the feeling that this is present more in folklore than in print (always a frustrating thing to hear). My impression is that the idea was first properly understood during the writing of

Martin Hyland, John Power, Pseudo-commutative monads and pseudo-closed 2-categories, Journal of Pure and Applied Algebra 175 (2002), 141-185.

However, the paper is written in greater generality, and the theory you need might not be totally apparent. A later paper is:

Oleksandr Manzyuk, Closed categories vs. closed multicategories, arXiv:0904.3137.

I haven’t read that.

Both build on the idea of closed category, introduced by Eilenberg and Kelly. (See either paper for the citation.) Whereas monoidal closed categories are equipped with a tensor product , a unit object I, and internal homs [,], closed categories don’t have the tensor product.

Attribution
Source : Link , Question Author : Inna , Answer Author : Tom Leinster

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