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 $$\mathrm{Hom}(\mathcal{E}_{k-1}, \mathcal{E}_k) \times\cdots\times \mathrm{Hom}(\mathcal{E}_0, \mathcal{E}_1) \longrightarrow \mathrm{Hom}(\mathcal{E}_0, \mathcal{E}_k).$$ (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?

  • $\begingroup$ Are you looking for references to this particular thing about the multicategory of Waldhausen categories, or references to multicategories with internal hom in general? $\endgroup$ – Tom Leinster Jan 19 '12 at 11:47
  • $\begingroup$ Either one. In the long term, I'm interested in how the K-theory functor interacts with these, but for now I'm just looking around to see what has been written. $\endgroup$ – Inna Jan 19 '12 at 14:50
  • $\begingroup$ This is besides the main point of the question, but are you sure that this works for levelwise cofibrations? It seems to me that in order for the composition to satisfy the pushout product property you need a stricter notion of cofibration. I mean the one where we require that in the naturality squares for cofibrations the induced map from the pushout is a cofibration. $\endgroup$ – Karol Szumiło Jan 19 '12 at 18:48
  • $\begingroup$ You're right; we need a "pushout square" condition on the cofibrations. $\endgroup$ – Inna Jan 20 '12 at 11:31

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 $\otimes$, a unit object $I$, and internal homs $[-,-]$, closed categories don't have the tensor product.

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  • $\begingroup$ I thought Manzyuk's paper was pretty good, though it was a while ago that I read it. It's unfortunate that the Eilenberg-Kelly paper is so hard to find nowadays (at least, it was when I was looking for it). $\endgroup$ – Mike Shulman Jan 20 '12 at 7:51

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