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When somebody asked me about the sSet enrichment of cosimplicial objects, I noticed that the nLab didn’t have the pointers. So I have now split off a stub cosimplicial object from simplicial object and added a bare minimum of pointers. No time for more at the moment.
There is no need to assume finite limits or colimits. I added an end formula for the hom-spaces: in effect, all we are doing is forming the totalisation of the cosimplicial simplicial set $\mathcal{C} (X^{\bullet}, Y^{\bullet})$.
Thanks!
Actually, the question was about the simplicial tensoring.
Ah. Then you will need colimits in $\mathcal{C}$.
I should have said it here as I did say it in the entry: what is called the “external simplicial structure” in Quillen 67, recalled for instance as Bousfield 03, section 2.10 involves the enrichment, the powering and the tensoring. That’s what I was pointing to. But I should have said so more clearly.
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