Ouch, no posts in 2 months? Shame on me. Let’s suppose I’ve been busy.

Today, and other scattered times in the past, I’ve been trying to come to grips with a proof concerning homotopy limits. Suppose and that for every . Suppose that is a (covariant) functor from to, say, spaces. Then the lemma I’ve been thinking about is that , and in fact the natural map from right to left gives the equivalence.

I’ve been studying Bousfield and Kan‘s proof a little, and trying to make it look like something I can think about. Their goes to simplicial sets, or so, and they have some assumption about it taking values in the collection of fibrant objects. I’m going to ignore that, because in Top all spaces are fibrant (right?), and because ignoring hypotheses can never lead you astray (right?).

So anyway, I thought I’d share something like an outline of a proof. I’ll write = where there probably should be instead. And that likely won’t be the worst of my transgressions.

Consider the bi-complex (I guess it’s a bi-cosimplicial space, or so) where

where the represent objects of , the come from , and arrows lie where they should (in particular, between primed elements, the arrows are in ). I’m going to get tired of that indexing, so I’ll let represent a chain of the form , and similarly is . In fact, I’m not going to put the subscripts, as the subscript on a will always be , and similarly for the primes. If I write , I mean that . So I can write

letting the ambiguation begin (or was that earlier?).

Next, let’s start thinking about what some homotopy limits are. I should probably be calling things Tot’s, for totalization, but my understanding is that they’re basically holim. Finally, “\holim” is not a built-in latex command. For convenience, I’m going to just write “lim”.

means below.

Ok, so, fix a . In what follows, denotes the mapping space, and represents the -th layer in the simplicial nerve of a category . The geometric realization of is denoted . Compute:

Finally, our assumption that these arrow categories are contractible lets us write this as simply . Which means that

.

That’s nice. Let’s do the homotopy limit the other way, fixing to begin.

Now , which almost looks close to what we’ve got above. To see that they are equivalent, notice that the arrow category has the initial object . This means that

We conclude that

Finally, apply Fubini’s theorem for homotopy limits, and conclude that

Now, to translate to the situation when my categories are internal categories in Top…

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