Homotopy Limits and Arrow Categories

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 \mathscr{C}'\subseteq \mathscr{C} and that |\mathscr{C}'\downarrow c|\simeq * for every c\in \mathscr{C}. Suppose that F is a (covariant) functor from \mathscr{C} to, say, spaces. Then the lemma I’ve been thinking about is that \text{holim}_{\mathscr{C}'}F\simeq \text{holim}_{\mathscr{C}}F, 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 F 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 \simeq instead. And that likely won’t be the worst of my transgressions.

Consider the bi-complex A_{i,j} (I guess it’s a bi-cosimplicial space, or so) where

\displaystyle A_{i,j}=\prod_{\substack{c_0'\leftarrow\cdots\leftarrow c_i'\\c_j\leftarrow c_0'\\c_0\leftarrow \cdots\leftarrow c_j}} F(c_0),

where the c_k represent objects of \mathscr{C}, the c_k' come from \mathscr{C}', and arrows lie where they should (in particular, between primed elements, the arrows are in \mathscr{C}'). I’m going to get tired of that indexing, so I’ll let \underline{c_j} represent a chain of the form c_0\leftarrow \cdots \leftarrow c_j, and similarly \underline{c_i'} is c_0'\leftarrow\cdots\leftarrow c_i'. In fact, I’m not going to put the subscripts, as the subscript on a \underline{c_j} will always be j, and similarly i for the primes. If I write \underline{c}\leftarrow \underline{c'}, I mean that c_j\leftarrow c_0'. So I can write

\displaystyle A_{i,j}=\prod_{\underline{c}\leftarrow \underline{c'}}F(c_0),

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”.

\lim means \text{holim} below.

Ok, so, fix a j. In what follows, Map(X,Y) denotes the mapping space, and N_n(\mathscr{D}) represents the n-th layer in the simplicial nerve of a category \mathscr{D}. The geometric realization of \mathscr{D} is denoted |\mathscr{D}|. Compute:

\begin{array}{rl} \lim_i A_{i,j} &= \displaystyle \lim_i \prod_{\underline{c}\leftarrow \underline{c'}} F(c_0) \\ &= \displaystyle \lim_i \prod_{\underline{c}}\prod_{c_j\leftarrow \underline{c'}} F(c_0) \\ &= \displaystyle \prod_{\underline{c}} \lim_i \prod_{c_j\leftarrow \underline{c'}}F(c_0) \\ &= \displaystyle \prod_{\underline{c}} \lim_i Map(N_i(\mathscr{C}'\downarrow c_j),F(c_0)) \\ &= \displaystyle \prod_{\underline{c}} Map(|\mathscr{C}'\downarrow c_j|,F(c_0))\end{array}

Finally, our assumption that these arrow categories are contractible lets us write this as simply \prod_c F(c_0). Which means that

\displaystyle \lim_{\mathscr{C}}F=\lim_j \prod_{\underline{c}} F(c_0) = \lim_j \lim_i A_{i,j}.

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

\begin{array}{rl} \lim_j A_{i,j} &= \displaystyle \lim_j \prod_{\underline{c}\leftarrow \underline{c'}}F(c_0) \\ &= \displaystyle \lim_j \prod_{\underline{c'}}\prod_{\underline{c}\leftarrow c_0'} F(c_0) \\ &= \displaystyle \prod_{\underline{c'}} \lim_j \prod_{\underline{c}\leftarrow c_0'} F(c_0) \end{array}

Now \lim_{\mathscr{C}'}F=\lim_i \prod_{\underline{c'}} F(c_0'), which almost looks close to what we’ve got above. To see that they are equivalent, notice that the arrow category c_0'\downarrow \mathscr{C} has the initial object c_0'\to c_0'. This means that

\displaystyle F(c_0')=\lim_{c_0'\downarrow \mathscr{C}}F=\lim_j \prod_{\underline{c}\leftarrow c_0'}F(c_0).

We conclude that

\displaystyle \lim_{\mathscr{C}'} F=\lim_i \lim_j A_{i,j}.

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

\displaystyle \lim_{\mathscr{C}}F=\lim_{\mathscr{C}'}F

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


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