Subsection 8.2. (0MTJ) · ScalingStacks

8.2. The classification space of a closed model category[0MTJ]

If ${\operatorname{\mathbf{M}}}$ is a simplicial model category, and $X$ and $Y$ objects in ${\operatorname{\mathbf{M}}}$ , we write $\map_{{\operatorname{\mathbf{M}}}}(X,Y)$ for the function complex from $X$ to $Y$ .

[05UZ]

Theorem 8.3. Let ${\operatorname{\mathbf{M}}}$ be simplicial closed model category, and let ${\operatorname{\mathbf{W}}}\subset{\operatorname{\mathbf{M}}}$ denote the subcategory of weak equivalences. Then $V=N^{f}({\operatorname{\mathbf{M}}},{\operatorname{\mathbf{W}}})$ is a complete Segal space. Furthermore, there is an equivalence of categories $\ho V\approx\ho{\operatorname{\mathbf{M}}}$ and there are weak equivalences of spaces $\map_{V}(X,Y)\approx\map_{{\operatorname{\mathbf{M}}}}(X,Y)$ .

We prove (8.3) below.

[05V0]

Remark 8.4. This result (8.3) presumably generalizes to an arbitrary closed model category, not necessarily simplicial; the function complex $\map_{V}(X,Y)$ would be taken to be one of those described by Dwyer and Kan in [DK80].

[05V1]

Remark 8.5. Note that any category $C$ having finite limits and colimits can be made into a closed model category in which the weak equivalences are precisely the isomorphisms (and all maps are fibrations and cofibrations). In this case $N(C)=N(C,\iso C)$ coincides with the classifying diagram construction described in (3.5), and we have noted (6.1) that this is already a complete Segal space.