equivalences in/of $(\infty,1)$-categories
Homotopy totalizations are a special case of homotopy limits, when the indexing diagram is $\Delta$, the category of simplices.
Homotopy totalizations can be defined in any relative category, just like homotopy limits, but practical computations are typically carried out in presence of additional structures such as model structures, in fact, enriched model categories are the most common setup.
In any $V$-enriched model category, the homotopy totalization of a cosimplicial object
can be computed in three different ways, all of which use the notion of enriched end of enriched functors (i.e., weighted limits).
Specifically, consider the functor
that takes the enriched end of functors, i.e., the weighted limit, where $V$ is the monoidal model category over which $C$ is enriched.
If we set the first argument to the constant functor with value $1$ (the monoidal unit of $V$), then the resulting functor is the limit functor $C^\Delta\to C$.
The functor $Hom$ becomes a right Quillen bifunctor if we equip $V^\Delta$ and $C^\Delta$ with one of the three following pairs of model structures:
Accordingly, the homotopy totalization of a cosimplicial object $X\colon \Delta\to C$ can be computed as follows.
Cofibrant resolutions in the injective model structure can be computed by appling some cofibrant resolution functor of $C$ objectwise.
Cofibrant resolutions in the Reedy model structure can be computed inductively, by repeatedly factoring the latching map of $X$ as a cofibration followed by a weak equivalence and adjusting $X$ accordingly.
Cofibrant resolutions in the projective model structure can be computed explicitly in some practical examples.
The inclusion of the category of semisimplices (i.e., fininite inhabited totally ordered sets and injective order-preserving maps) into the category of simplices (with the injectivity condition dropped) is a homotopy initial functor, i.e., restricting along this inclusion preserves homotopy limits.
Thus, homotopy totalizations can be computed as homotopy limits over the category of semisimplices. The latter category is a direct category, which makes cofibrancy conditions particularly easy.
A Reedy cofibrant replacement of the constant weight $1\colon \Delta\to sSet$ can be computed as the Yoneda embedding $Y_\Delta\colon\Delta\to sSet$.
Thus, the homotopy totalization of
can be computed as
where $R X$ is the Reedy fibrant replacement of $X$.
For chain complexes with quasi-isomorphisms (which we equip with the projective model structure on chain complexes), a computation analogous to the one for simplicial sets above Reedy cofibrantly resolves the constant weight as
where $\mathrm{N}$ denotes the normalized chains functor and $\mathbf{Z}[-]$ denotes the free simplicial abelian group functor.
All cosimplicial objects in chain complexes are Reedy fibrant because the matching map is a degreewise surjection of chain complexes.
Thus, the homotopy totalization of $X\colon \Delta \to Ch$ can be computed as
which is isomorphic to the direct product total complex of the double chain complex obtained by applying the Dold–Kan correspondence to the cosimplicial object $X\colon \Delta\to Ch$.
Consider topological spaces with weak homotopy equivalences. Below, we use the Serre model structure.
The topological simplex $\mathbf{\Delta}\colon \Delta\to Top$ is Reedy cofibrant as a cosimplicial topological space.
Not all cosimplicial objects in topological spaces are Reedy fibrant, since the matching map need not be a Serre fibration of topological spaces.
However, we can pass to the semisimplicial setting, as explained above. In this, case Reedy fibrancy boils down to the objectwise fibrancy, which is always true for topological spaces.
Thus, the homotopy totalization of $X\colon \Delta \to Top$ can be computed as
in the category of functors $\Delta_{inj}\to Top$. This can be seen as the totalization analog of the fat geometric realization. One could call it the fat geometric totalization.
Created on February 2, 2021 at 23:04:25. See the history of this page for a list of all contributions to it.