nLab homotopy totalization

Idea

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.

Computation

In any $V$-enriched model category, the homotopy totalization of a cosimplicial object

$$X\colon \Delta\to C$$

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

$$Hom\colon (V^\Delta)^{op} \times C^\Delta \to C$$

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:

• injective and injective;
• projective and projective;
• Reedy and Reedy.

Accordingly, the homotopy totalization of a cosimplicial object $X\colon \Delta\to C$ can be computed as follows.

• Cofibrantly resolve the constant weight $1$ in one of the three model structures listed above.
• Fibrantly resolve $X$ in the other model structure in the same pair.
• Compute $Hom(Q 1, R X)$, which is the homotopy totalization of $X$.

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.

Reduction to semisimplicial objects

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.

Examples

Simplicial sets

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

$$X\colon \Delta \to sSet$$

can be computed as

$$Hom(Y_\Delta, R X),$$

where $R X$ is the Reedy fibrant replacement of $X$.

Chain complexes of abelian groups

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

$$\mathrm{N} \mathbf{Z} [-]\colon \Delta \to Ch,$$

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

$$Hom(\mathrm{N} \mathbf{Z} [-], X),$$

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 $X\colon \Delta\to Ch$.

Topological spaces

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

$$Hom(\mathbf{\Delta}, X)$$

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.

Related notions

• totalization

• homotopy limit

• homotopy end

• homotopy pullback

• homotopy product

• homotopy equalizer

• homotopy realization

References

For the case of chain complexes, see

• Kensuke Arakawa, Homotopy Limits and Homotopy Colimits of Chain Complexes, arXiv:2310.00201.