nLab homotopy realization

Idea

Homotopy realizations are a special case of homotopy colimits, when the indexing diagram is $\Delta^{op}$, the opposite category of the category of simplices.

Homotopy realizations can be defined in any relative category, just like homotopy colimits, 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 realization of a simplicial object

$$X\colon \Delta^{op}\to C$$

can be computed in three different ways, all of which use the notion of tensor product of functors (i.e., weighted colimits).

Specifically, consider the functor

$$\otimes\colon V^\Delta \times C^{\Delta^{op}} \to C$$

that takes the tensor product of functors, i.e., the weighted colimit, 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 colimit functor $C^{\Delta^{op}}\to C$.

The functor $\otimes$ becomes a left Quillen bifunctor if we equip $V^\Delta$ and $C^{\Delta^{op}}$ with one of the three following pairs of model structures:

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

Accordingly, the homotopy realization of a simplicial object $X\colon \Delta^{op}\to C$ can be computed as follows.

• Cofibrantly resolve the constant weight $1$ in one of the three model structures listed above.
• Cofibrantly resolve $X$ in the other model structure in the same pair.
• Compute $Q 1 \otimes Q X$, which is the homotopy realization of $X$.

Cofibrant resolutions in the injective model structure can be computed by applying 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 the inclusion of opposite categories preserves homotopy colimits.

Thus, homotopy realizations can be computed as homotopy colimits over the opposite 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 $Y_\Delta\colon\Delta\to sSet$.

The Reedy model structure on simplicial objects in simplicial sets with simplicial weak equivalences coincides with the injective model structure, as explained in the article elegant Reedy category. In particular, all objects are cofibrant.

Thus, the homotopy realization of

$$X\colon \Delta^{op} \to sSet$$

can be computed as

$$Y_\Delta\otimes X,$$

which is isomorphic to the diagonal of $X$.

Chain complexes of abelian groups

For chain complexes with quasi-isomorphisms (which we equip with the injective 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.

Once again, all simplicial objects in chain complexes are Reedy cofibrant.

Thus, the homotopy realization of $X\colon \Delta^{op} \to Ch$ can be computed as

$$\mathrm{N} \mathbf{Z} [-] \otimes X,$$

which is isomorphic to the direct sum total complex of the double chain complex obtained by applying the Dold–Kan correspondence to the simplicial object $X\colon \Delta^{op}\to Ch$.

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 simplicial objects in topological spaces are Reedy cofibrant, since the latching map need not be a cofibration of topological spaces, i.e., a retract of a relative cellular map.

However, we can pass to the semisimplicial setting, as explained above. In this, case Reedy cofibrancy boils down to the objectwise cofibrancy.

Thus, the homotopy realization of $X\colon \Delta^{op} \to Top$ can be computed as

$$\mathbf{\Delta}\otimes Q X,$$

where $Q$ denotes objectwise cofibrant replacement. This is precisely the classical fat geometric realization of simplicial topological spaces.

Related notions

• realization

• homotopy colimit

• homotopy coend

• homotopy pushout

• homotopy coproduct

• homotopy coequalizer

• homotopy totalization

References

For the case of chain complexes, see

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