nLab totalization

Idea

The totalization of a cosimplicial object is the dual concept to the geometric realization of a simplicial object.

Definition

As an end

For $A : \Delta \to C$ a cosimplicial object in a category $C$ which is powered over simplicial sets and for

$$\Delta : [n] \mapsto \Delta[n]$$

the canonical cosimplicial simplicial set of simplices, the totalization of $A$ is the end

(1)$$\int_{[k]\in \Delta} (A_k)^{\Delta[k]} \,\,\, \in C \,.$$

As the homotopy limit

For a cosimplicial object $A \colon \Delta \to \mathcal{C}$ in a suitable model category such that $A$ is a fibrant object with respect to the Reedy model structure on $Func(\Delta, \mathcal{C})$, totalization in terms of the end-construction above in (1) is a model for the homotopy limit over $A$.

(Hirschhorn 15, theorem 9.2)

Properties

Homotopy and homology

The homotopy groups of the totalization of a cosimplicial space are computed by a Bousfield-Kan spectral sequence.

The homology groups by an Eilenberg-Moore spectral sequence.

Related concepts

Formally the dual to totalization is geometric realization: where totalization is the end over a powering with $\Delta$, realization is the coend over the tensoring.

But various other operations carry names similar to “totalization”. For instance a total chain complex is related under Dold-Kan correspondence to the diagonal of a bisimplicial set – see at Eilenberg-Zilber theorem. As discussed at bisimplicial set, this is weakly homotopy equivalent to the operation that is often called $Tot$ and called the total simplicial set of a bisimplicial set.

To a cosimplicial chain complex we can assign a double complex by taking the alternating sum of the coface maps. Then the totalization of this cosimplicial object and the totalization of the double complex as defined in homological algebra coincide. Moreover, the associated Bousfield-Kan spectral sequence and spectral sequence of a double complex coincide.

References

The concept for cosimplicial spaces originates with:

• Aldridge Bousfield, Daniel Kan, chapter X.3 of Homotopy limits, completions and localizations, Lecture Notes in Mathematics, 304 Springer (1972)

Introductory notes:

• Rosona Eldred, Tot primer (2008) [pdf]

Quick review:

• Daniel Dugger, section 5.3 of A primer on homotopy colimits (pdf)

The generalization to cosimplicial objects in more general model categories is discussed in

• Aldridge Bousfield, Cosimplicial resolutions and homotopy spectral sequences in model categories, Geom. Topol. 7 (2003) 1001-1053 (arXiv:math/0312531)

Review of this includes

• Marc Levine, The Adams-Novikov spectral sequence and Voevodsky’s slice tower, Geom. Topol. 19 (2015) 2691-2740 (arXiv:1311.4179)

Discussion of totalizations as homotopy limits includes

• Philip Hirschhorn, Section 9 of: The diagonal of a multicosimplicial object (arXiv:1506.06837)

• Akhil Mathew, Vesna Stojanoska, Fibers of partial totalizations of a pointed cosimplicial space, Proc. Amer. Math. Soc. 144 (2016), no. 1, 445–458 (arXiv:1408.1665)