nLab totalization



Category theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




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


As an end

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

Δ:[n]Δ[n] \Delta : [n] \mapsto \Delta[n]

the canonical cosimplicial simplicial set of simplices, the totalization of AA is the end

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

As the homotopy limit

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

(Hirschhorn 15, theorem 9.2)


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.

Formally the dual to totalization is geometric realization: where totalization is the end over a powering with Δ\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 TotTot 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.


The concept for cosimplicial spaces originates with:

Introductory notes:

Quick review:

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

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

Last revised on November 7, 2023 at 20:23:54. See the history of this page for a list of all contributions to it.