also added brief mentioning of Bousfield-Kan spectral sequence and Eilenberg-Moore spectral sequence.

(phew, that entry had been sitting there all the years lacking most of the homotopy theoretic information…)

]]>added statement and pointer to proof of the fact that the totalization, in the traditional sense, is a model for the homotopy limit over the cosimplicial objects, if the latter is Reedy fibrant.

]]>- Paul Bressler, Alexander Gorokhovsky, Ryszard Nest, Boris Tsygan,
*Deformation quantization of gerbes*, Advances in Mathematics 214 (2007) 230–266, pdf

reviews the constructon of the **totalization of a cosimplicial DGLA** in Section 3.4 (defined as certain colimit) and “prove that isomorphism classes of descent data of a cosimplicial DGLA are in one-to-one correspondence with isomorphism classes of Maurer–Cartan elements of its totalization”.

Whereas the diagonal of a bisimplicial set is actually its *geometric realization*, considered as a simplicial object in sSet!

Good point, the terminology is a bit inconclusive in some places of the literature, I think. I have added the following paragraph to totalization:

]]>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 Eilenberg-Zilber theorem. As discussed at

bisimplicial set, this is weakly homotopy equivalent to the operation that is often called $Tot$ and called thetotal simplicial setof a bisimplicial set.

Is this related to (one of the two) totalization(s) of a double complex in homological algebra and related totalizations in homological algebra ?

]]>stub for totalization

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