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stub for totalization
Is this related to (one of the two) totalization(s) of a double complex in homological algebra and related totalizations in homological algebra ?
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 the total simplicial set of a bisimplicial set.
Whereas the diagonal of a bisimplicial set is actually its geometric realization, considered as a simplicial object in sSet!
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”.
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…)
Thanks for the heads-up. I have added date and author-link (here).
I can’t recall why I wrote “some kind of notes” back in 2011, when this was the first and only reference for the entry, but I am sure it was not meant dismissively. Probably a combination of thinking this “might be lecture notes or might be a preprint, hard to tell” mixed with bad English skills.
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