homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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see also algebraic topology
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The totalization of a cosimplicial object is the dual concept to the geometric realization of a simplicial object.
For $A : \Delta \to C$ a cosimplicial object in a category $C$ which is powered over simplicial sets and for
the canonical cosimplicial simplicial set of simplices, the totalization of $A$ is the end
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 $A$.
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 $\Delta$, 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.
The concept cosimplicial spaces originates with
Quick review includes
The generalization to cosimplicial objects in more general model categories is discussed in
Review of this includes
Some kind of notes are in
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)