category theory

# Contents

## Definition

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

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

This is dual to geometric realization.

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.

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

Totalization of cosimplicial spaces is discussed in

and 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)

Some kind of notes are in

• Rosona Eldred, Tot primer (pdf)