totalization

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.

Some kind of notes are in

- Rosona Eldred,
*Tot primer*(pdf)

Revised on August 10, 2011 01:58:40
by Urs Schreiber
(89.204.137.110)