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.

## References

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)