nLab
totalization

Definition

For $A:\Delta \to C$ a cosimplicial object in a category $C$ which is powered over simplicial sets and for

$$\Delta :[n]\mapsto \Delta [n]$$\Delta : [n] \mapsto \Delta[n]

the canonical cosimplicial simplicial set of simplices, the totalization of $A$ is the end

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

This is dual to geometric realization.

Related concepts

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 $\mathrm{Tot}$ and called the total simplicial set of a bisimplicial set.

References

Some kind of notes are in

• Rosona Eldred, Tot primer (pdf)