nLab
totalization

Contents

Definition

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

Δ:[n]Δ[n] \Delta : [n] \mapsto \Delta[n]

the canonical cosimplicial simplicial set of simplices, the totalization of AA is the end

[k]Δ(A k) Δ[k]C. \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 Δ\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 TotTot 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)