nLab
multisimplicial set

Content

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Content

Idea

Multisimplicial sets are the analogs of simplicial sets with simplices replaced by multisimplices? (perhaps more appropriately called multiplexes).

Definition

Definition

An nn-fold multisimplicial set is a presheaf on the nn-fold multisimplex category? Δ n\Delta^n, that is, a functor X:(Δ n) opSetsX\colon(\Delta^n)^{op}\to Sets, equivalently a multisimplicial object? in the category Set of sets.

The category of nn-fold multisimplicial sets can be equipped with a model structure that turns it into a model category that is Quillen equivalent to the standard Kan?Quillen model structure? on simplicial sets.

An important operation on multisimplicial sets is the exterior product

Fun((Δ m) op,Set)×Fun((Δ n) op,Set)Fun((Δ m+n) op,Set)Fun((\Delta^m)^{op},Set)\times Fun((\Delta^n)^{op},Set)\to Fun((\Delta^{m+n})^{op},Set)

defined as the left Kan extension of the tautological functor

Δ m×Δ nΔ m+n.\Delta^m\times\Delta^n\to\Delta^{m+n}.

The exterior product is a left Quillen bifunctor whose left derived bifunctor? model the cartesian product in the ∞-category of spaces.

The exterior product is useful when it is desirable to have a product operation that does not require subdivision.

References

Last revised on July 19, 2015 at 13:57:30. See the history of this page for a list of all contributions to it.