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
Multisimplicial sets are the analogs of simplicial sets with simplices replaced by multisimplices? (perhaps more appropriately called multiplexes).
An $n$-fold multisimplicial set is a presheaf on the $n$-fold multisimplex category? $\Delta^n$, that is, a functor $X\colon(\Delta^n)^{op}\to Sets$, equivalently a multisimplicial object? in the category Set of sets.
The category of $n$-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
defined as the left Kan extension of the tautological functor
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.
Zouhair Tamsamani?, On non-strict notions of $n$-category and $n$-groupoid via multisimplicial sets (arXiv:alg-geom/9512006)
Gerd Laures? and James E. McClure,
Multiplicative properties of Quinn spectra (arXiv:0907.2367v2)
Yifeng Liu and Weizhe Zheng, Gluing restricted nerves of $\infty$-categories (arXiv:1211.5294)
Last revised on July 19, 2015 at 13:57:30. See the history of this page for a list of all contributions to it.