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
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
In homotopy theory, by an -space [Sagave & Schlichtkrull (2011)] one means a functor from
the category of finite sets (including the empty set) with injective maps between them;
a category such as TopologicalSpaces or SimplicialSets which carries a classical model category structure presenting plain homotopy types.
The functor category of -spaces then carries a model category-structure whose weak equivalences are those natural transformations which induce weak homotopy equivalences on homotopy colimits over ; and this turns out to be Quillen equivalent to the classical model structure via the colimit-functor
As such, -spaces are just another presentation of homotopy theory.
However, via Day convolution of the evident symmetric monoidal category-structure on (whose tensor product is disjoint union of finite sets) the functor category of -spaces becomes itself symmetric monoidal.
This monoidal structure on -spaces is more homotopy-truthful than the (cartesian) monoidal structure on plain spaces: strictly commutative monoids internal to -spaces serve to model all -spaces.
Discussion mostly in relation to symmetric spectra
and paramterized symmetric spectra:
Last revised on April 1, 2023 at 20:44:13. See the history of this page for a list of all contributions to it.