nLab I-space



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



Paths and cylinders

Homotopy groups

Basic facts


Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



In homotopy theory, by an \mathcal{I}-space [Sagave & Schlichtkrull (2011)] one means a functor 𝒮\mathcal{I} \to \mathcal{S} from

  1. the category \mathcal{I} of finite sets (including the empty set) with injective maps between them;

  2. a category such as TopologicalSpaces or SimplicialSets which carries a classical model category structure presenting plain homotopy types.

The functor category 𝒮 \mathcal{S}^{\mathcal{I}} of \mathcal{I}-spaces then carries a model category-structure whose weak equivalences are those natural transformations which induce weak homotopy equivalences on homotopy colimits over \mathcal{I}; and this turns out to be Quillen equivalent to the classical model structure via the colimit-functor

As such, \mathcal{I}-spaces are just another presentation of homotopy theory.

However, via Day convolution of the evident symmetric monoidal category-structure on \mathcal{I} (whose tensor product is disjoint union of finite sets) the functor category 𝒮 \mathcal{S}^{\mathcal{I}} of \mathcal{I}-spaces becomes itself symmetric monoidal.

This monoidal structure on \mathcal{I}-spaces is more homotopy-truthful than the (cartesian) monoidal structure on plain spaces: strictly commutative monoids internal to \mathcal{I}-spaces serve to model all E E_\infty -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.