nLab I-space

Idea

In homotopy theory, by an $\mathcal{I}$-space [Sagave & Schlichtkrull (2011)] one means a functor $\mathcal{I} \to \mathcal{S}$ 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 $\mathcal{S}^{\mathcal{I}}$ 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 $\mathcal{S}^{\mathcal{I}}$ 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 $E_\infty$-spaces.

Related concepts

• $E_\infty$-space

• model structure on parameterized spectra

References

• Steffen Sagave, Christian Schlichtkrull, §1.1 and §3 in: Diagram spaces and symmetric spectra, Advances in Mathematics 231 3-4 (2012) 2116-2193 [arXiv:1103.2764, doi:10.1016/j.aim.2012.07.013]

Discussion mostly in relation to symmetric spectra

• Stefan Schwede, Section 3.4 of: Symmetric spectra (2012) [pdf]

and paramterized symmetric spectra:

• Fabian Hebestreit, Steffen Sagave, Christian Schlichtkrull, Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces, Forum of Mathematics, Sigma 8 (2020) e16 [arXiv:1904.01824, doi:10.1017/fms.2020.11]