equivalences in/of $(\infty,1)$-categories
The collection of spectra forms an (∞,1)-category $Sp(\infty Grpd) =$ Spectra, which is in fact a stable (∞,1)-category. Indeed, it is the universal property stabilization of the $(\infty,1)$-category ∞Grpd, equivalently of the simplicial localization of the category Top at the weak homotopy equivalences.
$Sp(\infty Grpd)$ plays a role in stable homotopy theory analogous to the role played by the 1-category Ab of abelian groups in homological algebra, or rather of the category of chain complexes $Ch_\bullet(Ab)$ of abelian groups.
In the context of (∞,1)-categories a spectrum is a spectrum object in the (∞,1)-category $L_{whe} Top_*$ of pointed topological spaces.
Recall that spectrum objects in the (infinity,1)-category $C$ form a stable (∞,1)-category $Sp(C)$.
The stable (∞,1)-category of spectrum objects in $L_{whe} Top_*$ is the stable $(\infty,1)$-category of spectra
A sequence of morphisms of spectra $E \longrightarrow F \longrightarrow G$ is a homotopy fiber sequence if and only if it is a homotopy cofiber sequence:
In fact:
A homotopy-commuting square in Spectra is a homotopy pullback if and only it is a homotopy pushout.
This follows from Prop. by the fact that Spectra is additive (this Prop.).
See also arXiv:1906.04773, Prop. 6.2.11, MO:q/132347.
This property of Spectra (Prop. , Prop. ) reflects one of the standard defining axioms on stable (∞,1)-categories (see there) and on stable derivators (see there).
With the smash product of spectra $Sp(L_{whe}Top_*)$ becomes a symmetric monoidal (∞,1)-category.
an algebra object in $Sp(L_{whe}Top_*)$ with respect to this monoidal structure is an associative ring spectrum;
a commutative algebra object in $Sp(L_{whe}Top_*)$ with respect to this monoidal structure is a commutative ring spectrum;
The prime spectrum of a monoidal stable (∞,1)-category for p-local and finite spectra is labeled by the Morava K-theories. This is the content of the thick subcategory theorem.
There are several presentations of the $(\infty,1)$-category of spectra by model categories of spectra. In particular there are symmetric monoidal model categories where the smash product of spectra is presented by an ordinary tensor product, so that A-∞ rings, E-∞ rings and ∞-modules are presented by 1-categorical monoid objects and module objects, respectively (“brave new algebra”). See at:
model structure on spectra, symmetric monoidal smash product of spectra
The stable $(\infty,1)$-category of spectra is described in chapter 1 of
or section 9 of
Its monoidal structure is described in section 4.2
That this is a symmetric monoidal structure is described in section 6 of
Last revised on January 16, 2021 at 09:59:15. See the history of this page for a list of all contributions to it.