model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
Theorems
A stable model category is a 1-category structure used to present a stable (∞,1)-category in analogy to how a general model category encodes a (generally non-stable) (∞,1)-category.
A stable model category $\mathcal{C}$ is
such that the loop space object functor $\Omega$ and the suspension object functor $\Sigma$, are inverse equivalences on the homotopy category $Ho(C)$:
The homotopy category $Ho(\mathcal{C})$ of a stable model category, equipped with the reduced suspension functor $\Sigma \colon Ho(\mathcal{C})\overset{\simeq}{\to} Ho(\mathcal{C})$ is a triangulated category (Hovey 99, section 7).
Stabilization of model categories is a model for the abstractly defined stabilization in (infinity,1)-category theory (Robalo 12, prop. 4.15).
Let $\mathcal{C}$ be a stable model category that is in addition
then there is a chain of sSet-enriched Quillen equivalences linking $\mathcal{C}$ to the the spectrum-enriched functor category
equipped with the global model structure on functors, where $A_S$ is the $Sp$-enriched category whose set of objects is $S$
This is in (Schwede-Shipley, theorem 3.3.3)
An $Sp$-enriched category is a homotopy-theoretic analog of an Ab-enriched category, which may be thought of as a many-object version of a ring, a “ringoid”. Accordingly, an $Sp$-enriched category is an $A_\infty$-ringoid. It is has a single object then (as a pointed category) it is an A-infinity algebra.
Hence:
If if in prop. there is just one compact generator $P \in \mathcal{C}$, then there is a one-object $Sp$-enriched category, hence an A-infinity algebra $A$, which is the endomorphisms $A \simeq End_{\mathcal{C}}(P)$, and the stable model category is its category of modules:
This is in (Schwede-Shipley, theorem 3.1.1)
This may be thought of as a homotopy-theoretic analog of the Freyd-Mitchell embedding theorem for abelian categories.
One way to read this is that formal duals of presentable stable infinity-categories are a model for spaces in (“derived”) noncommutative geometry.
If $A$ is an Eilenberg-MacLane spectrum, then this identifies the corresponding stable model categories with the model structure on unbounded chain complexes.
This is (Schwede-Shipley 03, theorem 5.1.6).
The concept originates with
The classification theorems are due to
Discussion of the notion of stable model categories with the abstract notion of stabilization in (infinity,1)-category theory is in section 4.2 (prop. 4.15) of
Last revised on January 31, 2021 at 08:09:18. See the history of this page for a list of all contributions to it.