nLab linear model category

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Contents

Context

Model category theory

model category, model ∞ \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)(\infty,1)-categories

Model structures

for ∞\infty-groupoids

for ∞-groupoids

for equivariant ∞\infty-groupoids

for rational ∞\infty-groupoids

for rational equivariant ∞\infty-groupoids

for nn-groupoids

for ∞\infty-groups

for ∞\infty-algebras

general ∞\infty-algebras

specific ∞\infty-algebras

for stable/spectrum objects

for (∞,1)(\infty,1)-categories

for stable (∞,1)(\infty,1)-categories

for (∞,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (∞,1)(\infty,1)-sheaves / ∞\infty-stacks

Stable homotopy theory

Contents

Definition

A model category is called linear if it has a zero object (is a “pointed category” and hence a pointed model category) and for all of its objects XX, the unit

X⟶≃ΩΣX X \stackrel{\simeq}{\longrightarrow} \Omega \Sigma X

of the (reduced suspension ⊣\dashv loop space object)-adjunction is a weak equivalence.

(Schwede 97, def. 2.2.1)

References

Last revised on January 31, 2021 at 08:08:23. See the history of this page for a list of all contributions to it.

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