nLab
pointed model category

Contents

Context

Model category theory

model 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

specific

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

Contents

Definition

A model category is pointed if its underlying category is a pointed category, i.e., if the unique morphism from the initial object to the terminal object is an isomorphism, in which case both of them are denoted by 00 (the zero object).

In any pointed category, one has a canonical zero morphism between any objects AA and BB, given by the composition A0BA\to 0\to B.

The homotopy equalizer of f:ABf\colon A\to B and 0:AB0\colon A\to B is known as the homotopy fiber of ff.

The homotopy coequalizer of f:ABf\colon A\to B and 0:AB0\colon A\to B is known as the homotopy cofiber of ff.

In particuar there is the homotopy (co)-fiber of the zero object with itself, the loop space object- and reduced suspension-operation. Asking these operations to be equivalences in a suitable sense leads to the concept of linear model categories and stable model categories.

Examples

Model categories which are pointed without being linear or even stable:

Pointed model categories which are stable:

References

Last revised on February 10, 2021 at 07:54:50. See the history of this page for a list of all contributions to it.