nLab
left-determined 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

General

The following definition was introduced by Jeff Smith under the name of a minimal model structure.

Definition

(See Definition 2.1 in (RT 2003). A model structure on a category MM with a fixed class of cofibrations CC is left-determined if its class of weak equivalences is the smallest class of morphisms that is closed under retracts, satisfies the 2-out-of-3 property, contains all morphisms with a right lifting property with respect to CC, and whose intersection with CC is weakly saturated.

Note that a left-determined model structure is determined uniquely by its class of cofibrations.

Assuming the Vopěnka principle, the existence of left-determined model structures can be shown under very general conditions, see Theorem 2.2 in (RT 2003):

Theorem

The weak saturation of any set of morphisms in a locally presentable category is the class of cofibrations of a (unique) left-determined model structure.

Without the Vopěnka principle the best results known so far require additional restrictions on the underlying category:

Theorem

(Olschok 2009) The weak saturation of a set II of morphisms in a locally presentable category is the class of cofibrations of a (unique) left-determined model structure as long as the weak factorization system generated by II admits a very good cartesian cylinder (Definition 2.3 in (Gaucher 2015)) and all objects are cofibrant.

References

  • J. Rosický, W. Tholen, 2003, ‘Left-determined model categories and universal homotopy theories’, Transactions of the American Mathematical Society, vol. 355, no. 09, pp. 3611-3623: doi:10.1090/s0002-9947-03-03322-1

  • J. Rosický, W. Tholen, 2008, ‘Erratum to “ft-determined model categories and universal homotopy theories’’’, Transactions of the American Mathematical Society, vol. 360, no. 11, pp. 6179-6179: doi:10.1090/s0002-9947-08-04727-2

  • Marc Olschok, 2009, ‘Left Determined Model Structures for Locally Presentable Categories’, Applied Categorical Structures, vol. 19, no. 6, pp. 901-938: doi:10.1007/s10485-009-9207-2

  • Philippe Gaucher, Left determined model categories, New York Journal of Mathematics 21 (2015), 1093-1115 (nyjm:j/2015/21-50, arXiv:1507.02128)

Last revised on July 21, 2020 at 05:12:14. See the history of this page for a list of all contributions to it.