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
The following definition was introduced by Jeff Smith under the name of a minimal model structure.
(See Definition 2.1 in (RT 2003). A model structure on a category $M$ with a fixed class of cofibrations $C$ 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 $C$, and whose intersection with $C$ 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):
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:
(Olschok 2009) The weak saturation of a set $I$ 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 $I$ admits a very good cartesian cylinder (Definition 2.3 in (Gaucher 2015)) and all objects are cofibrant.
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 “Left-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)
Simon Henry, Minimal model structures, arXiv:2011.13408 (2020). (abstract)
Last revised on April 12, 2023 at 10:35:19. See the history of this page for a list of all contributions to it.