homotopy theory, (∞,1)-category theory, homotopy type theory
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The notion of a relative category is an extremely weak version of a category with weak equivalences, providing the bare minimum needed to present an (∞,1)-category. The idea was first explored in a series of papers Dwyer & Kan 1980 on simplicial localization (see there for more). A model category structure on relative categories was constructed by Barwick & Kan 2012, presenting the (∞,1)-category of (∞,1)-categories.
More generally, the notion of relative category captures the idea of having a distinguished wide subcategory of morphisms that should be preserved by functors. Relative categories are occasionally used for this purpose, unrelated to any higher categorical purpose.
A relative category is a pair , where is a category and is a wide subcategory. A morphism in is said to be a weak equivalence in . A relative functor is a functor that preserves weak equivalences in the obvious sense.
The homotopy category of a relative category is the ordinary category obtained from by freely inverting the weak equivalences in .
A semi-saturated relative category is a relative category such that every isomorphism in is a weak equivalence in .
A category with weak equivalences is a semi-saturated relative category such that has the two-out-of-three property.
A homotopical category is a relative category such that has the two-out-of-six property; note that this automatically makes a category with weak equivalences.
A saturated homotopical category is a relative category such that a morphism is a weak equivalence if and only if it is invertible in ; note that any such relative category must be a homotopical category in particular.
The semi-saturation condition is precisely the condition on the inclusion ensuring it is a monomorphism in the (2,1)-category of 1-categories.
Any ordinary category gives rise to three relative categories in a functorial way:
A minimal relative category , in which the only weak equivalences are identities.
A minimal homotopical category , in which the only weak equivalences are isomorphisms.
A maximal homotopical category , in which every morphism is a weak equivalence.
Let be the large category of small relative categories and relative functors. It is a locally finitely presentable cartesian closed category, and we refer to the exponential object as the relative functor category. is, in particular, a (strict) 2-category.
Let be the full subcategory of semi-saturated relative categories. The inclusion has a left adjoint that preserves finite products, so is a reflective exponential ideal of .
There are then the following strings of adjunctions:
Moreover, because embeds as a reflective exponential ideal in and in , the functor preserves finite products.
The application of relative categories as expressing the idea of having a subcategory of distinguished morphisms can be neatly packaged into viewing relative categories as enriched categories over the category of sets with distinguished subsets. An equivalent formulation is that of an M-category (see also at F-category for a 2-category-theoretic version of this idea).
Let be the cartesian closed category whose objects are pairs of small sets such that , and whose morphisms are functions with .
is isomorphic to the category of small -enriched categories.
With minimal rephrasing, a small -enriched category consists of the data
that is subject to identity and associativity relations. It’s immediate that this is exactly the same data as a relative category . And under this identification, enriched functors and relative functors are the same thing.
A relative category can be equipped with a weak model structure, left or right semimodel structure, or a model structure. (These are listed from less restrictive to more restrictive structures.)
These structures do not change the underlying (∞,1)-category. However, the do provide constructions to perform computations in the underlying (∞,1)-category. Different structures yield different constructions, but all resulting answers are weakly equivalent.
The theory of relative categories presents a theory of (∞,1)-categories in the following sense:
There exists an adjunction
such that every left Bousfield localization of the Reedy model structure on the category of bisimplicial sets induces a cofibrantly-generated left proper model structure on making the adjunction a Quillen equivalence. In particular, there exists a model structure on that is Quillen equivalent to the model structure for complete Segal spaces on .
This is discussed in further detail at model structure on categories with weak equivalences.
The terminology “relative categories” is due to
while the basic idea goes back, at least, to:
in the context of simplicial localization (cf. the references at “category with weak equivalences”).
Last revised on May 1, 2023 at 16:55:19. See the history of this page for a list of all contributions to it.