nLab
relative category

Contents

Context

Category theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

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.

Definition

Basic

A relative category CC is a pair (undC,weqC)(und C, weq C), where undCund C is a category and WW is a wide subcategory. A morphism in weqCweq C is said to be a weak equivalence in CC. A relative functor f:CDf : C \to D is a functor f:undCundDf \colon und C \to und D that preserves weak equivalences in the obvious sense.

The homotopy category of a relative category CC is the ordinary category HoCHo C obtained from undCund C by freely inverting the weak equivalences in CC.

Refinements

  • A semi-saturated relative category is a relative category CC such that every isomorphism in undCund C is a weak equivalence in CC.

  • A category with weak equivalences is a semi-saturated relative category CC such that weqCweq C has the two-out-of-three property.

  • A homotopical category is a relative category CC such that weqCweq C has the two-out-of-six property; note that this automatically makes CC a category with weak equivalences.

  • A saturated homotopical category is a relative category CC such that a morphism is a weak equivalence if and only if it is invertible in HoCHo C; note that any such relative category must be a homotopical category in particular.

Examples

Any ordinary category CC gives rise to three relative categories in a functorial way:

  • A minimal relative category minCmin C, in which the only weak equivalences are identities.

  • A minimal homotopical category min +Cmin^+ C, in which the only weak equivalences are isomorphisms.

  • A maximal homotopical category maxCmax C, in which every morphism is a weak equivalence.

Properties

Categories of relative categories

Let RelCat\mathbf{RelCat} 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 [C,D] h[C, D]_h as the relative functor category. RelCat\mathbf{RelCat} is, in particular, a (strict) 2-category.

Let SsRelCat\mathbf{SsRelCat} be the full subcategory of semi-saturated relative categories. The inclusion SsRelCatRelCat\mathbf{SsRelCat} \hookrightarrow \mathbf{RelCat} has a left adjoint that preserves finite products, so SsRelCat\mathbf{SsRelCat} is a reflective exponential ideal of RelCat\mathbf{RelCat}.

There are then the following strings of adjunctions:

minundmaxweq:RelCatCatmin \dashv und \dashv max \dashv weq : \mathbf{RelCat} \to \mathbf{Cat}
Homin +:CatRelCatHo \dashv min^+ : \mathbf{Cat} \to \mathbf{RelCat}
Homin +undmaxweq:SsRelCatCatHo \dashv min^+ \dashv und \dashv max \dashv weq : \mathbf{SsRelCat} \to \mathbf{Cat}

Moreover, because min +min^+ embeds Cat\mathbf{Cat} as a reflective exponential ideal in SsRelCat\mathbf{SsRelCat} and in RelCat\mathbf{RelCat}, the functor Ho:SsRelCatCatHo : \mathbf{SsRelCat} \to \mathbf{Cat} preserves finite products.

As enriched categories

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 PairSetPairSet be the cartesian closed category whose objects are pairs of small sets (X,A)(X, A) such that AXA \subseteq X, and whose morphisms (X,A)(Y,B)(X, A) \to (Y, B) are functions f:XYf : X \to Y with f(A)Bf(A) \subseteq B.

Proposition

RelCatRelCat is isomorphic to the category of small PairSetPairSet-enriched categories.

Proof

With minimal rephrasing, a small PairSetPairSet-enriched category consists of the data

  • A small set of objects CC
  • For objects X,YX,Y, a small set C(X,Y)C(X,Y) with a subset W(X,Y)W(X,Y)
  • For objects X,Y,ZX,Y,Z, a composition C(Y,Z)×C(X,Y)C(X,Z)C(Y,Z) \times C(X,Y) \to C(X,Z) that restricts to W(Y,Z)×W(X,Y)W(X,Z)W(Y,Z) \times W(X,Y) \to W(X,Z)
  • For objects XX, a choice of identity element id XW(X,X)id_X \in W(X,X)

that is subject to identity and associativity relations. It’s immediate that this is exactly the same data as a relative category (C,W)(C,W). And under this identification, enriched functors and relative functors are the same thing.

Model structures

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.

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

The theory of relative categories presents a theory of (∞,1)-categories in the following sense:

Theorem

There exists an adjunction

K ξN ξ:RelCatssSetK_\xi \dashv N_\xi : \mathbf{RelCat} \to \mathbf{ssSet}

such that every left Bousfield localization of the Reedy model structure on the category ssSet\mathbf{ssSet} of bisimplicial sets induces a cofibrantly-generated left proper model structure on RelCat\mathbf{RelCat} making the adjunction a Quillen equivalence. In particular, there exists a model structure on RelCat\mathbf{RelCat} that is Quillen equivalent to the model structure for complete Segal spaces on ssSet\mathbf{ssSet}.

This is discussed in further detail at model structure on categories with weak equivalences.

References

Last revised on June 25, 2021 at 07:00:44. See the history of this page for a list of all contributions to it.