nLab
relative category

Context

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

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 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

Idea

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 goes back to at least Dwyer and Kan [1980], but was first studied systematically by Barwick and Kan [2012].

Definition

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 : 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.

Remarks

Let RelCat\mathbf{RelCat} be the 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.

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

  • William Dwyer, Daniel Kan, Simplicial localizations of categories. Journal of Pure and Applied Algebra 17 (1980) pp. 267–284.

  • Clark Barwick, Daniel Kan, Relative categories: Another model for the homotopy theory of homotopy theories. Indagationes Mathematicae 23 (2012) pp. 42–68.

Last revised on April 9, 2013 at 09:43:54. See the history of this page for a list of all contributions to it.