relative category



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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 by Dwyer and Kan in 1980s. A model structure on relative categories was constructed by Barwick and Kan [2012].


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.


  • 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.

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.


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.


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:


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.


  • 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 May 8, 2020 at 00:52:11. See the history of this page for a list of all contributions to it.