nLab relative category

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

The homotopy category of a relative category $C$ is the ordinary category $Ho C$ obtained from $und C$ by freely inverting the weak equivalences in $C$.

Refinements

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

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

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

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

Remark

The semi-saturation condition is precisely the condition on the inclusion $weq C \to und C$ ensuring it is a monomorphism in the (2,1)-category of 1-categories.

Examples

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

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

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

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

Properties

Categories of relative categories

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

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

There are then the following strings of adjunctions:

$$min \dashv und \dashv max \dashv weq : \mathbf{RelCat} \to \mathbf{Cat}$$

$$Ho \dashv min^+ : \mathbf{Cat} \to \mathbf{RelCat}$$

$$Ho \dashv min^+ \dashv und \dashv max \dashv weq : \mathbf{SsRelCat} \to \mathbf{Cat}$$

Moreover, because $min^+$ embeds $\mathbf{Cat}$ as a reflective exponential ideal in $\mathbf{SsRelCat}$ and in $\mathbf{RelCat}$, the functor $Ho : \mathbf{SsRelCat} \to \mathbf{Cat}$ 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 $PairSet$ be the cartesian closed category whose objects are pairs of small sets $(X, A)$ such that $A \subseteq X$, and whose morphisms $(X, A) \to (Y, B)$ are functions $f : X \to Y$ with $f(A) \subseteq B$.

Proposition

$RelCat$ is isomorphic to the category of small $PairSet$-enriched categories.

Proof

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

• A small set of objects $C$
• For objects $X,Y$, a small set $C(X,Y)$ with a subset $W(X,Y)$
• For objects $X,Y,Z$, a composition $C(Y,Z) \times C(X,Y) \to C(X,Z)$ that restricts to $W(Y,Z) \times W(X,Y) \to W(X,Z)$
• For objects $X$, a choice of identity element $id_X \in W(X,X)$

that is subject to identity and associativity relations. It’s immediate that this is exactly the same data as a relative category $(C,W)$. 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 $(\infty,1)$-categories

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

Theorem

There exists an adjunction

$$K_\xi \dashv N_\xi : \mathbf{RelCat} \to \mathbf{ssSet}$$

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

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

Related concepts

• category with weak equivalences

• simplicial localization

• F-category

• M-category

• marked simplicial set

References

The terminology “relative categories” is due to

• Clark Barwick, Daniel Kan, Relative categories: Another model for the homotopy theory of homotopy theories. Indagationes Mathematicae 23 (2012) 42-68 [arXiv:1011.1691, doi:10.1016/j.indag.2011.10.002]

while the basic idea goes back, at least, to:

• William Dwyer, Daniel Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980) 267-284. [doi:10.1016/0022-4049(80)90049-3]

in the context of simplicial localization (cf. the references at “category with weak equivalences”).