Contents

category theory

# 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 $C$ is a pair $(und C, weq C)$, where $und C$ is a category and $W$ 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

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