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model category

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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 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].

Definition

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.

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.

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.

Remarks

Let $\mathbf{RelCat}$ 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 $[C, D]_h$ 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.

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 $\mathbf{ssSet}$.

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