homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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see also algebraic topology
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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 goes back to at least Dwyer and Kan [1980], but was first studied systematically by Barwick and Kan [2012].
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 $f : und C \to und D$ 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 $C$.
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.
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.
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:
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.
The theory of relative categories presents a theory of (∞,1)-categories in the following sense:
There exists an adjunction
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.
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.