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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 12th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea Partial model categories are one of the many intermediate notions between [[relative categories]] and [[model categories]]. They axiomatize those properties of [[model categories]] that only involve weak equivalences. ## Definition A __partial model category__ is a [[relative category]] such that its class of weak equivalences satisfies the [[2-out-of-6 property]] (if $s r$ and $t s$ are weak equivalences, then so are $r$, $s$, $t$, $t s r$) and admits a [[3-arrow calculus]], i.e., there are [[subcategories]] $U$ and $V$ (which can be thought of as analogues of acyclic cofibrations and acyclic fibrations) such that $U$ is closed under [[cobase changes]] (which are required to exist), $V$ is closed under [[base changes]], and any morphism can be functorially factored as the composition $v u$ for some $u\in U$ and $v\in V$. ## Properties If $(C,W)$ is a partial model category, then any Reedy fibrant replacement of the [[Rezk nerve]] $N(C,W)$ is a [[complete Segal space]]. ## Related concepts * [[model category]] * [[relative category]] * [[semimodel category]] * [[weak model category]] * [[premodel category]] * [[homotopical category]] ## References * [[Clark Barwick]], [[Daniel M. Kan]], _Partial model categories and their simplicial nerves_, [arXiv](https://arxiv.org/abs/1102.2512v2). <a href="https://ncatlab.org/nlab/revision/partial+model+category/1">v1</a>, <a href="https://ncatlab.org/nlab/show/partial+model+category">current</a>

   Created:

   Idea

   Partial model categories are one of the many intermediate notions between relative categories and model categories.

   They axiomatize those properties of model categories that only involve weak equivalences.

   Definition

   A partial model category is a relative category such that its class of weak equivalences satisfies the 2-out-of-6 property (if $s r$ and $t s$ are weak equivalences, then so are $r$, $s$, $t$, $t s r$) and admits a 3-arrow calculus, i.e., there are subcategories $U$ and $V$ (which can be thought of as analogues of acyclic cofibrations and acyclic fibrations) such that $U$ is closed under cobase changes (which are required to exist), $V$ is closed under base changes, and any morphism can be functorially factored as the composition $v u$ for some $u\in U$ and $v\in V$.

   Properties

   If $(C,W)$ is a partial model category, then any Reedy fibrant replacement of the Rezk nerve $N(C,W)$ is a complete Segal space.

   Related concepts

   • model category

   • relative category

   • semimodel category

   • weak model category

   • premodel category

   • homotopical category

   References

   • Clark Barwick, Daniel M. Kan, Partial model categories and their simplicial nerves, arXiv.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorHurkyl
   • CommentTimeJul 12th 2021
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexFunctoral factorization into vu is for weak equivalences only. <a href="https://ncatlab.org/nlab/revision/diff/partial+model+category/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/partial+model+category/2">v2</a>, <a href="https://ncatlab.org/nlab/show/partial+model+category">current</a>

   Functoral factorization into vu is for weak equivalences only.

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorHurkyl
   • CommentTimeJul 12th 2021
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexClarified $U, V$ are subcategories of the weak equivalences, and that (co)base changes are along arbitrary morphisms, not just weak equivlaences. <a href="https://ncatlab.org/nlab/revision/diff/partial+model+category/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/partial+model+category/2">v2</a>, <a href="https://ncatlab.org/nlab/show/partial+model+category">current</a>

   Clarified $U, V$ are subcategories of the weak equivalences, and that (co)base changes are along arbitrary morphisms, not just weak equivlaences.

   diff, v2, current