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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 11th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded publication data for: * [[Bertrand Guillou]], [[Peter May]], _Enriched model categories and presheaf categories_, New York J. Math. 26 (2020) 37–9 ([arXiv:1110.3567](http://arxiv.org/abs/1110.3567), [](https://www.emis.de/journals/NYJM/j/2020/26-3.html), [NYJM:2020/26-3](https://www.emis.de/journals/NYJM/j/2020/26-3.html)) <a href="https://ncatlab.org/nlab/revision/diff/enriched+model+category/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/enriched+model+category/21">v21</a>, <a href="https://ncatlab.org/nlab/show/enriched+model+category">current</a>

   added publication data for:

   • Bertrand Guillou, Peter May, Enriched model categories and presheaf categories, New York J. Math. 26 (2020) 37–9 (arXiv:1110.3567, , NYJM:2020/26-3)

   diff, v21, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 11th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Jacob Lurie]], Def. A.3.1.5 in: _[[Higher Topos Theory]]_, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf)) <a href="https://ncatlab.org/nlab/revision/diff/enriched+model+category/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/enriched+model+category/21">v21</a>, <a href="https://ncatlab.org/nlab/show/enriched+model+category">current</a>

   added pointer to:

   • Jacob Lurie, Def. A.3.1.5 in: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press 2009 (pup:8957, pdf)

   diff, v21, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 1st 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded ([here](https://ncatlab.org/nlab/show/enriched+model+category#MonoidalModelCategoryIsEnrichedModelOverItself)) the previously missing example that monoidal model categories are enriched over themselves, together with the Joyal-Tierney calculus that shows this. <a href="https://ncatlab.org/nlab/revision/diff/enriched+model+category/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/enriched+model+category/23">v23</a>, <a href="https://ncatlab.org/nlab/show/enriched+model+category">current</a>

   added (here) the previously missing example that monoidal model categories are enriched over themselves, together with the Joyal-Tierney calculus that shows this.

   diff, v23, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 4th 2021
   • (edited Oct 4th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexFollowing discussion in another thread ([here](https://nforum.ncatlab.org/discussion/13511/separation-axioms-in-terms-of-lifting-properties/?Focus=95634#Comment_95634)), I have adjusted notation for the lifting property (to "$&solb;$", using `&solb;`) in the proof ([here](https://ncatlab.org/nlab/show/enriched+model+category#MonoidalModelCategoryIsEnrichedModelOverItself)) that the pushout-product axiom implies the pullback-power axiom. Now it looks as follows: $$ \begin{aligned} \mathrm{C} \Box \mathrm{C} & \,\subset\, & \mathrm{C} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} \Box \mathrm{C} &\,\;&solb;\;\,& \mathrm{FW} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} & \,&solb;\, & \mathrm{FW}^{\Box \mathrm{C}} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{FW}^{\Box \mathrm{C}} & \,\subset\, & \mathrm{FW} \\ {\phantom{-}} \\ \mathrm{C} \Box \mathrm{CW} & \,\subset\, & \mathrm{CW} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} \Box \mathrm{CW} &\,\;&solb;\;\,& \!\!\mathrm{F}\;\;\; & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} & \;&solb;\; & \mathrm{F}^{\Box \mathrm{CW}} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{F}^{\Box \mathrm{CW}} & \,\subset\, & \mathrm{FW} \\ & & & & && & \;\;\; \Leftrightarrow \;\;\; & \mathrm{CW} & \;&solb;\; & \mathrm{F}^{\Box \mathrm{C}} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{F}^{\Box \mathrm{C}} & \,\subset\, & \mathrm{F} \end{aligned} $$ <a href="https://ncatlab.org/nlab/revision/diff/enriched+model+category/24">diff</a>, <a href="https://ncatlab.org/nlab/revision/enriched+model+category/24">v24</a>, <a href="https://ncatlab.org/nlab/show/enriched+model+category">current</a>

   Following discussion in another thread (here), I have adjusted notation for the lifting property (to “$⧄$”, using ⧄) in the proof (here) that the pushout-product axiom implies the pullback-power axiom.

   Now it looks as follows:

   $$\begin{aligned} \mathrm{C} \Box \mathrm{C} & \,\subset\, & \mathrm{C} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} \Box \mathrm{C} &\,\;⧄\;\,& \mathrm{FW} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} & \,⧄\, & \mathrm{FW}^{\Box \mathrm{C}} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{FW}^{\Box \mathrm{C}} & \,\subset\, & \mathrm{FW} \\ {\phantom{-}} \\ \mathrm{C} \Box \mathrm{CW} & \,\subset\, & \mathrm{CW} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} \Box \mathrm{CW} &\,\;⧄\;\,& \!\!\mathrm{F}\;\;\; & \;\;\; \Leftrightarrow \;\;\; & \mathrm{C} & \;⧄\; & \mathrm{F}^{\Box \mathrm{CW}} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{F}^{\Box \mathrm{CW}} & \,\subset\, & \mathrm{FW} \\ & & & & && & \;\;\; \Leftrightarrow \;\;\; & \mathrm{CW} & \;⧄\; & \mathrm{F}^{\Box \mathrm{C}} & \;\;\; \Leftrightarrow \;\;\; & \mathrm{F}^{\Box \mathrm{C}} & \,\subset\, & \mathrm{F} \end{aligned}$$

   diff, v24, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 19th 2023
   • (edited May 19th 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * [[Daniel Quillen]], chapter II, section 2 of: _[[Homotopical Algebra]]_, Lecture Notes in Mathematics __43__, Springer (1967) &lbrack;[doi:10.1007/BFb0097438](https://link.springer.com/book/10.1007/BFb0097438)&rbrack; which, even if stated for $sSet$-enrichment only, is clearly the origin of the general concept. <a href="https://ncatlab.org/nlab/revision/diff/enriched+model+category/27">diff</a>, <a href="https://ncatlab.org/nlab/revision/enriched+model+category/27">v27</a>, <a href="https://ncatlab.org/nlab/show/enriched+model+category">current</a>

   added pointer to

   • Daniel Quillen, chapter II, section 2 of: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer (1967) [doi:10.1007/BFb0097438]

   which, even if stated for $sSet$-enrichment only, is clearly the origin of the general concept.

   diff, v27, current