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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 11th 2021

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJun 11th 2021

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 1st 2021

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

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 4th 2021
• (edited Oct 4th 2021)

Following discussion in another thread (here), I have adjusted notation for the lifting property (to “$⧄$”, using &solb;) 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}
• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 19th 2023
• (edited May 19th 2023)

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