Have a look around on my n-Lab 'home page': https://ncatlab.org/timporter/show/HomePage and go down to the `resources'. There are various quite old sets of notes that look at simplicially enriched categories, homotopy coherence etc. and that may help you with homotopy limits.

With Cordier, I wrote a paper: Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54, which aimed to give the necessary tools to allow homotopy coherent ends and coends (and their applications) to be pushed through to the $\mathcal{S}$-enriched setting and so to be used `without fear' by specialists in alg. geometry, non-abelian cohomology, etc.

You can also find stuff in my Menagerie notes, mentioned on that Home Page.

Have a look around on my n-Lab 'home page': https://ncatlab.org/timporter/show/HomePage and go down to the `resources'. There are various quite old sets of notes that look at simplicially enriched categories, homotopy coherence etc. and that may help you with homotopy limits, homotopy coherent / $\infty$-category ends and coends, etc.

With Cordier, I wrote a paper: Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54, which aimed to give the necessary tools to allow homotopy coherent ends and coends (and their applications) to be pushed through to the $\mathcal{S}$-enriched setting and so to be used `without fear' by specialists in alg. geometry, non-abelian cohomology, etc.

You can also find stuff in my Menagerie notes, mentioned on that Home Page.

Have a look around on my n-Lab 'home page': https://ncatlab.org/timporter/show/HomePage and go down to the `resources'. There are various no quite old sets of notes that look at simpliciaily enriched categories, homotopy coherence etc. and that may help you with homotopy limits.

With Cordier I wrote a paper:Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54, which aimed to give the necessary tools to allow homotopy coherent ends and coends (and their applications) to be pushed through to the $\mathcal{S}$-enriched setting and so to be used `without fear' by specialists in alg. geometry, non-abelian cohomology, etc.

You can also find stuff in my Menagerie notes, mentioned on that Home Page.

Have a look around on my n-Lab 'home page': https://ncatlab.org/timporter/show/HomePage and go down to the `resources'. There are various quite old sets of notes that look at simplicially enriched categories, homotopy coherence etc. and that may help you with homotopy limits.

With Cordier, I wrote a paper: Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54, which aimed to give the necessary tools to allow homotopy coherent ends and coends (and their applications) to be pushed through to the $\mathcal{S}$-enriched setting and so to be used `without fear' by specialists in alg. geometry, non-abelian cohomology, etc.

You can also find stuff in my Menagerie notes, mentioned on that Home Page.