How to stop worrying about enriched categories? Recently I realized that ordinary category theory is not a suitable language for a big portion of the math I'm having a hard time with these days. One thing in common to all my examples is that they all naturally fit into the enriched categorical context.


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*2-Categories - Enriched in categories. Examples: Stacks ($BG$, $QCoh$) are 2-sheaves, 2-category of rings and bi-modules. 

*DG categories - Enriched in chain complexes. Prime example: The dg category of chain complexes of $\mathcal{O}_X$-modules over $X$.

*Topological/Enriched categories - Enriched in topological spaces/simplicial sets. Prime example: $\mathsf{Top}$.
I now have the impression that many of the difficulties I face in trying to learn about math that involves the three above originate in the gap between the ordinary categorical language and the enriched one. In particular, the natural constructions from ordinary category theory (limit, adjunctions etc.) are no longer meaningful and I'm practically blindfolded.
Is there a friendly introduction to enriched category theory somewhere where I can get comfortable with this general framework? Is it a bad idea to pursue this direction?
 A: 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. 
A: Try reading the intro and the first chapter of Lurie's "Higher Topos Theory" for a gentle introduction to various types of higher categories. Appendix A 1.4 provides an overview of the "classical" enriched category theory.
(By the way, 2-categories are not really the same categories enriched in categories.)  
