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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.

  1. 2-Categories - Enriched in categories. Examples: Stacks ($BG$, $QCoh$) are 2-sheaves, 2-category of rings and bi-modules.

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

  3. 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?

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    $\begingroup$ I don't know how friendly it is, but do you know Kelly's book? tac.mta.ca/tac/reprints/articles/10/tr10abs.html $\endgroup$ – Todd Trimble Mar 14 '16 at 16:24
  • $\begingroup$ @ToddTrimble I'm aware of it. It looks rather technical though, so I thought maybe it'd be a good idea to ask for advice. $\endgroup$ – Saal Hardali Mar 14 '16 at 16:27
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    $\begingroup$ The trick I use is not to start worrying about enriched categories! :-P $\endgroup$ – Asaf Karagila Mar 14 '16 at 16:29
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    $\begingroup$ It is not always necessary to understand the general structure to understand an example of it... $\endgroup$ – Thomas Rot Mar 14 '16 at 18:13
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    $\begingroup$ Maybe you want to look at Riehl's book math.jhu.edu/~eriehl/cathtpy.pdf $\endgroup$ – Tom Goodwillie Mar 14 '16 at 20:34
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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.

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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.)

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    $\begingroup$ On the contrary, that is the standard defn of 2-categories, from way back. $\endgroup$ – Peter May Mar 14 '16 at 23:46
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    $\begingroup$ I agree with Peter, but it depends what you mean. Some people mean 'bicategory' when they say 2-category, which is a slightly different concept. Every bicategory is bicategorically equivalent to a 2-category, but that doesn't mean we can go whole hog and say that the weak 3-category of bicategories is 3-equivalent to the 3-category $2$-Cat, so there's a little wobble there between the theory of bicategories and the theory of 2-categories (defined from the POV of Cat-enriched category theory). $\endgroup$ – Todd Trimble Mar 15 '16 at 13:14

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