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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
    Commented Mar 14, 2016 at 16:24
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    $\begingroup$ The trick I use is not to start worrying about enriched categories! :-P $\endgroup$
    – Asaf Karagila ♦
    Commented Mar 14, 2016 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
    Commented Mar 14, 2016 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
    Commented Mar 14, 2016 at 20:34
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    $\begingroup$ @TomGoodwillie Thanks! I can finally say in retrospect that this is indeed a tremendous book! I recommend it to anyone strugling with the same issues. Riehl does an amazing job at laying out slowly and clearly the many definitions involved so that everything is crystal clear once you get to the "meat" of the theory. $\endgroup$
    – Saal Hardali
    Commented Jun 11, 2016 at 21:53

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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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  • $\begingroup$ There is a related question from Saal Hardali at mathoverflow.net/questions/258859/… and I have put several direct links and comments that are relevant there as well. $\endgroup$
    – Tim Porter
    Commented Jan 6, 2017 at 17:40
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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
    Commented Mar 14, 2016 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
    Commented Mar 15, 2016 at 13:14

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