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.
There is a majestic paper by Mac Lane
MacLane, Saunders. "Possible programs for categorists." Category Theory, Homology Theory and their Applications I. Springer, Berlin, Heidelberg, 1969. 123-131.
whose opening line is one of the most beautiful I've ever read:
Communication among Mathematicians is governed by a number of unspoken rules. One of these specifies that a Mathematician should talk about explicit theorems or concrete examples, and not about speculative programs. I propose to violate this excellent rule.
I've often wondered what does remains of those suggestions, and I strongly recommend you (if you haven't already) to have a look at this inspiring note: it is a masterpiece of neat exposition and it is replete with the hope that category theory becomes deeper and stronger, with the passing of time.
It is organized in brief, lapidary short sections, and proposes several directions in which category theory can, should or will go: in a few words
- we shall find new general concepts,
- we shall polish and adapt old ones through (hard work and) time,
- we shall reach a deeper understanding of structured and low-dimensional higher categories (monoidal categories, bi- and tri-categories and their multiple applications),
- we shall link category theory to differential geometry, mathematical analysis and mathematical physics,
- we shall ground category theory on a real foundation (or even better we shall use it as a foundation).
I'll leave you the pleasure of reading the note for yourself. My opinion (which is only the humble feeling of a young craftman) is that there are few items we can feel we have completely solved, even 50 years later.
Of course, today we have more higher category theory than we could ever hope for. Of course, we have a few people working in axiomatic cohesion. Of course, we have people in type theory and in HoTT. And also, we are lucky because today few people question the "importance of being abstract". But there is so much still to do!
And the best way I can explain what I'm saying is by adding an item to the otherwise complete Mac Lane's list.
- We shall work together to let more and more mathematicians see how profound, and beautiful, and inspiring, and elegant category theory is.
Category theory is huge, but few people outside pure mathematics apply it. Many people know that it exists, but few people appreciate its elegant, tautological statements and try to apply it to different things (those who do it are outstanding mathematicians, way better than I will ever be). This is what makes pure mathematics vital: a bunch of flippant engineers and physicists and biologists shaking it, breaking it, deforming it. We shall give other people tools to package immensely deep ideas in an extremely low volume ("rings are spaces"; "homotopy theory is localization"; "the Yoneda lemma"...).
Last, but not least, I feel we shall communicate why we feel lucky: category theory is an island of beauty in the already beautiful land of mathematics, and we are in love with it. We shall communicate the bliss we feel when we do it.
Best Answer
One aspect of 2-category theory which I've sometimes found difficult or tricky is 2-limits (or variants thereof). If that is troubling you too, some of these papers (mentioned in the nLab article on 2-limits) could be helpful:
Ross Street, Limits indexed by category-valued 2-functors, Journal of Pure and Applied Algebra, Volume 8 No. 2 (June 1976), 149–181. link
Max Kelly, Elementary observations on 2-categorical limits, Bulletin of the Australian Mathematical Society (1989), 39: 301-317, link.
Ross Street, Fibrations in Bicategories, Cahiers de topologie et géométrie différentielle catégoriques, tome 21, no. 2 (1980), p. 111-160. numdam pdf. See also the Correction (same journal, Vol. 28 No. 1 (1987), 53-56). link
Steve Lack, A 2-categories companion arXiv:math.CT/0702535 (see section 6, page 37).
G.J. Bird, G.M. Kelly, A.J. Power, R.H. Street, Flexible limits for 2-categories, Journal of Pure and Applied Algebra, Vol. 61 No. 1, (November 1989), 1–27. link
Thomas Fiore, Pseudo Limits, Biadjoints, and Pseudo Algebras, arXiv:math/0408298; see chapters 3, 4, 5.
John Power, 2-categories, BRICS Notes Series NS-98-7, ISSN 0909-3206 (August 1998). pdf