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I study algebraic geometry / number theory and from time to time I stumble upon 2-categorical (co)limits. I have two main examples in mind:

Example 1) In étale cohomology, the (triangulated) derived category of $\overline{\mathbb{Q}_\ell}$-sheaves is defined as a 2-colimit of the derived categories of sheaves with coefficients in finite extensions of $\mathbb{Q}_\ell$.

Example 2) The fact that $\textsf{QCoh}\to \textsf{Sch}$ is a stack should mean that, given a scheme $X$ and a covering $\{U_i\to X\}$, the category $\textsf{QCoh}(X)$ is a 2-limit of the $\textsf{QCoh}(U_i)$.

While the second example is somewhat straightforward, given that we may describe the limit as a category of descent data, the first one is awkward.

For sure, I shouldn't need to understand a lot of 2-category theory to make sense of these (and all related) examples. There are a lot of intricacies in the 2-categorical world... For example, should we consider (co)limits in the (2,0)-category of categories or on the (2,1)-category of categories? Should we consider lax 2-functors or strict 2-functors? What changes with those choices? (Bear in mind that I know very little of all of this.)

All in all, how should an algebraic geometer approach these kinds of statements? Also, is there a quick reference for all of this?

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    $\begingroup$ If you ever find yourself thinking about stacks, you will encounter a fair bit of 2-category theory. I think some introductions to stacks might cover some of the relevant background. $\endgroup$
    – Wojowu
    Commented Mar 31, 2022 at 17:32
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    $\begingroup$ For everything you mentioned here (2,1)-categories are sufficient (where basically none of the subtleties of 2-category theory apply). In fact (∞,1)-category theory is much simpler than even 2-category theory... $\endgroup$
    – Denis Nardin
    Commented Mar 31, 2022 at 18:17
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    $\begingroup$ I'm not sure you wrote what you meant to write for example 1. That example is pretty subtle though $\endgroup$
    – Donu Arapura
    Commented Mar 31, 2022 at 21:30
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    $\begingroup$ @Gabriel Perhaps you meant to write "...the derived category of $\overline{\mathbb{Q}}_\ell$-sheaves...". As for the subtlety, see Ekedahl's paper The adic formalism. $\endgroup$
    – Donu Arapura
    Commented Apr 1, 2022 at 8:51
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    $\begingroup$ To be honest I think the question is too broad and vague. Or is this just about the two examples? In any case, have a look at this book: arxiv.org/abs/2002.06055 It is a really good intro to 2-categories. $\endgroup$
    – Martin Brandenburg
    Commented Nov 16, 2022 at 12:52

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A standard reference for category theory is Categories for the Working Mathematician (which I assume the OP knew about based on the question title). The closest reference I know to "2-categories for the working mathematician" is Steve Lack's A 2-categories companion, which presents the essential parts of the theory (including 2-limits, in Section 6) in only 73 pages. Lack is a famous 2-category theorist in Sydney, Australia, and throughout this monograph he emphasizes lax vs strict, pseudo vs strong, bicategories vs 2-categories, etc. He also discusses situations where the 2-cells are invertible. I mention this because of the comment telling the OP to look at (2,1)-categories.

Another nice (and much newer) reference is Elements of $\infty$-category theory by Emily Riehl and Dominic Verity. It's 606 pages, which might seem daunting, but Appendix B is an introduction to 2-category theory and is only 22 pages. An earlier draft of this book had the title $\infty$-categories for the working mathematician, and the book seems to be written with the goal of making the machinery easier to use, including to folks in algebraic geometry. Riehl and Verify are also world-class category theorists, and much of their approach to $\infty$-categories is 2-categorical in nature.

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