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?



2-categories for the working algebraic geometer
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](https://arxiv.org/abs/math/0702535v1)*, 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](https://emilyriehl.github.io/files/elements.pdf)* 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.


