Grothendieck universes and their connections to set theory and geometry I recently approached the notion of Grothendieck universe, but didn't find any "canonical" references about it. In particular, I would like to read some exhaustive explanation of its connections to 
a) large cardinals and consistency problems
b) geometry (algebraic geometry, topology, and so on); from the original ones (what are the concrete reasons that led Grothendieck to introduce his axiom?) to any other that could have arisen during the following decades.
Any books or notes about this matter are gladly welcome.
Thank you in advance!
 A: For simplicity, let's work in ZFC and conflate "universe" with "Grothendieck universe."

Recall that a universe is a transitive set $U$ containing $\omega$ which is closed under powersets, pairing, and "functional unions" (if $f:x\rightarrow U$ for $x\in U$, then $\bigcup_{i\in x}f(i)\in U$).
It's easy to show the following:

$U$ is a universe iff $U=V_\kappa$ for some strongly inaccessible $\kappa$.

The proof can be found here, but I'll include it for completeness. Showing that $V_\kappa$ is a universe whenever $\kappa$ is strongly inaccessible is trivial. In the other direction, supposing $U$ is a universe it suffices to show that if $\alpha=height(U)$ is the least ordinal not in $U$, then $\alpha$ is an uncountable regular strong limit cardinal (since then we can show by induction that $V_\beta\subseteq U$ for all $\beta<\alpha$, and this forces $U=V_\alpha$. 


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*Clearly $\alpha$ is a limit ordinal: if $\alpha=\beta+1$ and $\beta\in U$, then $\{\beta\}\in U$ by pairing ($\{\beta,\beta\}=\{\beta\}$); now $\{0, 1\}\in U$, so - mapping $0$ to $\beta$ and $1$ to $\{\beta\}$ - we get $\beta\cup\{\beta\}=\alpha\in U$ by functional unions. 

*Since $\alpha$'s a limit, cofinality makes sense, so let's talk about that. If $cf(\alpha)<\alpha$, then we'd have $cf(\alpha)\in U$ by definition of $\alpha$. Then we have a cofinal map $f: cf(\alpha)\rightarrow \alpha$, so $ran(f)\in U$. But now $\bigcup_{\beta<cf(\alpha)}f(\alpha)=\alpha\in U$. So $\alpha$ is regular - note that this means $\alpha$ is a cardinal.

*Since $\alpha$ is regular and $\omega\in U$, $\alpha$ must be uncountable. And since $U$ is closed under powersets, $\alpha$ is a strong limit: given $\beta<\alpha$ and fixing some bijection $f:2^\beta\rightarrow\vert 2^\beta\vert$ (we're distinguishing here between the set $2^\beta$, which is not an ordinal, and its cardinality, which is), by functional unions we have $\vert 2^\beta\vert\in U$ hence is $<\alpha$. So $\alpha$ is strongly inaccessible.
This addresses the set-theoretic issues completely, by reducing it to the study of strong inaccessibility (e.g. "every set is contained in a universe" = "there is a proper class of strongly inaccessible cardinals," etc.).
Note, and this is crucial, the use of external facts in the definition of a universe. We don't demand the "internal" property that $\mathcal{P}(x)\cap U$ be an element of $U$, for $x\in U$; instead we demand that the true powerset of $x$ be in $U$, and similarly for functional unions. This is the key which drives the strength of the principle. E.g. merely satisfying $V_\kappa\models$ZFC is the property of "worldliness," which is much weaker than strong inaccessibility (for instance, worldly ordinals need not even be regular).

As to the second half of your question, this moves out of my comfort zone, but I'm familiar with the basic picture (which hopefully someone will elaborate more on):


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*Grothendieck introduced universes to handle "size issues": e.g. while the category of groups is a large category (= proper class of objects), the category of $U$-groups for a universe $U$ is a small category. Now normally "chopping off" the category of groups at some arbitrary level might lose some interesting structure, but the definition of Grothendieck universe is designed to prevent this: broadly speaking, any phenomenon which happens in the category of groups happens already in the category of $U$-groups whenever $U$ is a universe. More specifically, universes are introduced to make e.g. definitions of cohomology theories which go through things like functor categories rigorous without needing to go beyond sets.

*That said, my impression is that it is generally agreed that all but the most extremely abstract results that can be proved via universes can also be proved without them. For example, my understanding is that the Stacks Project develops everything in ZFC, including complicated things like etale cohomology which are often exposited (or defined naively) using universes, without significant difficulty. More generally, there's a similarity with things like Scott's trick - see e.g. this MSE question - and this is one piece of evidence among many that universes are rarely necessary, merely efficient.
Others who are actually expert in algebraic geometry and category theory should say more about this; while I'm somewhat qualified to talk about set theory, I am in no way qualified to talk about algebraic geometry. But what I've said above reflects what a number of people who are so qualified have told me, so I'll stick with it until I hear otherwise.


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*Grothendieck was surely aware of this, so why introduce universes at all? I suspect the reason was ultimately a pragmatic one: the set-theoretic issues weren't ones he wanted to focus on (and strongly inaccessible cardinals in particular are pretty inoffensive); I believe he cared more about getting general results more than using weak axiom systems (and Brian Conrad discusses this a bit here). In particular, I'm not aware of any work by Grothendieck indicating "consistency concerns." EDIT: I'll quote a comment by user BCnrd, related to this point:



if one wants to write a treatise on a general theory of cohomology on all topoi, including operations like sheafification, enough injectives, derived categories, and Ext-sheaves, there needs to be a way to control the "size" of coverings which arise in these constructions (to replace the implicit role of "power set" for ordinary topology spaces). The universe stuff takes care of such matters in an elegant way, so one can focus attention on the more central aspects of the theory.


It's worth at this point looking at something like McLarty's article on the role of universes in Fermat's last theorem, carefully. McLarty is arguing that universes are "used" in FLT, for a very specific meaning of the word "use" (which has been criticized elsewhere). Ultimately McLarty's article is largely sociological - it's about, in McLarty's words, the "high-level" organization of mathematical concepts in their current deployment. 
