Exposition of Grothendieck's mathematics As Wikipedia says:

In Grothendieck's retrospective Récoltes et Semailles, he identified twelve of his contributions which he believed qualified as "great ideas". In chronological order, they are:

*

*Topological tensor products and nuclear spaces.

*"Continuous" and "discrete" duality (derived categories, "six operations").

*Yoga of the Grothendieck–Riemann–Roch theorem (K-theory, relation with intersection theory).

*Schemes.

*Topoi.

*Étale cohomology and l-adic cohomology.

*Motives and the motivic Galois group (Grothendieck ⊗-categories).

*Crystals and crystalline cohomology, yoga of "de Rham coefficients", "Hodge coefficients", ...

*"Topological algebra": ∞-stacks, derivators; cohomological formalism of topoi as inspiration for a new homotopical algebra.

*Tame topology.

*Yoga of anabelian algebraic geometry, Galois–Teichmüller theory.

*"Schematic" or "arithmetic" point of view for regular polyhedra and regular configurations of all kinds.


What are some modern, concise expository texts on these topics that students can use to learn the great ideas of Grothendieck? EGA, SGA, Les Dérivateurs, La Longue Marche, and so on don't count, because they are French, overwhelming, and maybe a bit outdated (I'm not sure) - anyway, it's not realistic for a student to go through them in their free time in addition to the courses they take for their degree. If you want to give a textbook, make sure it's as short as possible.
Please only one topic with expository text per answer.
Let me start by giving two examples of the kind of answers I am expecting:

*

*Leinster's An informal introduction to topos theory is an amazing introduction to the basic ideas surrounding topoi (and their connections to logic and geometry). And it is concise! This is idea 5.


*Milne's lecture notes on étale cohomology. Not as short as Leinster's paper, but better (for a student) than 1000 pages of SGA 4 and 4 1⁄2. This is idea 6.
 A: For topic 3, I would say Hartshorne's Appendix A is pretty good as a VERY brief introduction. After that, as also referred to there, I would recommend Manin's Lectures on the K-functor in algebraic geometry.
A: For topic 5, try Leinster's nice expository article https://arxiv.org/abs/1012.5647.
A: Topic 10 got developed further via the concept of o-minimality. See e.g. "Tame topology and o-minimal structures" by
Lou van den Dries, 1998 CUP, http://dx.doi.org/10.1017/CBO9780511525919
A shorter text is by Michel Coste: http://perso.univ-rennes1.fr/michel.coste/polyens/OMIN.pdf
See also nLab article on tame topology: http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/tame+topology
A: For learning about topoi, I'd strongly recommend Goldblatt's Topoi: The Categorical Analysis of Logic, which goes over the subject and a few interesting applications in a very approachable and pleasant manner. It's not particularly concise at 556 pages, though you needn't read the entire text to get a solid foundational concept of topoi.
A: For topic 6, there are Milne's notes: https://www.jmilne.org/math/CourseNotes/LEC.pdf
In the front matter Milne writes:

These are the notes for a course taught at the University of Michigan in 1989 and 1998.
In comparison with my book, the emphasis is on heuristic arguments rather than formal
proofs and on varieties rather than schemes.

A: For topic 7, Milne's article https://www.jmilne.org/math/xnotes/MOT.pdf could be useful.
A: A good roadmap for FGA (topic 4, with glimpses on topic 6) is
Fantechi, Barbara; Göttsche, Lothar; Illusie, Luc; Kleiman, Steven L.; Nitsure, Nitin; Vistoli, Angelo, Fundamental algebraic geometry: Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3541-6/hbk). x, 339 p. (2005). ZBL1085.14001.
A: For topic 4 (schemes), I'd suggest Schemes: The Language of Algebraic Geometry, by Eisenbud and Harris.  This book is out of print, and is generally regarded as having been "superseded" by its successor, The Geometry of Schemes. But for your purposes, I think the earlier book is actually better, since it's only 160 pages, and if you just want the main ideas, the first two chapters should suffice.  The authors specifically aimed at making the topic accessible to students encountering schemes for the first time.
A: Let me suggest having looking at Grothendieck's own article The cohomology theory of abstract algebraic varieties in the proceedings of the 1958 ICM. This is both short (15 pages) and in English. Although written quite early on, it explains the motivations and intuitions for many of the developments that lay in the future in a lucid way. For instance, he compares and contrast his notion of scheme (referred to here as a pre-schema) with the more classical notions. He  hints at "a definition of the Weil cohomology (involving both 'spatial' and Galois cohomology)...", which would eventually become étale cohomology.
A: 

*Infinity-stacks

Before learning about infinity stacks, one ought to learn about stacks. For this, look at section 4.1.1 of Vistoli's notes, Notes on Grothendieck topologies, fibred categories and descent theory, which is an exposition of one concrete case. This section is around a page and a half. He states:

that the fact we can glue continuous maps and topological spaces says $(Cont)$ is a stack over $Top$.

This in fact a generalisation of the clutching construction of bundles. And you can learn about bundles in the very readable first chapter of Grothendiecks Kansas notes on fibre bundles.
There is also an expository note by Barbara Fantechi, Stacks for Everybody.
A: For topic 1, the book by Diestel, Fourie, Swart: https://books.google.de/books?id=VBg5cGSngRoC&printsec=copyright&redir_esc=y#v=onepage&q&f=false provides a good introduction.
A: For 5. Topoi. There is a very readable notice of the AMS: What is... a topos?, Luc Illusie, and with M Raynaud about Schemes at Grothendieck and Algebraic Geometry. In fact, the webpage of Professor L. Illusie contains very valuable material regarding the question.
You can read (online) about various of the listed topics in Lectures grothendieckiennes Edited by Frédéric Jaëck
And find a lot of original material at Thèmes pour une Harmonie!
A: On topic 1, I'd like to recommend Ray Ryan's Introduction to tensor products of Banach spaces, Springer 2002. Zbl 1090.46001
