Why did Grothendieck invent schemes when they seem at odds with his philosophy?
From what I understand, a scheme is basically a sheaf on $\mathbf{Ring}^{\mathrm{op}}$ equipped with an appropriate Grothendieck topology, subject to an axiom that says that schemes look locally like commutative rings. If it weren't for this axiom, the category of schemes would be a topos, and hence very well behaved. But because of this axiom, the category of schemes isn't very well-behaved; for example, it's neither complete nor cocomplete. If so, then focusing on schemes seems to run at odds with Grothendieck's philosophy that it's better to have a good category with bad objects than a bad category consisting only of good objects. What gives here?
Solution 1:
1) Schemes were invented in the early 1960s, and Grothendieck wasn't the only one involved in their development (though he is usually credited with their invention) - I think it was Serre who first defined $\mathrm{Spec}(A)$ and its structure sheaf (hence affine schemes); Cartan came up with the idea of ringed spaces (at least this is what I gather from reading the excellent introduction to EGA I, Grundlehren edition). And of course, the fundamental analogy is: schemes / affine schemes = manifolds / euclidean spaces.
2) Topos theory was invented after schemes (of course), and it is (from a geometrical point of view) essentially the unsuccessful cousin of scheme theory. Schemes brought about a revolution and made it possible to resolve, say, the Weil conjectures and Fermat's last theorem. Can you name me a problem of similar importance that was solved by topos theory? The difference between topos theory and schemes is that the latter is geometrically meaningful. As with most mathematicians, Grothendieck's best work was in his youth (schemes, motives, etale cohomology), and in his youth he truly was doing geometry, and not merely category theory. With age Grothendieck lost a bit of his creativity and he mainly cared about excessive generalisations.
3) "Grothendieck's philosophy that it's better to have a good category with bad objects than a bad category consisting only of good objects." Can you give a reference showing that this is truly due to Grothendieck?