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]]>I disagree slightly but significantly with Harry about what is understood as a subject of classical algebraic geometry. It is true, it is an algebraic geometry of varieties over fields, mainly (but not only) algebraically closed. But when you come to a conference in algebraic geometry then you will often see talks using all kinds of modern machinery in your sense (I mean schemes, stacks, triangulated categories etc.) and saying we are proving such and such interesting result in classical algebraic geometry -- on curves, surfaces, intersection numbers and so on. Arguably, the works of Italian school center around things like calculating some sheaf cohomologies. They were doing resolvents in terms of construction successively families of various geometric objects ("pencils" and alike). I listened a modern exposition of a classical proof of Catelnuovo from around 1905 or so, it was one hour of complicated (even in modern language) cohomological constructions (the expositor was Makai).

On the other hand, calling Grothendieck school of algebraic geometry semi-classical does not do it right. First of all because motives which Grothendieck thought of at the end of 50-s, with all the philosophy behind it (correspondences, usage of Chow moving lemma to do deformation equivalences like in homotopy theory, Galois philosophy), and then the Grothendieck's works in his return manuscripts 1981 and 1984 (there main works: Esquizze, Pursuing stacks and Long march toward (higher) Galois theory) which sketched how to finish the program with mixing homotopy ideas (and contained, as Maltsiontis has shown, essentially also Batanin's def. of higher weak categories, the ideas on relations between homotopy types, higher Galois theory and higher descent). Simpson's school has realized part of that program at the end of 1990-s what postdated Voevodsky's ideas by several years which are already homotopical. Saying model category or saying quasi-category in working out concrete problems does not make it more or less modern a priori. Making it free of models in a way Grothendieck's homotopy theory of derivators (1992 and before) is arguably more modern in a way (and Cisinski is much beyond it now in pure homotopy theory, with new item "categories derivables" for which derivators are just an intermidiate notion).

I have trouble seeing periods, in various parts of the subject "modern" level of generality and point of view is taken at different points in time. Compare that Illusie had deformation complex in mid 1960s lead by Grothendieck, while this is just talking certain tangent spaces in derived geometry (and most of the simplicial machinery is there and it motivated Illusie to propose the weak equivalence part of model structure on simplicial presheaves later introduced in full by Joyal and then refined by Jardine; then see Brown's work in mid 1970-s and Beilinson's ideas on derived (category of qcoh sheaves) algebraic geometry from 1976/1977 taken up by Kapranov Bondal in mid 1990s. Notice that already such advanced homotopy/infty category constructions like Postnikov's towers for triangulated categories are used in algebraic geometry by Kapranov and Bondal in 1988/1989). In the same time in noncommutative geometry a la Connes, even the usual derived categories started being used (with long introductions and excuses at the conferences) only with the work of Nest and Ralph Meyer on KK-theory and Baum-Connes few years ago. Kontsevich on the other hand went immediately to deep applications without writing papers with big machinery of derived noncommutative algebraic geometry (base on A infty categories) which he had in his hands in early 1990s. Many top algebraic geometers were learning (in different terminology) some of the machinery on the fly. This has been mostly parallel to Simpson's school but in different way.

]]>So it's not really a word for "non-higher category theory" that you want, but rather a word for "algebraic geometry that uses category theory but not higher category theory"? I don't think I have much input on this; I have enough trouble finding good names for mathematical objects, without also worrying about finding names for styles of doing mathematics! If I had to refer to it, I'd probably use something along the lines of what Toby suggested.

]]>As an adjective to go before something like ‘algebraic geometry’, how about ‘1-category-theoretic’ (or ‘1-categorial’ or ‘1-categorical’)? Or does that put too much emphasis on the category-theoretic aspects?

]]>When we want to talk about, say, the algebraic geometry of 1-Stacks, this is "ordinary" algebraic geometry, since we're dealing with strict 2-categories and pseudofunctors and the like. This isn't modern algebraic geometry, since that role is already taken by the various forms of algebraic geometry on infinity categories. However, it's not classical algebraic geometry either. We can group the "time periods" of algebraic geometry by looking at the structures used. Classical algebraic geometry would be the algebraic geometry of Zariski, using varieties and such. We could call "semi-classical" algebraic geometry the algebraic geometry of the 1960s and 1970s, fueled by Grothendieck and his "school". This type of algebraic geometry utilizes ordinary category theory, which culminated with a general definition of descent for 1-stacks and a generalized notion of a "geometric space," a ringed topos satisfying certain requirements. The thing here is, since we have this generalized notion of a space, we can extend the terminology we'd been applying only to algebraic geometry to Differential geometry. That is, we have "semi-classical" differential geometry is the kind of thing discussed in SGA 4, where the definition of a manifold is brought into line with that of a scheme, and we have defined differential stacks as well. What I would consider "higher" or "modern" geometry would be the groupoidal categorification of semi-classical geometry, so something like the theory described in DAG 5 (Structured Spaces).

The main issue here is that "ordinary" geometry doesn't have the same meaning as "ordinary" category theory. Is it clearer now? ]]>

I'm having trouble parsing your first sentence; I don't see how "classical category theory" could mean the theory of 2-categories. It might be the theory of the 2-category of categories, but I think it's noticeably more than that.

Anyway, I often use a word like "ordinary" for "non-higher" category theory.

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