Homotopy Theory

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7:18 AM

@user616128 Often "geometry" (at least at the level of research mathematics) refers to things to are related to manifolds or "differential" geometry, in my experience. But I think this really depends on the circles you're traveling in. If you're talking to a number theorist for instance, Spec(Z) will definitely be "geometric," I think especially because it is considered as the counterpart to Z, which is "algebraic."

In other words, algebraic geometry is the process of studying algebra (commutative rings) via "geometric" objects (in this case spaces, or schemes).

But of course, the Zariski topology is not really "geometric" in the sense that it doesn't behave in any way like Euclidean space (the original home of "geometry").

But ultimately, I guess, whether or not you think that a derived category of sheaves is "geometric" depends on what you think the word "geometric" means. Or perhaps it depends on whatever the people around you think "geometric" means.

11 hours later…

Denis Nardin

6:30 PM

@JonathanBeardsley I'd object to that :P

Saal Hardali

7:04 PM

For a long time i've been carrying a question doe which i think this is an appropriate context for:

Let $k$ be an algebraically closed field for simplicity. Is there a definition of dimension of a functor $X:Alg_k\to Set$ at a point $x \in X(k)$ which specializes to the zariski dimension $dim_x(X)$ when $X$ is a scheme?
It seems this is one of the very few concepts in scheme theory in LRS language which I don't know how to define from a functor of points perspective.

1 hour later…