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Idea

Higher geometry studies vertical categorifications of the notions of space and geometry.

What exactly the rules of higher geometry are is still a bit in the making. A very encompassing framework has been proposed in

• Jacob Lurie, Structured Spaces .

Here the notion of geometry is formalized in the form of an (∞,1)-category $𝒢$ whose objects play the role of test-spaces on which all other spaces are modeled, in a hierarchy of generalized objects:

• test spaces $\hookrightarrow$ spaces locally equivalent to test spaces $\hookrightarrow$ concrete spaces with structure sheaves taking values in test spaces $\hookrightarrow$ spaces probeable by test spaces.

technically modeled by:

• geometry (for structured (∞,1)-toposes) $𝒢$ $\hookrightarrow$ generalized schemes $\hookrightarrow$ formal duals to $𝒢$-structured (∞,1)-toposes $\hookrightarrow$ (∞,1)-topos of ∞-stacks on $𝒢$.

A plethora of proposals for formalizations of higher geometry find their home in this pattern, for instance most of the concepts listed at generalized smooth space.

A notable exception to this is possibly the program by Maxim Kontsevich and others where under the term noncommutative geometry and derived noncommutative geometry spaces are modeled as the formal dual to A-∞-categories. But $A_{\mathrm{\infty }}$-categories are presentations for stable (∞,1)-categories and by the stable Giraud theorem presentable stable $(\mathrm{\infty }\mathrm{,1})$-categories play a very similar role to (unstable) ∞-stack (∞,1)-toposes. In particular they may be obtained from the latter by stabilization.

Derived geometry

Higher geometry has been particularly popularized in terms of derived geometry, notably derived algebraic geometry.

In typical usage the qualifier derived here (which otherwise can mean many things in homotopy theory and higher category theory) is meant to emphasize that the collection $𝒢$ of test spaces is a genuine (∞,1)-category instead of just a plain 1-category. Accordingly ∞-stacks on $𝒢$ are then called derived stacks (see there for more on this).

The point of amplifying this, historically, is that typically only with a suitble higher category $𝒢$ of test spaces, the limits and the intersection theory? of the generalized spaces behaves nicely.

The standard and motivating example for this to keep in mind is the case of derived algebraic geometry where the site $𝒢=$ CRing${}^{\mathrm{op}}$ of formal duals to ordinary commutative rings is generalized to the site $𝒢={\mathrm{sCRing}}^{\mathrm{op}}$ of formal duals to simplicial rings (and then further to that of formal duals of E-∞-rings):

where an ordinary affine scheme $\mathrm{Spec}R$ has a ring $R$ of functions on it, a derived scheme $\mathrm{Spec}{R}_{•}$ has a simplicial ring of functions on it. Under the monoidal Dold-Kan correspondence this is equivalently a non-positively graded cochain dg-algebra $N^{•}(R_{•})$:

If this complex has vanishing cohomology everywhere except in degree 0, it is a homological resolution of the quotient $R_0/d(R_1)$, which may not itself exist, dually, as a test object in $𝒢$.

In many fields it is a traditionally a well-known tool that such resolutions encode important information. Standard examples include the bar complex? used to define Hochschild homology or the homological resolutuions used in BV theory. Derived geometry may be understood from this perspective as providing a geometrical interpretation of these techniques.

For instance from the point of view of derived geometry, the Hochschild homology of a ring of functions on an object $X$ is the functions on the free loop space object $ℒX$ of $X$. And the corresponding cyclic cohomology is the $S^1$-equivariant part of this, where the circle acts in the natural way on the derived loop space $ℒX$. If nothing else, this geometric picture makes transparent all of the algebraic structures in Hochschild cohomology and cyclic cohomology. See there for more details.