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$\DeclareMathOperator\IndCoh{IndCoh}\DeclareMathOperator\QCoh{QCoh}$I've seen it written (for example, in Gaitsgory–Rozenblyum) that for a scheme $X$, the category $\IndCoh(X)$ is to be thought of as a categorificaiton of the space of distributions on $X$, just as the (derived) category of quasi-coherent sheaves is to be thought of as a categorification of functions on $X$.

Why exactly is $\IndCoh(X)$ to be thought of as dual to $\QCoh(X)$? The only relation between these categories that I know of is that $\IndCoh(X)$ is a module over $\QCoh(X)$. Is there some way that an ind-coherent sheaf can act on a quasi-coherent one to yield a categorical analog of a point (for example, an object in $\QCoh(\{ \mathrm{pt} \})$)?

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The connection is perhaps a bit more clear if you think about the corresponding $\infty$-categories of compact objects: the claim is that coherent sheaves behave like distributions while perfect sheaves behave like functions. This connection manifests itself in several ways:

  • Coherent sheaves pushforward along proper maps, while perfect complexes pullback along arbitrary maps. Similarly, distributions integrate along proper maps while functions pullback along arbitrary maps.
  • Skyscrapper sheaves at closed points are coherent but not perfect in general, in the same way that Delta functions are distributions but not functions.
  • When $X$ is proper, the $\infty$-category of coherent sheaves on $X$ is dual to the $\infty$-category of perfect complexes on $X$, see theorem 1.1.3 in https://arxiv.org/abs/1312.7164. Concretely, each coherent sheaf $\mathcal{G}$ defines a functional sending a perfect sheaf $\mathcal{F}$ to the global sections of $\mathcal{F} \otimes \mathcal{G}$.
  • There is a notion of singular support for coherent sheaves, which is a subset of the shifted cotangent bundle measuring how far the sheaf is from being perfect, see https://arxiv.org/abs/1201.6343. Similarly, distributions have a wave front set, which is a subset of the cotangent bundle measuring how far the distribution is from being a smooth function.
  • The Hochschild homology of the $\infty$-category of coherent sheaves on $X$ recovers the space of distributions on the loop space of $X$, while the Hochschild homology of the $\infty$-category of perfect sheaves on $X$ recovers the space of functions on the loop space $X$.
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  • $\begingroup$ Thanks for the thorough answer. Do you have a reference for the last bullet point? $\endgroup$
    – JustLikeNumberTheory
    Commented Feb 21 at 22:10
  • $\begingroup$ @JustLikeNumberTheory I guess that it is contained in Preygel's Thom–Sebastiani paper B.5.1–B.5.2. $\endgroup$
    – Z. M
    Commented Feb 21 at 22:33
  • $\begingroup$ In addition to Preygel, there is also arxiv.org/abs/1305.7175 (theorem 1.4 or proposition 5.3) which explains how Hochschild homology formulas can be deduced from the formulation of these sheaf theories in terms of functors out of $\infty$-categories of correspondences. $\endgroup$
    – G. Stefanich
    Commented Feb 22 at 1:24
  • $\begingroup$ This is a great answer, but I would like to highlight one place where the analogy falters: for smooth spaces both categories are the same (and the delta functions are actual functions). So maybe a tighter analogy would be that Perf is like cohomology/the de Rham complex, and Coh is like Borel-Moore homology/ the complex of currents. $\endgroup$
    – user19232801
    Commented Feb 22 at 3:39
  • $\begingroup$ Let me add to this great answer that for an automorphism $\gamma$ of $X$, its trace on Coh (Perf) gives the distributions (functions) on fixed points $X^\gamma$. For $\gamma=Id$ this is the last bullet point (or, taking S^1-Tate construction, the previous commenter's analogy). But for example taking $\gamma$=Frobenius acting on $X$=local systems on a curve over $\overline{\mathbb F}_q$ we get distributions / functions on local systems over $\mathbb F_q$ -- this is the coherent counterpart to the Grothendieck function/sheaf correspondence for etale sheaves. $\endgroup$
    – David Ben-Zvi
    Commented Mar 8 at 5:58

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