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At the heart of algebraic geometry lies the op-equivalence between commutative rings and affine schemes, i.e., $$\mathrm{CRing}^{\mathrm{op}}\simeq \mathrm{Aff\,Sch}.$$

At the heart of homotopy theory lies the equivalence of $\infty$-groupoids and spaces, i.e., $$\infty\text{-}\mathrm{Gpd}\simeq \mathrm{Spc}.$$

On the lefthand sides are (different but related) algebraic structures and on the right hand side are (different but related kinds of) spaces. Can something nontrivial (and cool) be said about the similarities here?

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    $\begingroup$ I think yes but only in a general, vague sense. Both are concerned with the bigger program of geometry/algebra duality in general, but I fail to see any deep connections... $\endgroup$
    – xuq01
    Commented Oct 31, 2022 at 16:04
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    $\begingroup$ The fact that the opposite category is involved in one but not the other seems problematic. $\endgroup$
    – Will Sawin
    Commented Oct 31, 2022 at 16:29
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    $\begingroup$ This reminds me of an old joke about the Grothendieck–Riemann–Roch theorem... $\endgroup$
    – Zhen Lin
    Commented Oct 31, 2022 at 22:43
  • $\begingroup$ groups and rings are related and so are affine schemes and spaces. $\endgroup$
    – Ola Sande
    Commented Nov 1, 2022 at 7:07

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