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LSpice
LSpice
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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 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?

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?

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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YCor
YCor
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Is the equivalence $CRing^$\mathrm{CRing}^{\mathrm{op}}\simeq AffSch$\mathrm{AffSch}$ related to the homotopy hypothesis?

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

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

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?

Is the equivalence $CRing^{op}\simeq AffSch$ related to the homotopy hypothesis?

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

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

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?

Is the equivalence $\mathrm{CRing}^{\mathrm{op}}\simeq \mathrm{AffSch}$ related to the homotopy hypothesis?

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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Ola Sande
Ola Sande
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Is the equivalence $CRing^{op}\simeq AffSch$ related to the homotopy hypothesis?

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

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

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?