# Isbell duality between algebras and sheaves

nLab says on Isbell duality, the following:

A general abstract adjunction $$(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves} \leftrightarrows \mathrm{Presheaves}$$ relates (higher) presheaves with (higher) copresheaves on a given (higher) category $$C$$: this is called Isbell conjugation or Isbell duality (after John Isbell).

To the extent that this adjunction descends to presheaves that are (higher) sheaves and copresheaves that are (higher) algebras this duality relates higher geometry with higher algebra.

I can not find a reference for this claim about sheaves/algebras. How are algebras defined here are these algebras for some functor or a monad? Anyone knows about a paper with a proof of this adjunction descending into sheaves and algebras?

Edit: I think I might have found the correct definition for the algebras as co-presheaves here: https://ncatlab.org/nlab/show/function+algebras+on+infinity-stacks#TAlgebras, but assuming this is correct still leaves open the following questions: For the copresheaves on $$T$$ (presheaves on $$T^{op}$$), that are function algebras, sheaves on which site should these be adjoint to? Or maybe is there a generalization that would work over any site?

• One example is the case where C is a lex category, which we can view as an "essentially algebraic theory". Then the presheaf category $[C^{op},Set]$ is the classifying topos of the algebraic theory, and on the other side the adjunction between $[C^{op},Set]$ and $[C,Set]^{op}$ descends to the category $Lex(C,Set)$ of C-algebras, ie points of $[C^{op},Set]$. Dec 19, 2022 at 21:21
• The section here, ncatlab.org/nlab/show/…, treating T-algebras as a full subcategory of copresheaves seems like a plausible candidate for the restriction of CoPresheaves, there they are called presheaves on C^op, still have to look into references or try working out the proof myself.
– Ilk
Dec 20, 2022 at 7:55