Moreover, when we consider an $R$-algebra $A$ s.t $A \models F$, then there's a unique map of rings into the Grothendieck ring of definable sets $K_0(F) \to K_0(Def_A)$* ; generally, this is not an isomorphism.

* $K_0(Def_M)$ is the Grothendieck ring of definable sets, see (https://arxiv.org/abs/math/0510133) for a precise definition.

defined by $\eta$ comes from a bijection between $\varphi$ and $\psi$-definable sets. We then can define the Grothendieck group of $F$, denoted $K_0(F)$, as a quotient of the free abelian group on $\sim$-equivalence classes of $\mathcal{L}$-formulae by its subgroup of formulae constructed as $[\varphi \wedge \psi] + [\varphi \vee \psi] − [\varphi] − [\psi]$ (where $\varphi, \psi$ have the same free vars). This becomes a ring by taking $[\varphi] \cdot [\psi]$ to be $[\varphi \wedge \psi]$.

Consider a ring $R$, and let $\mathcal{L}$ denote the language of rings over $R$ (e.g formulae composed of quantifiers and equalities of $R$-valued polynomials). It's possible to take a "Grothendieck group" of an $\mathcal{L}$-theory $F$.

First, define an equivalence relation $\sim$ on $\mathcal{L}$-formulae as $\varphi(x_1,..., x_n) \sim \psi(y_1,..., y_m) \iff$ there exists $\eta(x_1,...,x_n, y_1,..., y_m)$ s.t for every $R$-algebra $A$ where $A \models F$, the subset of $A^(m+n)$ ...

...giving an equivalence between the $\infty$-category of commutative algebra objects of $Shv_{Sp}(X)$ and the $\infty$-category of $E^{\infty}$-ring-valued sheaves on X.

- Given a sheaf of $E^\infty$-rings $\mathcal{O}$ on an $\infty$-topos X, we can take $f \circ \mathcal{O}$ to obtain a spectral sheaf on X, giving an equivalence between the $\infty$-category of commutative algebra objects of $Shv_{Sp}(X)$ and the $\infty$-category of E∞-ring-valued sheaves on X.

- It can be shown that f preserves small limits and is conservative...

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Joined Nov 2018