Nobody knows where $$NSOP_1$$ theories came from. The original definition was written down by Shelah at some point. When asked later why he came up with it, he had no idea.

Categorical theories, model categories, and categorical model theory — all distinct concepts.

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 (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 $A^(m+n)$ ...

Imagine mentioning "symplectic meanders associated to seaweed" to someone, and imagine them believing it is not part of an elaborate joke. This cannot be done.

Watching a finch outside my window repeatedly pick up/chew/spit out pebbles. "He clearly has not learned" I think to myself. Later, the finch sits on my windowsill as I attempt to prove something about $NSOP_1$ theories, and thinks the same.

Moral of this? Eat pebbles in the garden.

Rodoljub the sparrow, who has made many of my diagrams commute.

...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.

The forgetful functor $f:E^\infty Ring \to Spectra$ has at least two interesting uses that come to mind:
- 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...

A Mastodon instance named Mathtodon, where you can post toots with beautiful mathematical formulae in TeX/LaTeX style.