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It seems to me that there is a difference in the treatment of "partial" elements in boolean-valued models in set theory vs topos theory: in set theory, one usually only considers "global" elements of …

Let $\mathbb{T}$ be a finitary algebraic theory. For each $\mathbb{T}$-algebra $A$, let $Q (A)$ be the join semilattice of finitely generated congruences on $A$. There is an evident pushforward opera …

The Keisler–Shelah theorem implies that the following are equivalent: $M$ and $N$ are elementarily equivalent. For some set $X$ and some ultrafilter $U$ on $X$, $M^X / U$ and $N^X / U$ are isomorphi …

My question is ultimately about the model theory of $L_{\infty, \omega}$, but it is more convenient to phrase it in terms of category theory. Suppose I have finitely accessible categories $\mathcal{C} …

Let $\Sigma$ be a one-sorted first-order signature, let $A$ be a $\Sigma$-structure, and let $B \subseteq A$ be a $\Sigma$-substructure. Fix a class $\mathcal{L}$ of formulae over $\Sigma$. We say an …