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The answer to your question really depends on just how closely you want the theory of "algebraic theories relative to $\mathcal M$" to mirror the theory of classical ($S$-sorted) algebraic theories. P …

In fact, the converse holds: every cartesian closed locally presentable category is a reflective subcategory of a presheaf category whose reflector preserves finite products. First, a locally presenta …

The original reference for the essential algebraicity of elementary toposes is Freyd's Aspects of topoi (in which the notion of essentially algebraic theory, which is equivalent to that of a finite li …

It appears that the definition of 2-category was introduced independently by two authors, both of whom independently introduced the modern notion of enriched category, for which 2-categories appeared …

Regarding monadicity (rather than comonadicity), the (2-categorical variant of the) question is answered in Bunge–Carboni's The symmetric topos. In their paper, $\mathbf A$ denotes the 2-category of l …

In Di Liberti–Ramos González's Gabriel-Ulmer duality for topoi and its relation with site presentations, they state (Theorem 3.17) that $\mathbf{Ind}(\mathscr C)$ is a Grothendieck topos if and only i …

There are two concepts here, which are tightly connected. Logically, this corresponds to the distinction between $\vdash$ and $\Rightarrow$. (A) Morphisms $t : \Gamma \to A$ represent (well-formed, ty …