2
$\begingroup$

The title pretty much says it all: I've read somewhere of the fact that algebras (i.e. models in $\mathbf{Set}$) for an equational theory can be seen as lex functors from the opposite category of finitely presented algebras to $\mathbf{Set}$. However, I wasn't able to find any proof of this assertion anywhere.

$\endgroup$
6
$\begingroup$

What you want is a corollary to Gabriel-Ulmer duality. For example, your result is Theorem 10 (presented without much proof) in Lack's and Powers' Gabriel-Ulmer Duality and Lawvere Theories Enriched over a General Base. Algebraic Theories of Quasivarieties by Adámek and Porst gives a detailed proof for Gabriel-Ulmer duality. However, that itself requires knowing about locally finitely presentable categories and that finitely presentable objects in a category of models of an algebraic theory are, indeed, finitely presented algebras.

Basically, Gabriel-Ulmer duality states that $\mathbf{Lex}^{op}\simeq\mathbf{LFP}$ where this is a biequivalence of 2-categories. $\mathbf{Lex}$ is the 2-category of small finitely complete categories, finite limit preserving functors, and natural transformations, and $\mathbf{LFP}$ is the 2-category of locally finitely presentable categories, filtered colimit preserving right adjoint functors, and natural transformations. One direction of the biequivalence is via the embedding $\mathbf{Lex}(-,\mathbf{Set}) : \mathbf{Lex}^{op}\to\mathbf{LFP}$.

A locally finitely presentable category is one that has colimits and is generated by a set of finitely presentable objects via filtered colimits, i.e. every object is a filtered colimit of finitely presentable objects. The skeleton of the full subcategory of finitely presentable objects is a small(!) finitely cocomplete category, so its opposite is finitely complete. This is (part of) the part of the biequivalence going from $\mathbf{LFP}\to\mathbf{Lex}^{op}$. Let $\mathcal{C}$ be a locally finitely presentable category, and call $\mathcal{C}_{fp}$ the skeleton of the full subcategory of finitely presentable objects described above. Then $\mathcal{C}_{fp}^{op}\in\mathbf{Lex}$ and $\mathcal{C}\simeq\mathbf{Lex}(\mathcal{C}_{fp}^{op},\mathbf{Set})$ which is the result you want (given that finitely presentable objects live up to their name in categories of models of an algebraic theory).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.