Gabriel-Ulmer duality

Gabriel–Ulmer duality

The idea

Gabriel–Ulmer duality says that there is an equivalence of 2-categories (or in other words, a biequivalence)

Lex op LFP C Lex(C,Set) \begin{matrix} Lex^{op} & \to & LFP \\ C & \mapsto & Lex(C, Set) \end{matrix}

where Lex is the 2-category of:

and LFP is the 2-category of

The idea is that an object CLexC \in Lex can be thought of as an essentially algebraic theory, which has a category of models Lex(C,Set)Lex(C,Set). Gabriel–Ulmer duality says that this category of models is locally finitely presentable, all LFP categories arise in this way, and we can recover the theory CC from its category of models. There are similar dualities for other classes of theory such as regular theories.

A version of Gabriel–Ulmer duality for enriched category theory was proved by Max Kelly (see LackTendas). Let 𝒱\mathcal{V} be a symmetric monoidal closed complete and cocomplete category which is locally finitely presentable as a closed category. Then let 𝒱\mathcal{V}-LexLex be the 2-category of finitely complete 𝒱\mathcal{V}-categories (𝒱\mathcal{V}-categories with finite weighted limits), finite limit preserving 𝒱\mathcal{V}-functors, and 𝒱\mathcal{V}-natural transformations, and 𝒱\mathcal{V}-LFPLFP the 2-category of locally finitely presentable 𝒱\mathcal{V}-categories, right adjoint 𝒱\mathcal{V}-functors that preserve filtered colimits, and 𝒱\mathcal{V}-natural transformations. Then there is a biequivalence

𝒱Lex op 𝒱LFP C Lex(C,𝒱). \begin{matrix} \mathcal{V}-Lex^{op} & \to & \mathcal{V}-LFP \\ C & \mapsto & Lex(C, \mathcal{V}). \end{matrix}

For instance, in the truth value-enriched case, the duality is between meet semilattices and algebraic lattices.

Gabriel-Ulmer duality is a duality exhibited by the 2-Chu construction, Chu(Cat,Set)Chu(Cat,Set).


The original source is:

A careful discussion and proof of the biequivalence is in

Some other general treatments of Gabriel-Ulmer duality (and generalizations to other doctrines):

For a 2-dimensional analogue see the slides from a 2010 talk by Makkai: pdf

A formal-categorical account using Yoneda structures can be found in

For a discussion of Gabriel–Ulmer duality and related dualities in the context of enriched category theory see

This discusses (see Theorem 2.1) Kelly’s original result for VV-enriched categories, where VV is a closed symmetric monoidal category whose underlying category V 0V_0 is locally small, complete and cocomplete, in section 9 (cf. theorem 9.8) of

For a connection to Tannaka duality theory see

For a discussion of an \infty-version of Gabriel-Ulmer duality between finitely complete and idempotent complete (,1)(\infty, 1)-categories and locally finitely presentable (,1)(\infty, 1)-categories see this MO discussion.