# nLab Gabriel–Ulmer duality

Contents

### Context

category theory

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

# Contents

## The idea

### Plain case

Gabriel–Ulmer duality says that there is an equivalence of 2-categories

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

where Lex is the 2-category of:

and LFP is the 2-category of

The idea is that an object $C \in Lex$ can be thought of as an essentially algebraic theory, which has a category of models $Lex(C,Set)$.
Gabriel–Ulmer duality says that this category of models is locally finitely presentable, all LFP categories arise in this way, and that we can recover the theory $C$ from its category of models.

This duality may be exhibited as the 2-Chu construction $Chu(Cat,Set)$.

There are similar dualities for other classes of theory such as regular theories.

### Enriched case

A version of Gabriel-Ulmer duality for enriched category theory was proved by Max Kelly (see Lack & Tendas 2020):

For base of enrichment $\mathcal{V}$ which is a symmetric monoidal closed category which is complete and cocomplete and locally finitely presentable as a closed category, consider:

1. $\mathcal{V}$-$Lex$ the 2-category of

2. $\mathcal{V}$-$LFP$ the 2-category of

Then there is a biequivalence

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

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

## References

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):

A 2-category theoretic analogue:

A formal category theoretic account using KZ-doctrines:

Discussion the context of enriched category theory:

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

• Max Kelly, Structures defined by finite limits in the enriched context, Cahiers de Topologie et Géométrie Différentielle catégoriques, 23 1 (1982) 3-42 [numdam, MR648793]

For an extension of Gabriel–Ulmer duality to a duality between Cauchy-complete clans and locally finitely presentable categories equipped with a well-behaved kind of weak factorization system see

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 $(\infty, 1)$-categories and locally finitely presentable $(\infty, 1)$-categories see this MO discussion.

Last revised on June 22, 2024 at 13:14:43. See the history of this page for a list of all contributions to it.