nLab Gabriel–Ulmer duality

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:

• smallfinitely complete categories,

• finite limit$\:$ preserving functors, and

• natural transformations,

and LFP is the 2-category of

• locally finitely presentable categories,

• finitary right adjoint functors and

• natural transformations.

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

   • finitely complete $\mathcal{V}$-categories ($\mathcal{V}$-categories with finite weighted limits),

   • finite limitpreserving $\mathcal{V}$-functors,

   • $\mathcal{V}$-natural transformations,

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

   • locally finitely presentable$\mathcal{V}$-categories,

   • rightadjoint $\mathcal{V}$-functors that preserve filtered colimits,

   • $\mathcal{V}$-natural transformations.

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:

• Peter Gabriel, Friedrich Ulmer, Lokal präsentierbare Kategorien, Lecture Notes in Mathematics 221, Springer (1971) [doi:10.1007/BFb0059396]

A careful discussion and proof of the biequivalence is in

• Jiří Adámek, Hans-Eberhard Porst, Algebraic Theories of Quasivarieties, J. Algebra 208 (1998) 379-398 [doi:10.1006/jabr.1998.7499, pdf]

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

• C. Centazzo, Enrico M. Vitale, A duality relative to a limit doctrine, Theory and Appl. of Categories 10 20 (2002) 486-497 lbrack;tac:10-20, pdf]

• Stephen Lack, John Power, Gabriel-Ulmer duality and Lawvere Theories enriched over a general base, Journal of Functional Programming 19 3-4 (2009) 265-286 [doi:10.1017/S0956796809007254, pdf]

• Michael Makkai, Andrew Pitts, Some results on locally finitely presentable categories, Trans. Amer. Math. Soc. 299 (1987) 473-496 [doi:10.2307/2000508, pdf, MR88a:03162]

A 2-category theoretic analogue:

• Michael Makkai, Gabriel-Ulmer duality, talk at ASL annual meeting (2010) [scan: pdf, pdf]

A formal category theoretic account using KZ-doctrines:

• Ivan Di Liberti, Fosco Loregian, Accessibility and Presentability in 2-Categories, Journal of Pure and Applied Algebra 227 1 (2023) [arXiv:1804.08710, doi:10.1016/j.jpaa.2022.107155]

Discussion the context of enriched category theory:

• Stephen Lack, Giacomo Tendas, Enriched regular theories, J. Pure Appl. Alg. 226, n. 6 (2020) 106268, (arXiv:1907.02301, doi:10.1016/j.jpaa.2019.106268)

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

• Jonas Frey, Duality for Clans: a Refinement of Gabriel-Ulmer Duality (arXiv:2308.11967)

For a connection to Tannaka duality theory see

• nCafé discussion here

• Brian Day, Enriched Tannaka duality, JPAA 108 (1996) pp.17-22, MR97d:18008 doi

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.