nLab Gabriel–Ulmer duality

The idea

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

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

where Lex is the 2-category of:

• small finitely 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 we can recover the theory $C$ 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 $\mathcal{V}$ be a symmetric monoidal closed complete and cocomplete category which is locally finitely presentable as a closed category. Then let $\mathcal{V}$-$Lex$ be the 2-category of finitely complete $\mathcal{V}$-categories ($\mathcal{V}$-categories with finite weighted limits), finite limit preserving $\mathcal{V}$-functors, and $\mathcal{V}$-natural transformations, and $\mathcal{V}$-$LFP$ the 2-category of locally finitely presentable $\mathcal{V}$-categories, right adjoint $\mathcal{V}$-functors that preserve filtered colimits, and $\mathcal{V}$-natural transformations. Then there is a biequivalence

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

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, $Chu(Cat,Set)$.

References

The original source is:

• Peter Gabriel, Friedrich Ulmer, Lokal Praesentierbare Kategorien, Springer Lecture Notes in Mathematics 221, Berlin, 1971.

A careful discussion and proof of the biequivalence is in

• Jiri Adamek, Hans-Eberhard Porst, Algebraic Theories of Quasivarieties , J. Algebra 208 (1998) pp.379-398.

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

• C. Centazzo, E. M. Vitale, A duality relative to a limit doctrine, Theory and Appl. of Categories 10, No. 20, 2002, 486–497, pdf

• Stephen Lack, John Power, Gabriel–Ulmer duality and Lawvere Theories enriched over a general base, pdf

• M. Makkai, A. Pitts, Some results on locally finitely presentable categories, Trans. Amer. Math. Soc. 299 (1987), 473-496, MR88a:03162, doi, pdf

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

A formal-categorical account using KZ-doctrines can be found in

• Ivan Di Liberti, Fosco Loregian, Accessibility and Presentability in 2-Categories , (arXiv:1804.08710)

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

• 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 no. 1 (1982), pp. 3-42, MR648793,numdam

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.