Gabriel-Ulmer duality

GabrielUlmer 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.


The original source is:

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 Yoneda structures can be found in

  • Ivan Di Liberti, Fosco Loregian, Accessibility and Presentability in 2-Categories , arXiv:1804.08710 (2018). (abstract)

The version for VV-enriched categories, where VV is closed symmetric monoidal category whose underlying category V 0V_0 is locally small, complete and cocomplete is in section 9 (cf. theorem 9.8)

  • G. M. 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 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.

Last revised on May 17, 2018 at 15:00:34. See the history of this page for a list of all contributions to it.