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Gabriel-Ulmer duality

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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 C∈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.

References

The original source is:

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

The version for V-enriched categories, where V is closed symmetric monoidal category whose underlying category V 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, I. Cahiers de Topologie et Géométrie Différentielle catégoriques, 23 no. 1 (1982), p. 3-42, MR648793,numdam

For a connection to Tannaka duality theory see

Revised on November 15, 2012 14:26:26 by Tim Porter (95.147.237.28)
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