category theory

# 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:

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 we can recover the theory $C$ from its category of models.

## References

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

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, Cahiers de Topologie et Géométrie Différentielle catégoriques, 23 no. 1 (1982), pp. 3-42, MR648793,numdam

For a connection to Tannaka duality theory see

Revised on May 8, 2016 13:23:32 by Thomas Holder (176.7.89.208)