Gabriel–Ulmer duality says that there is an equivalence of 2-categories (or in other words, a biequivalence)
where Lex is the 2-category of:
and LFP is the 2-category of
finitary right adjoint functors and
The idea is that an object can be thought of as an essentially algebraic theory, which has a category of models . 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 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 be a symmetric monoidal closed complete and cocomplete category which is locally finitely presentable as a closed category. Then let - be the 2-category of finitely complete -categories (-categories with finite weighted limits), finite limit preserving -functors, and -natural transformations, and - the 2-category of locally finitely presentable -categories, right adjoint -functors that preserve filtered colimits, and -natural transformations. Then there is a biequivalence
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, .
The original source is:
A careful discussion and proof of the biequivalence is in
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
For a discussion of Gabriel–Ulmer duality and related dualities in the context of enriched category theory see
This discusses (see Theorem 2.1) Kelly’s original result for -enriched categories, where is a closed symmetric monoidal category whose underlying category is locally small, complete and cocomplete, in section 9 (cf. theorem 9.8) of
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 -version of Gabriel-Ulmer duality between finitely complete and idempotent complete -categories and locally finitely presentable -categories see this MO discussion.
Last revised on April 18, 2023 at 07:56:36. See the history of this page for a list of all contributions to it.