Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Gabriel–Ulmer duality says that there is an equivalence of 2-categories
where Lex is the 2-category of:
finite limit$\:$ preserving functors, and
and LFP is the 2-category of
finitary right adjoint functors and
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 that we can recover the theory $C$ from its category of models.
This duality may be exhibited as the 2-Chu construction $Chu(Cat,Set)$.
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 Lack & Tendas 2020):
For base of enrichment $\mathcal{V}$ which is a symmetric monoidal closed category which is complete and cocomplete and locally finitely presentable as a closed category, consider:
$\mathcal{V}$-$Lex$ the 2-category of
finitely complete $\mathcal{V}$-categories ($\mathcal{V}$-categories with finite weighted limits),
$\mathcal{V}$-$LFP$ the 2-category of
Then there is a biequivalence
For instance, in the truth value-enriched case, the duality is between meet semilattices and algebraic lattices.
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, Enrico M. Vitale, A duality relative to a limit doctrine, Theory and Appl. of Categories 10 20 (2002) 486-497 lbrack;tac:10-20, pdf]
Stephen Lack, John Power, Gabriel-Ulmer duality and Lawvere Theories enriched over a general base, Journal of Functional Programming 19 3-4 (2009) 265-286 [doi:10.1017/S0956796809007254, pdf]
Michael Makkai, Andrew Pitts, Some results on locally finitely presentable categories, Trans. Amer. Math. Soc. 299 (1987) 473-496 [doi:10.2307/2000508, pdf, MR88a:03162]
A 2-category theoretic analogue:
A formal category theoretic account using KZ-doctrines:
Discussion the context of enriched category theory:
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
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
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.
Last revised on June 22, 2024 at 13:14:43. See the history of this page for a list of all contributions to it.