Quantales and quantaloids
A (unital) quantale is a monoid object in the monoidal category of sup-lattices . Equivalently, it is a closed monoidal preorder having all joins. Quantales occur often as a base for enriched categories. Modules over quantales, such as Hilbert Q-modules , are also useful.
A quantaloid is the horizontal categorification of a quantale, specifically a category enriched in sup-lattices. Thus, a quantale is a quantaloid with one object.
Examples
Quantales
- Booleans \(\mathbb{B} = (\{\top, \bot\}, \leq, \wedge, \top)\)
- Extended nonnegative reals \(\overline{\mathbb{R}}_+ = ([0,\infty], \geq, +, 0)\)
- Extended reals \(\overline{\mathbb{R}} = ([-\infty, \infty], \geq, +, 0)\)
- Binary relations \(\mathbf{Rel}(X,X) = (\mathcal{P}(X \times X), \subseteq, \cdot, 1_X)\) on any set \(X\)
Quantaloids
- Category of relations \(\mathbf{Rel}\)
- For any quantale \(Q\), the category of \(Q\)-valued matrices
Literature
Books
- Joyal & Tierney, 1984: An extension of the Galois theory of Grothendieck
(doi)
- Regarded as a seminal work on Grothendieck toposes
- According to Paseka, the “categorical approach to quantales” starts here
- Rosenthal, 1990: Quantales and their applications
- Rosenthal, 1996: The theory of quantaloids
- Eklund et al, 2018: Semigroups in complete lattices: quantales, modules, and related topics (doi)
- Fong & Spivak, 2019: Seven sketches in compositionality, Ch. 2: Resource theories: Monoidal preorders and enrichment
Surveys
- Paseka & Rosicky, 2000: Quantales (doi)
- Resende, 2000: Quantales and observational semantics (doi)
- Kruml & Paseka, 2008: Algebraic and categorical aspects of quantales (doi),
chapter in Handbook of Algebra, Vol 5
- Covers quantales and modules over quantales, including free quantales and free quantale modules
- Stubbe, 2014: An introduction to quantaloid-enriched categories (doi, pdf)
- Fujii, 2019: Enriched categories and tropical mathematics (arxiv)
- See also enriched categories and
Limits and colimits
- Shen & Tholen, 2015: Limits and colimits of quantaloid-enriched categories and
their distributors (pdf, arxiv)
- Abstract: “It is shown that, for a small quantaloid \(Q\), the category of small \(Q\)-categories and \(Q\)-functors is total and cototal, and so is the category of \(Q\)-distributors and \(Q\)-Chu transforms.”
Bilinear maps and inner products