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 B=({⊤,⊥},≤,∧,⊤)\mathbb{B} = (\{\top, \bot\}, \leq, \wedge, \top)
  • Extended nonnegative reals R‾+=([0,∞],≥,+,0)\overline{\mathbb{R}}_+ = ([0,\infty], \geq, +, 0)
  • Extended reals R‾=([−∞,∞],≥,+,0)\overline{\mathbb{R}} = ([-\infty, \infty], \geq, +, 0)
  • Binary relations Rel(X,X)=(P(X×X),⊆,⋅,1X)\mathbf{Rel}(X,X) = (\mathcal{P}(X \times X), \subseteq, \cdot, 1_X) on any set XX

Quantaloids

  • Category of relations Rel\mathbf{Rel}
  • For any quantale QQ, the category of QQ-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)

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 QQ, the category of small QQ-categories and QQ-functors is total and cototal, and so is the category of QQ-distributors and QQ-Chu transforms.”

Bilinear maps and inner products

  • Paseka, 1999: Hilbert Q-modules and nuclear ideals in the category of V-semilattices with a duality (doi)
    • Paseka, 1999: Hermitian kernels, Hilbert Q-modules, and Ando dilation (ps )
    • Paseka, 2000: Interior tensor product of Hilbert modules (ps )
  • Resende, 2004: Sup-lattice 2-forms and quantales (doi, arxiv)
  • Resende & Rodrigues, 2010: Sheaves as modules (doi, arxiv)
    • Heymans & Stubbe, 2009: Modules on involutive quantales: canonical Hilbert structure, applications to sheaf theory (doi, arxiv)
  • Resende, 2012: Groupoid sheaves as quantale sheaves (doi, arxiv)