nLab
groupoid quantale

Idea

Let GG be a group or a discrete groupoid. Recall that an associative ring is a semigroup (if unital then monoid) in the category of abelian groups and a quantale a semigroup in the category of sup-lattices.

While group(oid) ring G\mathbb{Z}G is the free abelian group with basis GG (resp. Mor(G)Mor(G)) with convolution product making it a ring, the groupoid quantale is the free sup-lattice with basis GG and the convolution product making it a quantale.

Construction-definition

The free sup-lattice on a set XX is the power set P(X)P(X) (set of subsets of XX) with the infinite union and finite intersection as sup \Vee (infinite join) and meet \wedge respectively; or equivalently the set 2 X2^X of boolean functions on XX. If X=Mor(C)X = Mor(C) is a morphism set of a category (special cases: semigroup, group, groupoid) then P(Mor(C))P(Mor(C)) has the multiplication

AB={ab|aA,bB,aandbcompose} A B = \{ a b \,|\, a\in A, b\in B, a\,\,\, and\,\,\, b\,\,\, compose \}

or, in terms of functions (the characteristic function of the subsets of CC) of f,g2 Mor(C)f,g\in 2^{Mor(C)}, the convolution product

(f*g)(c)={f(a)f(b)|c=ab} (f \ast g)(c) = \Vee \{ f(a)\wedge f(b)\,|\, c = a b \}

This way, 2 Mor(C)2^{Mor(C)} becomes a quantale denoted simply 2 C2^C, which may be called a category quantale. This quantale is unital; namely the unit in P(Mor(C))P(Mor(C)) is the set of units Ob(C)=C 0Ob(C) = C_0 of the quantale which is canonically identified with a subset of Mor(C)Mor(C). The unit for the convolution is, equivalently, the characteristic function of C 0C_0.

If C=GC = G is a groupoid, then there is an additional involution \star on P(Mor(G))P(Mor(G)) where

X ={g 1|gX} X^\star = \{ g^{-1} \,|\, g\in X \}

for XMor(G)X \subset Mor(G). In terms of the convolution quantale 2 G2^G,

f *(g)=f(g 1),f2 Mor(G),gG. f^*(g) = f(g^{-1}), \,\,\,\,f\in 2^{Mor(G)}, g\in G.

This groupoid quantale is therefore an involutive unital quantale.

Literature

  • Pedro Resende, Étale groupoids and their quantales, Advances in Mathematics 208 (2007) 147–209 doi

Created on December 6, 2015 at 06:20:53. See the history of this page for a list of all contributions to it.