2.6 Diagrams and colimits
2.6.2 Diagrams of C ∗ -correspondences
In this section, we describe diagrams inCorr, transformation between such diagrams and modification between transformations. Our reference for this is [11, §4], where the same is worked out for the bicategory of C∗-algebras with nondegenerate
∗-homomorphisms as arrows and unitary intertwiners as 2-arrows.
Proposition 2.44. A functor C →Corrconsists of
• C∗-algebras Ax for all objects x of C;
• correspondences Eg:Ax→Ay for all arrows g:x→y in C;
• isomorphisms of correspondences µg,h: Eh ⊗Ay Eg → Egh for all pairs of composable arrows g:y→z,h:x→y in C;
such that
(1) E1x is the identity correspondence onAx for all objects x of C;
(2) µ1y,g:Eg⊗AyAy → Eg and µg,1x:Ax⊗AxEg → Eg are the canonical
Here g02:=g12◦g01, g13:=g23◦g12, and g03:=g23◦g12◦g01.
The diagram (2.31) commutes automatically if one of the arrows g01, g12
or g23 is an identity arrow.
Proposition 2.45. Let (A0x,Eg0, µ0g,h) and (A1x,Eg1, µ1g,h) be two functors from C to Corr. A transformation between them consists of
• correspondences γx from A0x to A1x for all objects x of C;
xγx is the canonical isomorphism for each object x in C;
The diagram (2.32) commutes automatically ifg or h is an identity arrow.
Proposition 2.46. Let(A0x,Eg0, µ0g,h)and (A1x,Eg1, µ1g,h)be functors fromCtoCorr and let (γx1, Vg1) and (γx2, Vg2) be transformations between them. A modification from(γx1, Vg1)to(γx2, Vg2)consists of isomorphisms of correspondencesWx:γx1 →γx2 for all objects x in C such that the diagrams
γx1⊗A1
commute for all arrows g:x→y inC. This diagram commutes automatically if g is an identity arrow.
2.6 Diagrams and colimits
2.6.3 Examples
Group actions by homeomorphisms
Lemma 2.47. Let X be a locally compact space and let G be a discrete group.
There is an equivalence between G-actions on X by homeomorphisms and mor-phismsG→Gr which map the unique object in Gto X.
Proof. First, let α:G → Homeo(X) be a group action. That is, αg := α(g) are homeomorphisms on X such that αgαh = αgh for all g, h ∈ G. Let Xg be the correspondence given by Xg := X as a topological space, with anchor mapsrg, sg:Xg →X given by
rg(x) :=x, sg(x) :=αg(x).
for allx∈X. To define a functorG→Grwe still need to define the multiplication isomorphisms; these are given by
µαg,h:Xg×sg,X,rhXh−∼→Xhg, (x, αg(x))7→x.
It is routine to check that (Xg, µαg,h) is indeed a functor G→Gr.
Conversely, let (Xg, µαg,h) be a functor G → Gr with X1 = X. That is Xg:X →X are correspondences andµg,h:Xg◦XXh −→ X∼ hg are isomorphisms of correspondences. The correspondenceXg is an equivalence for each g ∈ G. By Theorem 2.30, it is a Morita equivalence over the space X. This implies that bothr:Xg →X and s:Xg →X are homeomorphisms. Let
αg :=sgr−1g for allg∈G. Then
G→Homeo(X), g7→αg,
defines an action of Gon the spaceX. Moreover, the maps Vg :=sg:Xg →Xg, whereXg:X→X are the correspondences as in the first part of the proof, form an invertible transformation from the functor (Xg, µg,h) to the functor associated to an action ofGon X by homeomorphisms.
Lemma 2.48. Let α:G→Homeo(X) be a group action on a spaceX by homeo-morphism. The colimit of the action inGr is the transformation groupoid GnX.
Proof. Let (Xg, µαg,h) be the functor associated to α: G → Homeo(X) as in Lemma 2.47. Let D be a groupoid and let (Y, Vg) be a transformation from (Xg, µαg,h) to constD. The group Gacts on Y by
g·y :=Vg(rY(y), y). (2.34)
Together with the D-equivariant maprY ∗:Y/G →X, the group action ofGonY gives an action of the transformation groupoidGnX onY that commutes with theD-action (see [31, Proposition 4.10]). This gives a correspondenceY:GnX→ D. Conversely, let Y:GnX → Dbe a correspondence. By [31, Proposition 4.10]
this is equivalent to an action of G onY with a G-map rY:Y → X. It follows thatY:X→ Dis a correspondence using the anchor map rY. We define
Vg(rY(y), y) :=g·y. (2.35) for all g ∈ G and ally ∈ Y. Then Vg: X×id,X,rY Y → Y is an isomorphism of correspondences and (Y, Vg) satisfies the coherence condition in(2.29)and therefore gives a transformation from (Xg, µαg,h) to constD. The above two constructions are inverse to each other and natural in the formal sense. So they show thatGoX is the colimit.
3 Actions of Ore monoids
In this chapter, we study Ore monoid actions in the bicategory of groupoid correspondences. The most basic example of an Ore monoid is the monoid of natural numbersN. An action of N inGr by proper correspondences is equivalent to a single correspondenceX:G → G for a groupoid G. To associate a C∗-algebra to the action given by X, we may first use the functor Gr → Corr to obtain a C∗-correspondence E := C∗(X) : C∗(G) → C∗(G) and then take the Cuntz–
Pimsner algebraOE.
Now we know from [2] that OE is the colimit of the action given by E in the correspondence bicategoryCorr. We will show that the action given byX also has a colimit groupoidHinGr. Theorem 3.36 shows thatH is a groupoid model of the Cuntz–Pimsner algebraOE, that is,OE ∼= C∗(H).
Our construction of the groupoid colimit of an action of an Ore monoidP by correspondence has two steps. We first show that an Ore monoid action by tight correspondences always has a colimit. This is done inSection 3.2. InSection 3.3 we show how to approximatean Ore monoid action by correspondences through an action by tight correspondences in a way that does not change the colimit, see Theorem 3.30.
3.1 Ore monoids
In this section, we introduce Ore monoids and their actions in the bicategory of groupoid correspondencesGr.
Definition 3.1([27, Section IX.1]). A categoryCis filtered if it is nonempty and (1) for any two objects x, y∈ C0, there arez∈ C0,g∈ C(x, z) and h∈ C(y, z);
(2) for any two parallel arrows g, h∈ C(x, y), there are z ∈ C0 and k∈ C(y, z) withkg=kh.
Let P be a monoid and let CP be the category with object set P and arrow setP×P, where (p, q) is an arrow frompto pq, and where (pq, r)·(p, q) := (p, qr) for allp, q, r∈P. The categoryCP is filtered if and only ifP satisfies the following Ore conditions:
(Ore1) for allx1, x2 ∈P, there arey1, y2 ∈P withx1y1 =x2y2;
(Ore2) ifxy1=xy2 fory1, y2, x∈P, then there isz∈P withy1z=y2z.
Definition 3.2. We callP aright Ore monoid if the categoryCP is filtered. We callP a left Ore monoid ifPop is aright Ore monoid.
Condition(Ore2)follows ifP has cancellation. Both hold ifP ⊆Gfor a groupG with P P−1 =G.
Let P be an Ore monoid. We may construct a group completion G= G[P] of P by taking equivalence classes of formal fractions pq−1 := (p, q) ∈ P ×P, where (p1, q1) ∼ (p2, q2) if there are elements a1, a2 ∈ P with (p1a1, q1a1) = (p2a2, q2a2). The product of the elements [p1, q1] and [p2, q2] is given by
[p1, q1][p2, q2] := [p1t1, q2t2]
where t1, t2 ∈P are such that q1t1 =p2t2. The groupG is determined by the monoidP, and every monoid morphism ϕ:P →H for a groupH extends to a group homomorphism ˜ϕ:G→H. The group homomorphism is simply given by
ϕ[p, q] =˜ ϕ(p)ϕ(q)−1 for all [p, q]∈G.
Example 3.3. Commutative monoids are Ore monoids. Normal monoids are also Ore; here we say that a monoidP is normal ifsP =P s for alls∈P. Non-abelian free monoids are examples of monoids which do not satisfy the Ore conditions.
An action of an Ore monoid P in the bicategory Gris a functor F:Pop → G (see Proposition 2.40). Here we think of an Ore monoid as a category with one object (the identity) and elements of the monoid as arrows; the product of two arrowsp and q is pq.
We include the following definition for convenience.
Definition 3.4. Let G be a groupoid and let P be an Ore monoid. An action of Pop onG by correspondences consists of the following data:
• correspondences Xp:G → G forp∈P\ {1};
• isomorphismsσp,q:Xpq→ Xp◦GXq forp, q∈P\ {1}.
We assume that X1 =G is the identity correspondence and that σp,1 and σ1,q are the canonical homeomorphismsXp ∼=Xp◦GG and Xq∼=G ◦GXq forp, q∈P. For an action ofP, we also require the diagram
Xp◦GXq◦GXt Xpq◦GXt
Xp◦GXqt Xpqt σp,q◦GidXt
idXp◦Gσq,t σpq,t
σp,qt
(3.1)