aa r X i v : . [ m a t h . C T ] F e b Slices of groupoids are group-like
Nicholas Cooney Jan E. Grabowski ‡ February 10, 2020
Abstract
Given a category, one may construct slices of it. That is, one builds a new category whose objectsare the morphisms from the category with a fixed codomain and morphisms certain commutativetriangles. If the category is a groupoid, so that every morphism is invertible, then its slices are(connected) groupoids.We give a number of constructions that show how slices of groupoids have properties even closerto those of groups than the groupoids they come from. These include natural notions of kernels andcoset spaces.MSC (2020): 18B40 (Primary), 20N02 (Secondary)
A groupoid is a category in which every morphism is invertible. Since a one-object groupoid is naturallyidentified with a group, groupoids are regarded as a many-object generalisation of a group. Howevergroupoids do not have a completely parallel theory to that of groups. For example, a groupoid does nothave an identity element.In this short note, we observe that by passing from a groupoid to one of its slices, one finds a situationcloser to that of groups. A slice of a category C at an object X is a category whose objects are morphismsof C with codomain X and morphisms certain commutative triangles.In particular, since any object has an identity map id X : X → X , the slice of a groupoid has adistinguished object id X which, of course, has the properties of an identity.Given a functor F between groupoids G and H , we have an induced functor between G /X and H / F X .With some relatively mild assumptions, this induced functor behaves like a group homomorphism: wemay define its kernel, prove an analogue of the correspondence between cosets of a kernel and elementsof the image and show that there is a natural bijection between pre-images.In a slightly different direction, we give a “coset space” construction associated to the action of asubgroupoid on either the whole groupoid or a slice of it.These constructions were found in the course of work on the paper [CG]. However they are much moregeneral than the setting of that work and we hope they may be of independent interest. Let G be a groupoid. We first recall the definition of the slice groupoid, G /X , for X ∈ G . The objectsof G /X are all morphisms in G with codomain X . A morphism from f : Y → X to f ′ : Z → X is acommutative triangle Y ZXgf f ′ ‡ Email: [email protected] . Website: g ∈ Mor( G ) exists. Let us write g ∇ : f → f ′ for such a triangle.Composition of morphisms is given by horizontal concatenation of triangles: Y Y ′ Xgf f ′ ◦ Y ′ Y ′′ Xg ′ f ′ f ′′ = Y Y ′ Y ′′ Xf f ′ f ′′ g g ′ = Y Y ′′ Xg ′ ◦ gf f ′′ which we will write in our more compact notation as g ∇ ◦ g ′ ∇ = g ′ ◦ g ∇ . This is the natural order to writecomposition in the slice category, although it conflicts with the “right to left” convention for composingmorphisms.Note that since G is a groupoid, for any f, f ′ , there exists a morphism f → f ′ in G /X induced by g = ( f ′ ) − ◦ f . Moreover, since every morphism in G is invertible, the every morphism in G /X is too, sothat G /X is a connected groupoid. In fact, more is true. Lemma 1.
In the slice groupoid G /X , we have | Hom G /X ( f, f ′ ) | = 1 for all objects f, f ′ of G /X .Proof: A morphism from f to f ′ in the slice category is a triangle g ∇ with g such that f = f ′ ◦ g . If g ′ ∇ isanother morphism, so f = f ′ ◦ g ′ , then since we are in a groupoid, f ′ ◦ g = f = f ′ ◦ g ′ implies g = g ′ .We also see that id X : X → X is a zero object in G /X , that is, it is both initial and terminal. For wehave the morphisms g − ∇ : id X → g and g ∇ : g → id X , which are unique by uniqueness of inverses. We willshortly show that id X is the natural analogue of the identity element of a group. Indeed, if G is a one-object groupoid with object ∗ , then Mor G ( ∗ , ∗ ) is a group with identity element id ∗ : ∗ → ∗ . Consideredin the slice G / ∗ , this element is precisely the zero object just discussed.Now let H be another groupoid and consider F : G → H a functor. For each X , there is a inducedfunctor F X : G /X → H / F X given on objects of G /X by F X f def = F f and on morphisms by F X g ∇ def = F g ∇ .Note that the codomain of F X f = F f is indeed F X and also that F g ∇ is precisely the commutative trianglein H given by taking the image of g ∇ under F .Note that F X is necessarily essentially surjective, since H / F X is a connected groupoid; in the groupoidsetting, “up to isomorphism” statements are extremely weak.Let us denote by G X the connected component of G containing X . The connected component H F X consists of all objects Y of H such that there exists a morphism h : Y → F X . (Since H is a groupoid,it is not necessary to also consider the objects Z for which there is a morphism h ′ : F X → Z , as such amorphism exists if and only if ( h ′ ) − : Z → F X exists.)Note that connected components are, by definition, taken to be full subcategories and are thereforealso groupoids. Note too that the slices G /X and G X /X are naturally identified: taking the slice at X disregards any morphism whose domain is not in the connected component containing X . This showsthat if we are interested only in slices of groupoids, we may assume our groupoids are connected withoutloss of generality.If X ′ is an object in G X , so that there exists a morphism g : X ′ → X , then F X ′ is an object of H F X ,as witnessed by F g . Then if g : X ′ → X ′′ is a morphism in G X , F g : F X ′ → F X ′′ is a morphism in H F X ,since it is a morphism between objects of H F X and the latter is a full subcategory of H .We may therefore consider the restriction of F to G X as a functor F| G X : G X → H F X . Since H F X isa connected groupoid, F| G X is also essentially surjective.The slice H / F X has its own identity element, the zero object of H / F X , id F X . It is then a naturalquestion whether or not we have F id X = id F X . From the lemma below, we see that this holds providedthe functor F| G X is “full at X ” (it need not be full, although this is certainly sufficient). Lemma 2.
Assume that the restriction F| G X of F to the connected component G X has the property thatthe induced function Mor G X ( X, X ) → Mor H F X ( F X, F X ) is surjective. Then F id X = id F X . roof: Let g ∈ Mor H ( F X, F X ). By the assumption, there exists f ∈ Mor G ( X, X ) such that g = F f .Since for all f ∈ Mor G ( X, X ), id X satisfies id X ◦ f = f = f ◦ id X , we have that F id X ◦ g = g = g ◦ F id X for all g ∈ Mor H ( F X, F X ). Hence F id X is an identity element in the group Mor H ( F X, F X ) and so isequal to id F X .Now we may define a notion of kernel for functors between slices of groupoids. Definition 3.
Let G and H be groupoids, F : G → H a functor between them and X an object of G . Let F X : G /X → H / F X be the induced functor.Define ker F X = { f : Y → X | F X f = id F X } to be the collection of objects of G /X whose image under F X is the object id F X of H / F X .The kernel of F X may in principle be an empty collection, but provided F is full at X (i.e. satisfiesthe assumption in Lemma 2), we have id X ∈ ker F X and a non-empty kernel.One may easily verify the following properties of objects of the the kernel. • For any f : Y → X belonging to ker F X , we have F Y = F X . • Given f ∈ ker F X , the morphism f ∇∈ Mor G /X ( f, id X ) satisfies F X f ∇ = id F X ∇ (noting that id F X ∇ : id F X → id F X is the identity morphism for the object id F X in H / F X ). • More generally, given f, g ∈ ker F X , we have a (unique) morphism in G /X given by g − ◦ f ∇ : f → g and this morphism satisfies F X g − ◦ f ∇ = id F X ∇ .Note that in the last of these properties, g − ◦ f is not necessarily itself in ker F X , since the codomain of g − ◦ f is not necessarily X .Arguably the most important property of the kernel of a group homomorphism f : G → H is that itinduces a partition of G whose parts are in bijection with the elements of im f , and such that the parts arein pairwise bijection with each other (being cosets of the kernel). Let us call a partition whose parts are inpairwise bijection an equipartition . The next result shows that the kernel ker F X induces an equipartitionof the objects of G /X .Let us denote by im F X the image of F X , as a collection of objects of H / F X . (Note that at present,both ker F X and im F X are not being considered as categories, only as subcollections of the objects of therespective groupoids.) Proposition 4.
Let G and H be groupoids, F : G → H a functor between them and X an object of G . Let F X : G /X → H / F X be the induced functor.For any g ∈ im F X , define F − X ( g ) = { f : Y → X | F f = g } . Then (i) the collection P ( F X ) def = {F − X ( g ) | g ∈ im F X } is a partition of the collection of objects of G /X ; (ii) there is a bijection between P ( F X ) and im F X ; and (iii) for any g ∈ im F X , there is a bijection between F − X ( g ) and ker F X .That is, P ( F X ) determines an equipartition of G /X .Proof: That P ( F X ) defines a partition is immediate, as is the existence of a bijection with im F X .Fix g ∈ im F X . Let l, m ∈ F − X ( g ) so that F X l = F X m , or equivalently F l = F m . Then, as H is agroupoid, it follows that F X ( m ◦ l − ) = F ( m ◦ l − ) = id F X and so m ◦ l − ∈ ker F X .In particular, taking l = g , for any m ∈ F − X ( g ) there exists k m ∈ ker F X such that m ◦ g − = k m , orequivalently m = k m ◦ g . Now it is straightforward to check that m k m is a bijection between F − X ( g )and ker F X , noting that at various points we rely on the fact that we are working in groupoids to invertmorphisms as required. 3onsider the full subgroupoid K of G /X on ker F X . That is, K is the groupoid whose objects are thosebelonging to ker F X and Mor K ( f, f ′ ) = Mor G /X ( f, f ′ ). Then the observations above imply that for anymorphism g ∇ in K , we have F X ( g ∇ ) = id F X ∇ , so that F X (ker F X ) is the trivial groupoid with one object,id F X , and one morphism, id F X ∇ .Observe that if G and H are one-object groupoids, the above definitions and results reduce to exactlythe classical constructions and statements for groups. We will now explain how to construct the groupoid version of a coset space of a group with respect to asubgroup, and the natural group action on this.
The action groupoid
Let G be a groupoid. There is a groupoid G // G , the action groupoid with respect to the (right) action of G on itself, via the functor Hom G (? , − ) : G →
Set given byHom G (? , − )( X ) = Hom G ( X, − ) def = G Y ∈G Hom G ( X, Y )Hom G (? , − )( f : X → Y ) : Hom G ( Y, − ) → Hom G ( X, − ) , ( g : Y → Z ) ( g ◦ f : X → Z )Let us write ρ G for Hom G (? , − ).The groupoid G // G has objects f : X → Y ∈ G X G Y Hom G ( X, Y ) = Mor( G )and morphisms ( g : Y → Z, f : X → Y ) : g → g ◦ f. Composition in G // G is given by(( g ′ , f ′ ) ◦ ( g, f ) : g → g ◦ f ◦ f ′ ) = ( g, f ◦ f ′ )where this is defined, i.e. when g ′ = g ◦ f . We have id g = ( g, id s ( g ) ).We have that slice groupoids embed naturally in the action groupoid, as follows. Lemma 5.
There is a fully faithful contravariant functor that is injective on objects, ι X : G /X → G // G given by ι X ( f : Y → X ) = fι X ( g ∇ : f → f ′ ) = ( f ′ , g ) : f ′ → f Proof:
For functoriality, let g ∇ : f → f ′ , g ′ ∇ : f ′ → f ′′ . Then we have ι X ( g ∇ ◦ g ′ ∇ ) = ι X ( g ′ ◦ g ∇ )= ( f ′′ , g ′ ◦ g ) : f ′′ → ( f ′′ ◦ g ′ ◦ g = f )= ( f ′ , g ) ◦ ( f ′′ , g ′ )= ι X ( g ∇ ) ◦ ι X ( g ′ ∇ ) . It is straightforward to see that ι X respects identity morphisms. Despite appearances, this shows that ι X is a contravariant functor: our notational convention for composition in the slice category is unfortunatelymisleading at this point. 4he functor ι X is faithful as if ι X ( g ∇ : f → f ′ ) = ι X ( h ∇ : f → f ′ ), then ( f ′ , g ) = ( f ′ , h ). Thennecessarily g = h .For fullness, take ( f ′ , g ) : ι X ( f ′ ) → ι X ( f ). Then by the definition of morphisms in G // G , we have ι X ( f ) = f ′ ◦ g and hence f = f ′ ◦ g . Then there exists a triangle Y ZXgf f ′ g ∇ =such that ι X ( g ∇ : f → f ′ ) = ( f ′ , g ) as required.Lastly, it is immediate that ι X is injective on objects.Later, we will have need of the corresponding covariant functor ι ∗ X : ( G /X ) op → G // G given by ι X ( f : X → Y ) = fι X ( g ∇ : f ′ → f ) = ( f ′ , g ) : f ′ → f This functor is also fully faithful and injective on objects.
An analogue of the coset space
Now let H be a subgroupoid of G . We will assume that H is wide , that is, Obj( H ) = Obj( G ); this is a verymild restriction with respect to what follows, and is made so that additional clauses of the form “when X is also an object of H ” can be omitted. We will construct a groupoid corresponding to the action of G on the right cosets of H .In what follows, we use the “source” function s : Mor( G ) → Obj( G ), s ( f : X → Y ) = X , and later thecorresponding “target” function t : Mor( G ) → Obj( G ), t ( f : X → Y ) = Y .First, we define a relation ∼ H on Mor( G ) by( g : X → Y ) ∼ H ( g ′ : X ′ → Y ′ ) ⇔ ( s ( g ) = s ( g ′ ) and ∃ h ∈ Mor H ( Y, Y ′ ) such that g ′ = h ◦ g ) . It is straightforward to check that H being a groupoid implies that ∼ H is an equivalence relation. For g ∈ Mor( G ), denote by [ g ] its ∼ H -equivalence class.Note that by the definition of ∼ H , s : (Mor( G ) / ∼ H ) → Obj( G ), s ([ g ]) = s ( g ) is well-defined. Lemma 6.
There is a contravariant functor ρ H : G →
Set , defined by ρ H ( X ) = { [ g ] | s ( g ) = X } ρ H ( g : X → Y ) = ρ H ( g ) : ρ H ( Y ) → ρ H ( X ) , ρ H ( g )([ f ]) = [ f ◦ g ] Proof:
We first check that ρ H ( g ), and hence ρ H , is well-defined. Assume that [ f ] = [ f ′ ], so there exists h ∈ Mor( H ) such that f ′ = h ◦ f . Then f ′ ◦ g = h ◦ f ◦ g , so [ f ′ ◦ g ] = [ f ◦ g ]. Then ρ H ( g )([ f ]) = [ f ◦ g ] = [ f ′ ◦ g ] = ρ H ( g )([ f ′ ])and ρ H ( g ) is well-defined for all g .We have ρ H (id X : X → X )([ f ]) = [ f ◦ id X ] = [ f ] for all f such that s ( f ) = X , so ρ H (id X ) = id ρ H ( X ) as required. Then given g : X → Y , g ′ : Y → Z and [ f ] ∈ ρ H ( Z ), we have ρ H ( g ′ ◦ g )([ f ]) = [ f ◦ g ′ ◦ g ] = ρ H ( g )([ f ◦ g ′ ]) = ( ρ H ( g ) ◦ ρ H ( g ′ ))([ f ])and so ρ H is functorial.Next, we give the definition of the “groupoid coset space”.5 efinition 7. Let G be a groupoid and H a subgroupoid of G . Define the H -coset space action groupoid( H : G ) // G to be the groupoid with objects the elements of Mor( G ) / ∼ H and morphisms([ g ] , f ) : [ g ] → [ g ◦ f ]defined when t ( f ) = s ([ g ]) = s ( g ). Composition of morphisms is given by([ g ′ ] , f ′ ) ◦ ([ g ] , f ) = ([ g ] , f ◦ f ′ )when this is defined, i.e. when [ g ′ ] = [ g ◦ f ]. We have id [ g ] = ([ g ] , id s ([ g ]) ) = ([ g ] , id s ( g ) ). Proposition 8.
There is a full (covariant) functor that is surjective on objects, π H : G // G → ( H : G ) // G given by π H ( f ) = [ f ] π H ( g, f ) = ([ g ] , f ) Proof:
Recall that in G // G , we have id g : g → g given by id g = ( g, id s ( g ) ). Then π H ( g, id s ( g ) ) = ([ g ] , id s ( g ) ) = id π H ( g ) . Consider the composable morphisms ( g ◦ f, f ′ ), ( g, f ) in G // G . We have π H (( g ◦ f, f ′ ) ◦ ( g, f )) = π H ( g, f ◦ f ′ )= ([ g ] , f ◦ f ′ )= ([ g ◦ f ] , f ′ ) ◦ ([ g ] , f ) π H ( g ◦ f, f ′ ) ◦ π H ( g, f )so that π H is (covariantly) functorial.Given ([ g ] , f ) ∈ Mor(( H : G ) // G ), we have π H ( g, f ) = ([ g ] , f ), so that π H is full.That π H is surjective on objects is clear from the definitions.The “source” function can be upgraded to a (forgetful) functor on each of the categories consideredthus far, as follows: • s : ( G /X ) op → G , s ( f : Y → X ) = s ( f ) = Y , s ( g ∇ : f ′ → f ) = g − , where g ∇ is the triangle aspreviously but now considered in ( G /X ) op and hence as a morphism from f ′ to f ; • s : G // G → G , s ( g ) = s ( g ), s ( g, f ) = f − ; • s : ( H : G ) // G → G , s ([ g ]) = s ( g ), s ([ g ] , f ) = f − .Indeed, there is a commutative diagram of groupoids( G /X ) op G // G ( H : G ) // GG s s sι ∗ X π H Let us consider the functor π X H def = π H ◦ ι ∗ X . Explicitly, we have π X H ( f : Y → X ) = π H ( f ) = [ f ] π X H ( g ∇ : f ′ → f ) = π H (( f ′ , g )) = ([ f ′ ] , g ) : [ f ′ ] → [ f ]6et us define ( H : G /X ) // G to be the full image of π H ◦ ι ∗ X . Then ( H : G /X ) // G is a connectedsubgroupoid of ( H : G ) // G ; its fullness as a subcategory corresponds to being “closed under the G -action”,or a “ G -submodule” of the “ G -module” ( H : G ) // G .The groupoid ( H : G ) // G is obtained by an equivalence relation on morphisms of G coming from thesubgroupoid H : somewhat loosely, morphisms in H become equivalent to identity morphisms in ( H : G ).The construction is done in a G -equivariant way, hence the “coset space” ( H : G ) still admits a G -action,giving rise to ( H : G ) // G .The groupoid ( H : G /X ) // G is a “sliced” version of this: we restrict attention to those ∼ H -equivalenceclasses of morphisms that contain at least one morphism whose codomain is the fixed object X . Just aswith the usual slice category, by slicing we focus on a connected groupoid with a “base point” an objectof interest to us, disposing of other components of G , of which there may be many.Our G -action works by pre-composition, hence is compatible with the slicing at X . The upshot ofthe above constructions is therefore that we have a groupoid whose objects are ∼ H -equivalence classescontaining certain morphisms from our original groupoid, with a G -action by pre-composition.Note that unless H contains only the identity morphisms, ∼ H -equivalence classes will necessarilycontain morphisms with codomains not equal to X : s ([ f ]) is well-defined (by construction) but t ([ f ]) isin general not.We conclude with a question: Question.