Action representability of the category of internal groupoids
aa r X i v : . [ m a t h . C T ] S e p ACTION REPRESENTABILITY OF THE CATEGORYOF INTERNAL GROUPOIDS
MARINO GRAN AND JAMES RICHARD ANDREW GRAY
Abstract.
When C is a semi-abelian category, it is well knownthat the category Grpd ( C ) of internal groupoids in C is again semi-abelian. The problem of determining whether the same kind ofphenomenon occurs when the property of being semi-abelian is re-placed by the one of being action representable (in the sense ofBorceux, Janelidze and Kelly) turns out to be rather subtle. Inthe present article we give a sufficient condition for this to be true:in fact we prove that the category Grpd ( C ) is a semi-abelian actionrepresentable algebraically coherent category with normalizers ifand only if C is a semi-abelian action representable algebraically co-herent category with normalizers. This result applies in particularto the categories of internal groupoids in the categories of groups,Lie algebras and cocommutative Hopf algebras, for instance. Preliminaries
In this paper C will always denote a semi-abelian category (in thesense of Janelidze, M´arki and Tholen [26]), usually satisfying someadditional axioms. Recall that a category C is semi-abelian if it is • finitely complete, finitely cocomplete and pointed, with zeroobject 0; • (Barr)- exact [1]; • (Bourn)- protomodular [5], which in the pointed case can be ex-pressed by the validity of the Split Short Five Lemma in C .There are plenty of interesting algebraic categories which are semi-abelian. For example, any variety of algebras whose algebraic theoryhas among its operations and identities those of the theory of groupsis semi-abelian (see [10] for a precise characterization). As a conse-quence, the categories Grp of groups, Ab of abelian groups, Rng of(not necessarily unitary) rings,
Lie R of Lie algebras over a commuta-tive ring R , XMod of crossed modules (of groups), are all semi-abeliancategories. In addition any category of compact Hausdorff models ofa semi-abelian algebraic theory [3], such as the category
Grp ( Comp )of compact Hausdorff groups, is semi-abelian, as is also the category
Hopf K, coc of cocommutative Hopf algebras over a field K [18]. The dualcategory Set op ∗ of the category Set ∗ of pointed sets is semi-abelian [26].In any semi-abelian category C there is a natural notion of centralityof arrows [24, 7] (in fact weaker assumptions on the base category canbe required [2], but in this work we shall always ask C to be at leastsemi-abelian). Given two morphisms f : A → B and g : C → B withthe same codomain, they are said to commute in the sense of Huq [24]if there is a (necessarily unique) arrow c : A × C → B making thediagram commute A (1 A , / / f " " ❋❋❋❋❋❋❋❋❋ A × C c (cid:15) (cid:15) C (0 , C ) o o g | | ①①①①①①①①① B, where (1 A ,
0) and (0 , C ) are the unique morphisms induced by theuniversal property of the product A × C . When this is the case theunique arrow c : A × C → B is called the cooperator of f and g . Oneusually writes [ f, g ] Huq = 0, or simply [ f, g ] = 0, when this is thecase. Given two subobjects f : A → B and g : C → B with the samecodomain, the Huq commutator [ f, g ], usually denoted by [ A, C ] (ifthere is no risk of confusion), is the smallest normal subobject D of B with the following universal property: in the quotient π : B → BD theregular images π ( A ) and π ( C ) (of f : A → B and g : C → B along π )commute in the sense above.Given a morphism f : A → B in a semi-abelian category C we willdenote by z f : Z B ( A, f ) → B the centralizer of f in B , i.e. the terminalobject in the category of morphisms that commute with f , wheneverit exists (see e.g. [11, 19]). In this case z f is always a monomorphism,and we write Z B ( A, f ) or Z B ( A ) (when there is no risk of confusion) forthe corresponding subobject of B . For a monomorphism f : A → B the normalizer of f is the terminal object in the category with ob-jects triples ( N, n, m ) where n is a normal monomorphism and m amonomorphism such that nm = f [20].Recall that a split extension is a diagram in C X κ / / A α / / B β o o (1) CTION REPRESENTABILITY OF THE CATEGORY OF INTERNAL GROUPOIDS3 where κ is the kernel of α and αβ = 1 B . A morphism of split extensionsis a diagram in C X κ / / u (cid:15) (cid:15) A α / / v (cid:15) (cid:15) B β o o w (cid:15) (cid:15) X ′ κ ′ / / A ′ α ′ / / B ′ β ′ o o where the top and bottom rows are split extensions (the domain andcodomain, respectively), and vκ = κ ′ u , vβ = β ′ w and wα = α ′ v . Letus write SplExt( C ) for the category of split extensions in C , and write P, K : SplExt( C ) → C for the functors sending a split extension toits codomain and to the (object part of the) kernel , respectively. Thecategory C can be equivalently defined to be action representable, inthe sense of Borceux, Janelidze, Kelly [4] when each fiber of the functor K has a terminal object. This means that for each X in C there existsa split extension X k / / [ X ] ⋉ X p / / [ X ] , i o o called the generic split extension with kernel X , with the universalproperty that there exists a unique morphism to it from each splitextension in the fiber K − ( X ), that is, there is a unique morphism,which is the identity on kernels, to it from each split extension (1) withkernel X : X κ / / A α / / (cid:15) (cid:15) B β o o (cid:15) (cid:15) X k / / [ X ] ⋉ X p / / [ X ] . i o o For instance, in the category
Grp of groups, the generic split extensionwith kernel a group X is given by the split extension X k / / Aut ( X ) ⋉ X p / / Aut ( X ) , i o o where Aut ( X ) is the group of automorphisms of X and the action of Aut ( X ) on the group X is given by the evaluation.More generally, for X an object in C , a split extension in K − ( X )is called faithful if there is at most one morphism to it from each splitextension in K − ( X ). The category C is called action accessible [11] iffor each X in C each split extension in K − ( X ) admits a morphism toa faithful split extension in K − ( X ). The category C is algebraicallycoherent [14] when the change of base functors of fibers of P preserve MARINO GRAN AND JAMES RICHARD ANDREW GRAY joins. This is equivalent (in the pointed protomodular context) torequiring that for each cospan of monomorphisms of split extensions X κ / / u (cid:15) (cid:15) A α / / v (cid:15) (cid:15) B β o o X κ / / A α / / B β o o X κ / / u O O A α / / v O O B β o o if the morphisms v and v are jointly strongly epimorphic in C , then soare the morphisms u and u . Recall that a semi-abelian algebraicallycoherent category C with normalizers has the following properties: • C is action accessible [11] (see also [21]), and hence centralizersof normal monomorphisms exist and are normal (Proposition5.2 of [9]). • Huq commutators distribute over joins of subobjects [22]: giventhree subobjects A → C , A → C and B → C of the sameobject C the Huq commutator satisfies the following identity:[ A ∨ A , B ] = [ A ∨ B, A ∨ B ] . • The Jacobi identity holds for normal subobjects (Theorem 7.1[14]): if
K, L, M are normal subobject of an object C , then[ K, [ L, M ]] ≤ [[ K, L ] , M ] ∨ [[ M, K ] , L ] . (2) • a split extension in C X κ / / A α / / B β o o is faithful if and only if Z A ( X, κ ) ∧ B = 0 (see [6] Corollary 4.1).2. Reflexive graphs and groupoids
It is well-known that a reflexive graph G σ / / τ / / G e o o (3)in any category C with equalizers can be equivalently seen as a triple( G, s : G → G, t : G → G ) where G is an object in C , and s , t aremorphisms in C satisfying the identities st = t , ts = s (so that inparticular ss = s and tt = t ). To see this, given a reflexive graph (3),one simply defines s = eσ and t = eτ . Conversely, given such a triple CTION REPRESENTABILITY OF THE CATEGORY OF INTERNAL GROUPOIDS5 ( G, s : G → G, t : G → G ), the “object of objects” G of the associatedgraph is the (domain) of the equalizer of s and 1 G (equivalently, of t and 1 G ).When C is a semi-abelian action accessible category, an internalgroupoid in C can be equivalently presented as a triple ( G, s : G → G, t : G → G ) as above such that, moreover, the commutator of theirkernels is trivial: [ker( s ) , ker( t )] = 0 . (4)This follows from the results in [12, 11] (the Smith commutator andthe Huq commutator coincide in this context), and[ker( σ ) , ker( τ )] = [ker( s ) , ker( t )] = 0 , since e is a monomorphism.The category of groupoids in C is then equivalent to the categorywhose objects are triples ( G, s : G → G, t : G → G ) as above with[ker( s ) , ker( t )] = 0, and arrows f : ( G, s : G → G, t : G → G ) → ( H, s ′ : H → H, t ′ : H → H )those f : G → H in C such that s ′ f = f s and t ′ f = f t .With a slight abuse of notation, since C will always be assumed tobe semi-abelian, from now on we shall write Grpd ( C ) for this latterequivalent category, and also call its objects internal groupoids. Weshall also denote by ker( s ) : ∗ G → G and ker( t ) : G ∗ → G , the kernelof s : G → G and t : G → G , respectively. The notation ∗ G forthe domain of the kernel of s intuitively reminds one of the fact thatits “elements” are the (internal) arrows of G whose “source” is the“zero element in G ”, whereas the arrows in G ∗ have as “target” thissame “zero element”. We shall also simply write [ ∗ G, G ∗ ] to denote thecommutator [ker( s ) , ker( t )].The remaining part of this section consists largely of a series of lem-mas building up to our main results: Theorems 2.8 and 2.10, and Corol-lary 2.11. Let us first recall the following known result (see Lemma 2.6in [13]): Lemma 2.1.
Let C be a semi-abelian category. Consider a split ex-tension as in the bottom row of the diagram K / / k (cid:15) (cid:15) K ∨ Z (cid:15) (cid:15) Z o o X x / / Y f / / Z s o o in C , with the property that xk : K → Y is a normal monomorphism.Then this split extension lifts along k : K → X to yield a normal MARINO GRAN AND JAMES RICHARD ANDREW GRAY monomorphism of split extensions, where K ∨ Z is the join of the sub-objects K and Z of Y . Lemma 2.2.
Let C be a semi-abelian action accessible algebraicallycoherent category. For each split extension X κ / / A α / / B β o o of internal reflexive graphs, there exists a largest sub-reflexive-graph ˜ B of B such that [ ∗ ˜ B, X ∗ ] = 0 = [ ˜ B ∗ , ∗ X ] in C .Proof. We will show that˜ B = (( Z A ( ∗ X, κ ker( s )) ∧ B ∗ ) ∨ B ) ∧ (( Z A ( X ∗ , κ ker( t )) ∧ ∗ B ) ∨ B )is the largest sub-reflexive-graph of B satisfying the desired property.Since any subobject of B containing B (the “object of objects” ofthe reflexive graph B ) is a sub-reflexive graph of B , we know that˜ B is a sub-reflexive-graph of B . To see that it satisfies the desiredproperty let Z = Z A ( ∗ X, κ ker( s )) ∧ B ∗ , and note that, since ∗ X = X ∧ ∗ A , it follows that ∗ X is a normal subobject of A and hence so is Z A ( ∗ X, κ ker( s )). Accordingly, Z is a normal subobject of B , and thisimplies that ( Z ∨ B ) ∗ = Z - just note that we are in the situationof Lemma 2.1, with Y = B , X = B ∗ and Z = B . From this we thendeduce that ˜ B ∗ ≤ ( Z ∨ B ) ∗ = Z ≤ Z A ( ∗ X, κ ker( s )) . A similar argument shows that ∗ ˜ B ≤ Z A ( X ∗ , κ ker( t )). Now, let B ′ be a sub-reflexive-graph of B satisfying the desired property. Clearly B ′∗ ≤ B ∗ and B ′∗ ≤ Z A ( ∗ X, κ ker( s )), and hence B ′∗ ≤ Z A ( ∗ X, κ ker( s )) ∧ B ∗ . Since B ′ ≤ B and B ′ = B ′∗ ∨ B ′ (by protomodularity), it follows that B ′ = B ′∗ ∨ B ′ ≤ ( Z A ( ∗ X, κ ker( s )) ∧ B ∗ ) ∨ B . By exchanging the rolesof s and t of the internal reflexive graphs, we have that B ′ ≤ ( Z A ( X ∗ , κ ker( t )) ∧ ∗ B ) ∨ B , from which the claim easily follows. (cid:3) Lemma 2.3.
Let C be a semi-abelian action accessible algebraicallycoherent category. For each split extension X κ / / A α / / B β o o CTION REPRESENTABILITY OF THE CATEGORY OF INTERNAL GROUPOIDS7 of internal reflexive graphs, if X is a groupoid and [ ∗ B, X ∗ ] = 0 = [ B ∗ , ∗ X ] , then [ X, [ A ∗ , ∗ A ]] = 0 . Proof.
Note that since C is protomodular we have A ∗ = X ∗ ∨ B ∗ and ∗ A = ∗ X ∨ ∗ B . Note that trivially [ X, B ∗ ] ≤ X , but also that [ X, B ∗ ] ≤ A ∗ meaning that [ X, B ∗ ] ≤ X ∧ A ∗ = X ∗ . We have[ X, A ∗ ] = [ X, X ∗ ] ∨ [ X, B ∗ ] ≤ X ∗ ∨ X ∗ = X ∗ . Therefore [ ∗ A, [ X, A ∗ ]] ≤ [ ∗ A, X ∗ ]= [ ∗ X, X ∗ ] ∨ [ ∗ B, X ∗ ]= 0 , where the last equality follows from the equality [ ∗ X, X ∗ ] = 0 (since X is a groupoid) and the assumption [ ∗ B, X ∗ ] = 0.This and its dual (i.e. swapping s and t ) mean that[ A ∗ , [ X, ∗ A ]] = 0 = [ ∗ A, [ X, A ∗ ]] . The Jacobi identity (2) now implies that[ X, [ A ∗ , ∗ A ]] ≤ [[ X, A ∗ ] , ∗ A ] ∨ [ A ∗ , [ X, ∗ A ]] = 0 . (cid:3) Lemma 2.4.
Let C be a semi-abelian action accessible algebraicallycoherent category. For each split extension X κ / / A α / / B β o o of internal reflexive graphs. If X and B are groupoids and [ ∗ B, X ∗ ] = 0 = [ B ∗ , ∗ X ] , then A is a groupoid.Proof. We have[ A ∗ , ∗ A ] = [ X ∗ ∨ B ∗ , ∗ X ∨ ∗ B ]= [ X ∗ , ∗ X ] ∨ [ X ∗ , ∗ B ] ∨ [ B ∗ , ∗ X ] ∨ [ B ∗ , ∗ B ]= 0 . (cid:3) MARINO GRAN AND JAMES RICHARD ANDREW GRAY
Lemma 2.5.
Let C be a semi-abelian action accessible algebraicallycoherent category. If S is sub-reflexive-graph of A , then the centralizerof S in A has underlying object Z ∧ s − ( Z ) ∧ t − ( Z ) where Z is thecentralizer of the underlying subobject inclusion of S in A .Proof. The claim follows from Theorem 2.1 of [23], since the categoryof internal reflexive graphs in C is a functor category of C . (cid:3) Lemma 2.6.
Let C be a semi-abelian action accessible algebraicallycoherent category. A split extension X κ / / A α / / B β o o of internal graphs is faithful if and only if Z A ( X, κ ) ∧ s − ( Z A ( X, κ )) ∧ t − ( Z A ( X, κ )) ∧ B = 0 in C .Proof. Since according to Corollary 2.3 of [23] the category of internalreflexive graphs in C is action accessible as soon as C is, the claimfollows from the previous lemma via the last bullet of Section 1. (cid:3) Lemma 2.7.
Let C be a semi-abelian action accessible algebraicallycoherent category. For each faithful split extension X κ / / A α / / B β o o of internal reflexive graphs, B is a groupoid if and only if [ X, [ B ∗ , ∗ B ]] = 0 in C .Proof. If B is a groupoid then this is trivially the case. The con-verse follows from Lemma 2.6 and the fact that [ B ∗ , ∗ B ] is always in s − ( Z A ( X, κ )) and t − ( Z A ( X, κ )) (because s ([ B ∗ , ∗ B ]) = 0 and simi-larly for t ). (cid:3) Theorem 2.8.
Let C be a semi-abelian action accessible algebraicallycoherent category. For each faithful split extension X κ / / A α / / B β o o of reflexive graphs with X a groupoid, there exists a largest sub-split-extension of groupoids with kernel X . CTION REPRESENTABILITY OF THE CATEGORY OF INTERNAL GROUPOIDS9
Proof.
By Lemma 2.2 there is a largest sub-reflexive-graph ˜ B of B suchthat [ ∗ ˜ B, X ∗ ] = 0 = [ ˜ B ∗ , ∗ X ]. We will prove that the split extension atthe top of the diagram X ˜ κ / / ˜ A (cid:15) (cid:15) ˜ α / / ˜ B (cid:15) (cid:15) ˜ β o o X κ / / A α / / B β o o obtained by pulling back along ˜ B → B is the desired split extensions ofgroupoids. According to Lemma 2.3 [ X, [ ˜ A ∗ , ∗ ˜ A ]] = 0 which means that[ X, [ ˜ B ∗ , ∗ ˜ B ]] = 0. Therefore, since a sub-split-extension of a faithfulextension is faithful, it follows from Lemma 2.7 that ˜ B is a groupoid.The final claim then follows from Lemma 2.4. (cid:3) We will also need the following proposition which shows that a core-flective subcategory closed under certain limits admits generic splitextensions whenever the category it is coreflective in does. Recall thata functor between pointed categories is protoadditive [16] when it pre-serves split exact sequences.
Proposition 2.9.
Let X and Y be semi-abelian categories, and let I : X → Y be a full and faithful protoadditive functor with right adjoint R : Y → X . If Y has generic split extensions, then so does X .Proof. Suppose X is an object in X and suppose that I ( X ) k / / [ I ( X )] ⋉ I ( X ) p / / [ I ( X )] i o o is the generic split extension with kernel I ( X ) in Y . Let η be the unit ofthe adjunction I ⊣ R which is an isomorphism. The claim now followsby observing that (i) the lower part of (6) (below) is a split extension;(ii) for each split extension (1) the upper part of (5) (below) is a splitextension, and the adjunction produces a bijection between morphismsof split extensions of the form I ( X ) I ( κ ) / / I ( A ) (cid:15) (cid:15) ✤✤✤ I ( α ) / / I ( B ) I ( β ) o o (cid:15) (cid:15) ✤✤✤ I ( X ) k / / [ I ( X )] ⋉ I ( X ) p / / [ I ( X )] i o o (5) in Y and morphisms of split extensions of the form X κ / / A (cid:15) (cid:15) ✤✤✤✤ α / / B β o o (cid:15) (cid:15) ✤✤✤✤ X R ( k ) η X / / R ([ I ( X )] ⋉ I ( X )) R ( p ) / / R ([ I ( X )]) R ( i ) o o (6)in X . (cid:3) Theorem 2.10.
A category C is a semi-abelian action representablealgebraically coherent category with normalizers if and only if the cat-egory Grpd ( C ) of internal groupoids in C is a semi-abelian action rep-resentable algebraically coherent category with normalizers.Proof. For the “if” part suppose that
Gpd ( C ) is a semi-abelian alge-braically coherent category with normalizers. Noting that functor from C to Gpd ( C ), sending an object in C to the discrete groupoid in Gpd ( C )embeds C as a full reflective and coreflective subcategory of Gpd ( C )(closed in Grpd ( C ) under quotients and subobjects), it follows that C is a semi-abelian algebraically coherent category with normalizers. Ac-tion representablity now follows from the previous proposition. For the“only if” part suppose that C is a semi-abelian algebraically coherentcategory with normalizers. It follows from [20] that the category ofreflexive graphs being a functor category is action representable. For agroupoid X , applying the previous theorem to the generic split exten-sion of reflexive graphs with kernel X it easily follows that the largestsub-split extension of groupoids with kernel X is the generic split exten-sion of groupoids with kernel X . The fact that Grpd ( C ) is semi-abelianwhen C is semi-abelian follows from Lemma 4 . Grpd ( C ) is algebraically coherent, and hence itremains to show that Grpd ( C ) has normalizers. However, by Corollary2.3 of [23] the category RG ( C ) of reflexive graphs in C has normalizersand hence so does Grpd ( C ) being closed under subobjects and finitelimits in RG ( C ). (cid:3) Corollary 2.11. If C is a semi-abelian action representable algebraicallycoherent category with normalizers, then so is the category Grpd n ( C ) of n -fold internal groupoids in C . Examples 2.12.
The results in this article apply to some importantalgebraic categories, such as the categories
Grpd ( Grp ), Grpd ( Lie R ), or Grpd ( Hopf K , coc ), of internal groupoids in the categories of groups, Liealgebras over a commutative ring R , or cocommutative Hopf algebras CTION REPRESENTABILITY OF THE CATEGORY OF INTERNAL GROUPOIDS11 over a field K , respectively. For the fact that Hopf K , coc is an action rep-resentable semi-abelian category the reader is referred to [18], whereasthe fact that it is algebraically coherent is explained in Example 4.6 in[14]. To see that Hopf K , coc has normalizers recall that:(a) Hopf K , coc is equivalent to the category Grp ( Coalg
K,coc ) of internalgroups in the finitely complete cartesian closed category
Coalg
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