Crossed Modules of Monoids II: Relative Crossed Modules

This is the second part of a series of three strongly related papers in which three equivalent structures are studied: Internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans. Crossed modules of monoids relative to this class of spans. Simplicial monoids of so-called Moore length 1 relative to this class of spans. The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids). In this second part we define relative crossed modules of monoids and prove their equivalence with the relative categories of Part I. Internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans. Crossed modules of monoids relative to this class of spans. Simplicial monoids of so-called Moore length 1 relative to this class of spans.


Introduction
Since their appearance in [25], crossed modules of groups have been intensively studied and applied in various contexts; see e.g. the reviews [19,21,22] and the references in them. They admit several different descriptions: a simplicial group whose Moore complex is concentrated in degrees 1 and 2 turns out to be the internal nerve of an internal category in the category of groups (which is necessarily an internal groupoid, a.k.a. strict 2-group or Cat 1 -group) and the Moore complex yields a crossed module. These constructions establish, in fact, equivalences between these three notions.
The first (to our knowledge) proofs of the equivalence between crossed modules and strict 2-groups can be found in [6]-where it is referred also to the unpublished proof [7]-and in [17]. Based on the fact that groups constitute a semi-abelian category in the sense of [15], another short and deeply conceptual proof is due to Janelidze [14]. It also leads to a broad generalization describing the equivalent notions of crossed modules and internal categories Communicated by George Janelidze. of all of Lie algebras, -groups in the sense of [13], Heyting semi-lattices, the dual of the category of pointed sets and much more. Thus by working in an arbitrary semi-abelian category, not only a more transparent proof is obtained, but also a much wider generality, also unifying earlier results in [16,23].
More recently, however, some results on, and certain applications of crossed modules of groups were extended to crossed modules of ordinary monoids [20], groupoids [5] and of Hopf algebras [1,8,9,18,25]. From this list only cocommutative Hopf algebras over a field are known to constitute a semi-abelian category [11,12]. Hence Janelidze's proof can not be applied directly to the rest of these generalizations. Our aim is therefore to develop a wider theory of crossed modules of monoids in more general monoidal categories which are not expected to have all pullbacks (not even along split epimorphisms). We have the above two main examples in mind: -Categories of spans whose monoids are small categories, including groupoids in particular. -Categories of comonoids in symmetric monoidal categories whose monoids are bimonoids including Hopf monoids in particular.
In the first part [3] of this series of papers we discussed classes of spans satisfying appropriate conditions; and relative pullbacks with respect to them. Assuming that such pullbacks exist-as they do in our key examples-we introduced a monoidal category with monoidal product provided by these pullbacks. We defined a relative (to the chosen class of spans) category as a monoid in this monoidal category. It is given by the usual data In the current article we make the next step and prove the equivalence of the following categories for a fixed class of suitable spans in a monoidal category: -The category of relative categories in the category of monoids, -The category of relative crossed modules of monoids.
Our methodology is inspired by Janelidze's paper [14]. In Sect. 1 we investigate first some category of the category of split epimorphisms of monoids. We obtain an equivalent description of a split epimorphism of monoids B i A s in terms of a distributive law which allows for handy characterizations of possible morphisms t and d in ( * ). This is used in Sects. 2 and 3, respectively, to present equivalent descriptions of some reflexive graphs of monoids in terms of relative pre-crossed modules of monoids; and of relative category objects ( * ) in categories of monoids in terms of relative crossed modules of monoids. Applying our results to categories of spans and to categories of comonoids, respectively, we re-obtain the definitions of crossed modules of groupoids in [5] and of crossed modules of Hopf monoids in [25], respectively.
Our next aim is to extend to our setting the equivalence of the category of strict 2-groups (that is, internal groupoids in the category of groups) and the category of crossed modules of groups to the further category of simplicial groups whose Moore complex has length 1. This will be achieved in Part III of this series [2].

Split Epimorphisms of Monoids Versus Distributive Laws
We freely use definitions, notation and results from [3]. Throughout, the composition of some morphisms A g B and B f C in an arbitrary category will be denoted by A f ·g C . Identity morphisms will be denoted by 1 (without any reference to the (co)domain object if it causes no confusion). In any monoidal category C the monoidal product will be denoted by juxtaposition and the monoidal unit will be I . For the monoidal product of n copies of the same object A also the power notation A n will be used. For any monoid A in C, the multiplication and the unit morphisms will be denoted by A 2 m A and I u A , respectively. If C is also braided, then for the braiding the symbol c will be used.
Recall that a class S of spans in an arbitrary category is said to be admissible if it satisfies the following two properties in [ [3,Assumption 4.1] asserts that there exists the S-relative pullback of those cospans whose legs are in an admissible class S. Under this assumption it was proven in [3,Corollary 4.6] i is a common section of s and t (that is, B i A s t is a reflexive graph).
A class S of spans in a monoidal category is said to be monoidal if it satisfies the following two conditions in [3, Definition 2.5].
(UNITAL) For any morphisms f and g whose domain is the monoidal unit I , It is discussed in [3, Example 2.8] that a monoidal admissible class S of spans in a braided monoidal category C induces a monoidal admissible class of spans in the category of monoids in C; and it is shown in [3,Example 4.4] that if S satisfies [3, Assumption 4.1] then so does the induced class in the category of monoids. This allows for the discussion of S-relative categories in the category of monoids. In this paper we will be interested mainly in these relative categories of monoids. They contain, in particular, a split epimorphism of monoids (consisting of the morphisms i and s of ( * ) in the Introduction). So we start with the analysis of the following category of split epimorphisms of monoids. Proof We prove the theorem by constructing mutually inverse equivalence functors. The first one Let us see that the object map is meaningful. Then by [3,Proposition 3.5 (2)] it is invertible with the inverse q 1 in the second diagram. Both morphisms q −1 1 and q 1 are well-defined by the commutativity of the first diagram of (1.1); see [3,Proposition 3.5 (1)]. This proves that the object map of our candidate functor is meaningful. Concerning the morphism map, a 1 is a well-defined morphism in C by the assumption that b·s = s ·a (see [3, Proposition 3.5 (1)]) and it is a monoid morphism by [3,Proposition 3.7 (2)]. Condition p I · (a 1) = p I holds by construction and the other equality holds since the commutativity of the first diagram of implies the commutativity of the second diagram.
In the opposite direction DistLaw S (C) → SplitEpiMon S (C) we define a functor sending Here Y B is considered with the monoid structure induced by the distributive law x, see [ whose unlabelled regions commute since e : Y → I is a monoid morphism.) The rows are split epimorphisms (of monoids) by the unitality of the monoid morphism e. By (a') and the implies that the bottom row is the inverse of the isomorphism f 1 in the left column hence it is invertible. This proves that the object map is well defined.
Concerning the morphism map, it follows by the assumption yb · x = x · by that yb is a monoid morphism, see So we have well-defined functors in both directions, it remains to see that their composites are naturally isomorphic to the identity functors. The composite We claim that a natural isomorphism from this to the identity functor has the components Naturality with respect to any morphism ( B b B , A a A ) in SplitEpiMon S (C) follows by the commutativity of the first diagram of (1.2). Composing our functors in the opposite order we obtain the functor sending We claim that a natural isomorphism from this to the identity functor has the invertible Finally, the naturality with respect to an arbitrary morphism (

Example 1.2
For any fixed set X , the category C of spans over X is monoidal via the pullback over X . A monoid in C is a small category with the object set X and a monoid morphism is a functor acting on the objects as the identity map. Moreover, C has all pullbacks (computed in the underlying category of sets). So taking as S the class of all spans in C, from Theorem 1.1 we obtain the equivalence of the following categories (from now on we shall denote by s the source map and by t the target map of any category).
SplitEpiMon(C) whose objects are pairs of identity-on-objects functors B ι A σ between categories of the common object set X such that the composite σ ι is the identity functor, and the map is invertible. (The map of (1.3) is invertible e.g. if B is a groupoid; then its inverse takes a morphism a to (a · ι(σ (a) −1 ), t(a), σ (a)). ) morphisms are pairs of identity-on-objects functors ( A α A , B β B ) for which αι = ι β and βσ = σ α.
DistLaw(C) whose objects consist of categories B and Y with the common object set X such that Y has no morphisms between non-equal objects (that is, its source map s and target map t coincide; using the terminology of [4] this means that Y is a totally disconnected category); and an action B morphisms are pairs of identity-on-objects functors ( Only the above description of an object in DistLaw(C) requires some explanation. The monoidal unit of C is the trivial span X X X . Its trivial monoid structure yields the discrete category D(X ). An identity-on-objects functor Y e D(X ) as in Theo- is clearly a pullback of X -spans for any span morphism g. A δ A 2 f g XY is a comonoid morphism, which holds if and only if c · f g · δ = g f · δ. Thanks to the symmetry of M, its monoidal structure is inherited by C. A monoid A in C is known as a bimonoid in M. Recall that the monoidal structure of M is lifted to the category of (left or right) modules over the monoid A in M. A monoid (respectively, a comonoid) in the category of A-modules is known as an A-module monoid (respectively, A-module comonoid).

Recall from [3, Example 3.3] that for a cospan
of comonoids whose legs are in S, the S-relative pullback is given by the so-called cotensor product, defined as the equalizer This concise description of DistLaw S (C) requires a proof. Note that the monoidal unit I is now a terminal object in C; the unique morphism Y → I is the counit ε. It obviously sat- The other condition B B B ∈ S in (a') of Theorem 1.1 reduces to the requirement that the comonoid B is cocommutative.
Next we establish a bijective correspondence between distributive laws BY → Y B satisfying property (b') of Theorem 1.1 and left actions BY → Y as in the description above.
It is a unital action by the left unitality of x and it is associative by the left multiplicativity of x: By the right unitality of x the unit I u Y is a morphism of B-modules and by the right (note that here we also used the comultiplicativity of x). The condition that the counit Y ε I is a morphism of B-modules coincides with the counitality of l and also with the l is comultiplicative by the comultiplicativity of x: Conversely, given an action l as above, put x : clearly satisfies (b ) by the counitality of l hence it is counital. It is comultiplicative by the comultiplicativity of l: where the top-left region commutes by the coassociativity and cocommutativity of the comonoid B. This morphism x is a distributive law. Indeed, the left unitality and the left multiplicativity follow by the unitality and the associativity of the action l, respectively: and the right unitality and the right multiplicativity of x follow using that the unit and the multiplication of Y are B-module morphisms: The above correspondences between l and x are bijective by the commutativity of for a comultiplicative morphism x satisfying (b') and any morphism l.
is invertible without any further assumption; its inverse is constructed as In order to see that it is the inverse, indeed, recall that by [3, Hence the following diagrams commute.
This completes the characterization of the objects of DistLaw S (C). Concerning the mor- , the first condition in Theorem 1.1 is the counitality of the bimonoid morphism y hence it identically holds. The second condition in Theorem 1.1 is equivalent to y ·l = l · by by the commutativity of Example 1. 4 We can apply Example 1.3 to the particular case of a finitely complete category M regarded with the cartesian monoidal structure. Then the category C of comonoids in M is isomorphic to M. Since in this case the monoidal unit I of M is a terminal object, with the trivial monoid structure it becomes the zero object in the category of monoids in M. Then for any morphism  [20]. That is, for each element a of A, there is a unique element z a in the kernel of s such that a = z a · is(a). (Indeed, this condition clearly implies the surjectivity of q. For its injectivity assume and the uniqueness part of the Schreier property we infer z = z . Conversely, if q is invertible then its inverse a → (z a , s(a)) defines the required element z a of the kernel.) On the other hand, in this case an object of DistLaw S (M) reduces to a monoid morphism from B to the monoid of monoid endomorphisms of Y .
Recall that a bimonoid B-with monoid structure (m, u) and comonoid structure (δ, ε)-is a Hopf monoid provided that there exists a morphism B z B -the so-called antipode- If the antipode exists then it is unique. It is a monoid morphism from B to the monoid with the opposite multiplication m · c and comonoid morphism from B to the comonoid with the opposite comultiplication c · δ. Following ideas in [24], we use the antipode z of B and the image of the equalizer (1.5) under the functor −B to construct the inverse: This definition works because the horizontal morphism equalizes the parallel morphisms of the fork on the right; see Fig. 1. The so constructed morphism q −1 is the inverse of q by the commutativity of the diagrams of Fig. 2 (in the second case we also need to use that the columns are equal monomorphisms).
The top right region commutes by the Hopf monoid identity δ · z = zz ·c ·δ and the assumed cocommutativity of A. The bottom right region commutes since any bimonoid morphism s commutes with the antipodes.

Remark 1.7
There are particular symmetric monoidal categories M whose cocommutative Hopf monoids constitute semi-abelian categories Hopf(M); e.g. the category of sets (which is cartesian monoidal hence the Hopf monoids are the groups, all of them cocommutative) or the category of vector spaces over an arbitrary field [12] (for the particular case of an algebraically closed field see [11]). In such cases the equivalence of Proposition 1.5 (2)

Reflexive Graphs of Monoids Versus Pre-crossed Modules
Consider a monoidal admissible class S of spans in a monoidal category C for which The correspondence is given by Combining this observation with the equivalence of Theorem 1.1, next we present an equivalent description of a suitable category of reflexive graphs of monoids. This leads to the notion of pre-crossed module over a monoid.  1.1 exists).
PreX S (C) whose such that e · y = e, k · y = b · k and x · by = yb · x.
Proof We show that the equivalence functors of Theorem 1.1 lift to the equivalence of the claim. In the direction ReflGraphMon S (C) → PreX S (C) we send

holds by the commutativity of
Thus using again the proof of Theorem 1.1 we conclude that this functor is well-defined.
In the opposite direction the functor PreX S (C) → ReflGraphMon S (C) is defined by PreX(C) whose objects consist of categories B and Y of the common object set X such that Y is totally disconnected (in the sense of [4]); an action (cf. Example 1.2) B X Y Y and an identity-on-objects functor Y κ B such that

Relative Categories of Monoids Versus Crossed Modules
Consider again a monoidal admissible class S of spans in a monoidal category C for which [ Whenever the morphism is invertible, we infer from [3, Corollary 1.7] that there exists at most one monoid morphism d rendering commutative which is our candidate to serve as the composition morphism of a relative category. Note that if there is a monoid morphism d rendering commutative the diagram of (3.2), then it satisfies So if q 2 is invertible, then the only candidate is d = m · p A 1 · q −1 2 . By this motivation, in this section first we investigate the condition that q 2 of (3.1) is invertible. Assuming so, next we show that whenever the morphism d of (3.2) exists, it makes the object B i A s t of ReflGraphMon S (C) to an S-relative category. Finally, based on Theorem 2.1, we give an equivalent description of the category of S-relative categories in the category of monoids in C, in terms of crossed modules introduced hereby. (3) For a common section i of s and t, consider the morphism 1.1 (b)). If q n+1 is invertible for some n, then q k is invertible for all 0 < k ≤ n. Hence by the evident commutativity of the exterior of the diagram in part (2), universality of the S-relative pullback in its codomain implies the existence of the unique morphism h n .

4) (it is well-defined by [3, Proposition 3.5] and q 1 is equal to q in Theorem
(3) For some positive integer n assume that q n+1 is invertible. Then so is q n with the inverse    For any natural number n the following assertions hold.
has its legs in S (hence there exists its S-relative pullback Y B B Y n B).
(2) There exists a unique morphism b n+1 of spans (for the spans (3.11)) rendering commutative (3) If b n+1 in part (2) is an isomorphism then also b k is an isomorphism for all 0 < k ≤ n.
(4) For the morphism where f is the isomorphism in Theorem 1.1 (c').

Proof (1) By definition the first two spans in
belong to S hence so does the last one by the multiplicativity of S. Again, by definition the second and the third spans of (3.12) belong to S hence by the multiplicativity of S so does the first one in Then the second span of (3.13) is in S by (POST).
(2) Since the first span of (3.12) and the second span of (3.13) are in S, the multiplicativity of S implies that so is So by the evident commutativity of the exterior of the diagram of part (2) the stated morphism b n+1 exists. It is a morphism of spans (for the spans (3.11)) by the commutativity of the following diagrams.
(3) Since for a positive integer n, Y n−1 B 1...1u1 Y n B is a morphism between the spans of (3.11), the morphism in the top row of the following diagram is well-defined by [3, 632 G. Böhm 1k...k1 By their commutativity we infer b n+1 · 1 . . . 1u1 = (1 1 . . . 1u1) · b n Similarly, since for n > 0 also Y n B 1...1m1 Y n−1 B is a morphism between the spans of (3.11), the morphism in the top row of the following diagram is well-defined by [3,Proposition 3.5].
By their commutativity, b n ·1 . . . 1m1 = (1 1 . . . 1m1)·b n+1 . It follows from these identities and the unitality of the monoid Y that whenever b n+1 is invertible then so is b n with the inverse (4) We proceed by induction in n. For n = 0 the diagram in the claim reduces to the diagram whose upper half part commutes by construction (see part (2)) and the lower half part commutes since f 1 and q 1 are mutual inverses (see the proof of Theorem 1.1).
For any positive value of n, denote the top-right path in the diagram of the claim by b n+1 and the bottom row by q n+1 . Then the diagram takes the form The region at the bottom left corner commutes if the claim holds for n − 1; and the commutativity of the large region is proven in Fig. 4. (5) By Theorem 1.1 q 1 is an isomorphism without any further assumption; it is the inverse Assume that b l is iso for some l > 1. Take the diagram of part (4) for n = 1; it says b 2 = q 2 · f 11. Since f is an isomorphism by definition and b 2 is an isomorphism by part (3), also q 2 is an isomorphism. If l = 2 then this completes the proof. If l > 2 then take next the diagram of part (4) for n = 2; it says (1 b 2 ) · b 3 = q 3 · 1q 2 · f f 11. All of the occurring morphisms but q 3 are known to be isomorphisms proving that so is q 3 . Repeating this reasoning for all n ≤ l we conclude that q n is an isomorphism for all 0 < n ≤ l.
The opposite implication is proven by the same steps. Assume that q l is iso for some l > 1. Take the diagram of part (4) for n = 1; it says b 2 = q 2 · f 11. Since f is an isomorphism by definition and q 2 is an isomorphism by Lemma 3.1 (3), also b 2 is an isomorphism. If l = 2 then this completes the proof. If l > 2 then take next the diagram of part (4) for n = 2; it says (1 b 2 ) · b 3 = q 3 · 1q 2 · f f 11. All of the occurring morphisms but b 3 are known to be isomorphisms proving that so is b 3 . Repeating this reasoning for all n ≤ l we conclude that b n is an isomorphism for all 0 < n ≤ l.

(2) is invertible by Lemma 3.(5). Since the diagram
commutes, we conclude that the morphism in its bottom-right path-involving the equalizer j as in (1.5)-is the inverse of b n .
Let S be a monoidal admissible class of spans in a monoidal category C for which [3, We infer again by [3, Proposition 3.6] that the S-relative pullback in the second diagram of (3.14) exists and the top row of of the second diagram is an isomorphism.
Proof Assertion (1) follows by the commutativity of the diagrams and part (2) follows by the commutativity of the diagrams The following assertions hold.

The Composition Morphism of a Relative Category of Monoids
(1) There is at most one monoid morphism d rendering commutative The monoid morphism d of part (1) exists if and only if the following diagram commutes (recall that q 2 is invertible by Lemma 3.1 (3)).
(3) Whenever the monoid morphism d of part (1)  Since a monoid morphism d as in part (1) obviously renders commutative (3.15), this proves its uniqueness.
(2) By [3, Corollary 1.7] commutativity of the diagram of part (2) is equivalent to the existence of a (unique) monoid morphism making (3.15) commute. Since a monoid morphism d in part (1) provides such a morphism, its existence implies commutativity of the diagram of part (2).
In order to prove the converse implication, we show that any monoid morphism d making The stated expression of d immediately follows from [3, Corollary 1.7] (see also (3.3)).
(3) In order to see that the monoid morphism d in part (1) commutes since a a is multiplicative by [3,Proposition 3.7 (2)] and by the functoriality of ; see [3,Proposition 3.5 (2)]. It is used to prove the commutativity of the second diagram.

The Equivalence Between Relative Categories and Crossed Modules of Monoids
such that e · y = e, k · y = b · k and x · by = yb · x.
Proof It follows by Propositions 3.8 and 3.9 that CatMon S (C) is a full subcategory of ReflGraphMon S (C) and obviously Xmod S (C) is a full subcategory of PreX S (C). Below we show that the mutually inverse functors of Theorem 2.1 restrict to functors between these subcategories thus establishing the stated equivalence.
Regarding an object B i A proves (q q) · b 2 = q 2 · 1q. (Here the bottom-right region of the first diagram commutes since the lower half of the diagram of (3.9) commutes and the bottom-right region of the second diagram commutes since the lower half of the diagram of (3.10) commutes.) By the associativity of the monoid A and the multiplicativity of A B I p A A also the following diagram commutes.
With the help of these identities and Lemma 3.7, and using that the region marked by ( * ) commutes by Proposition 3.8 (2), the diagram of Fig. 5 is seen to commute. This proves that the stated object belongs to Xmod S (C) indeed. In The region at the top-right corner is the commutative diagram of Lemma 3.4 (4) for n = 1. The region bounded from below by the curved arrows commutes by Lemma 3.7. The region marked by (d') coincides with the diagram of part (d') hence it commutes.

Example 3.11
As in Example 1.2, take the (evidently admissible and monoidal) class of all spans in the category C of spans over a given set X . Then the equivalent categories of Theorem 3.10 take the following forms.
CatMon(C) whose objects are the double categories with the object set X and only identity horizontal morphisms and such that the morphism (1.3) is invertible. (This last condition holds e.g. if the vertical edge category is a groupoid.) morphisms are the double functors which are identities on the objects (and hence on the horizontal morphisms). Xmod(C) whose objects consist of categories B and Y with the common object set X such that Y is totally disconnected (in the sense of [4]); an action (see Example 1.2) B X Y Y and an identity-on-objects functor Y κ B such that and (κ(y) y ) · y = y · y for all morphisms b in B and y, y in Y for which s(b) = t(y) = t(y ). morphisms are the same as the morphisms in PreXMon(C), see Example 2.2.
Note that these equivalent categories have equivalent full subcategories whose objects are such that the category B is a groupoid; and other equivalent full subcategories whose objects are such that both of the occurring categories are groupoids. In the latter case these are the category of categories in the category of groupoids; and the category of crossed modules of groupoids in [5, Definition 1.2], respectively.

Example 3.12
In the setting of Example 1.3, the equivalent categories of Theorem 3.10 take the following explicit forms.
CatMon S (C) whose objects are S-relative categories B i A s t A B A d in the category of monoids in C-that is, in the category of bimonoids in M-such that the morphism q of Theorem 1.1 (b) is invertible. morphisms are S-relative functors in the category of monoids in C-that is, in the category of bimonoids in M. Xmod S (C) whose objects consist of a bimonoid Y and a cocommutative bimonoid B together with a left action BY l Y which makes Y both a B-module monoid and a B-module comonoid and a bimonoid morphism Y k B for which the following diagrams commute.
The third condition appears in [25,Definition 12 (v)] under the name Peiffer condition (motivated by the terminology for groups).
morphisms are pairs of bimonoid morphisms ( B b B , Y y Y ) such that k · y = b · k and l · by = y ·l.
These equivalent categories are equivalent furthermore to the full subcategory of ReflGraphMon S (C) of Example 2.3 for whose objects B i A s t the following diagrams commute.
(3.17) The above description of CatMon S (C) requires no further explanation. In the description of Xmod S (C) we need to show that the third diagram (the Peiffer condition) is equivalent to the diagram of Theorem 3.10 (d') in the current setting. The path on the right hand side of the diagram of Theorem 3.10 (d') appears as the left bottom path of the commutative diagram Hence it can be replaced by the top right path yielding the equivalent form (3.18) of the diagram of Theorem 3.10 (d'). The first diagram of Fig. 6 shows that if the diagram of (3.18) commutes then the Peiffer condition in the above presentation of Xmod S (C) holds. The opposite implication is proven by the second diagram of Fig. 6. In order to justify the further equivalent characterization of these categories as a full subcategory of ReflGraphMon S (C), we need to see the equivalence of the diagram of Proposition 3.8 (2) in the current setting and the diagram of (3.17). This follows by noting that the top row of the diagram of Proposition 3.8 (2) in the current setting appears in the left-bottom G. Böhm    [20]. Hence it extends the main result of [20]. • The full subcategory of Xmod S (C) for whose objects (B, Y , BY l Y , Y k B ) the bimonoid B in M is a Hopf monoid.
• The full subcategory of ReflGraphMon S (C) for whose objects B i A s t the following conditions hold.
-B is a Hopf monoid (with antipode z) -t1 · δ = t1 · c · δ -for the morphisms Proof The only ingredient that requires a proof is the equivalence of diagrams (3.17) and (3.19) in the case when B has an antipode z. The proof will repeatedly use the identity on − → s encoded in the following commutative diagram. I . Indeed, g A · p A = 1ε.q −1 · q · 1u = 1ε · 1u = 1.
On the other hand, since in Proposition 1.5 the inverse q −1 was constructed as the unique solution of p A 1 · q −1 = − → s s · δ, also the equality p A · g A = p A · 1ε.q −1 = 1ε. p A 1 · q −1 = 1ε · − → s s · δ = − → s holds, proving that − → s is idempotent.
Pre-composing both paths around (3.17) with the split epimorphism 1g A , we obtain the equivalent diagram m (3.21) Its rightmost region commutes by (3.20) and the fact that − → s is idempotent.
The morphism around the right hand side of (3.21) occurs as the left-bottom path of the commutative diagram Hence it can be replaced by the top-right path yielding the equivalent form of (3.21).
Finally, observe that for any morphisms A 2 φ,ψ A the following diagrams are equivalent: m (3.23) Indeed, the first diagram below shows that if the first diagram of (3.23) commutes then so does the second one; and the opposite implication follows by the second diagram below.
in whose objects they are both cocommutative Hopf monoids. In this way, Proposition 3.13 includes [25,Proposition 11] and [25,Theorem 14] about the equivalence between the category of so-called Cat 1 -Hopf algebras and the category of crossed modules over Hopf algebras; hence in particular the equivalence between the category of Cat 1 -groups and the category of crossed modules over groups in [6, Theorem 1]-where it is also referred to unpublished works by Verdier and Duskin-, [17, Lemma 2.2] and [14, Section 3.9] (whose language is most similar to ours).