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Journal of Algebra Torsors, herds and flocks
Torsors, herds and flocks
Thomas Booker, Ross Streetयह पुस्तक आपको कितनी अच्छी लगी?
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2011
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10.1016/j.jalgebra.2010.12.009
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Journal of Algebra 330 (2011) 346–374 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra Torsors, herds and ﬂocks Thomas Booker ∗,1 , Ross Street 2 Macquarie University Sydney, Faculty of Science, Sydney, NSW, Australia a r t i c l e i n f o Article history: Received 12 January 2010 Available online 15 January 2011 Communicated by Michel Van den Bergh a b s t r a c t This paper presents noncommutative and structural notions of torsor. The two are related by the machinery of Tannaka–Krein duality. © 2010 Elsevier Inc. All rights reserved. MSC: 18D35 18D10 20J06 Keywords: Torsor Herd Flock Monoidal category Descent morphism 1. Introduction Let us describe brieﬂy how the concepts and results dealt with in this paper are variants or generalizations of those existing in the literature. While aﬃne functions between vector spaces take lines to lines, they are not linear functions. In particular, translations do not preserve the origin. This phenomenon appears at the basic level with groups: left multiplication a− : G → G by an element a of a group G is not a group morphism. However, left multiplication does preserve the ternary operation q : G 3 → G deﬁned by q(x, y , z) = xy −1 z. Clearly, the operation q, together with a choice of any element as unit, determine the group structure. * 1 2 Corresponding author. Email address: thomas.booker@students.mq.edu.au (T. Booker). The author was supported by an Australian Postgraduate Award. The author gratefully acknowledges the support of an Australian Research Council Discovery Grant DP0771252. 00218693/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2010.12.009 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 347 A herd is a set A with a ternary operation q satisfying three simple axioms motivated by the group example. References for this term go back a long way, originating with the German form “Schar”: see [13,1,8]. Because Hopf algebras generalize groups, i; t is natural to consider the corresponding generalization of herds. Such a generalization, involving algebras with a ternary “cooperation”, is indeed considered in [11]; for later developments see [2] and references there. Moreover, using the term “quantum heaps” of his 2002 thesis, Škoda [14] proved an equivalence between the category of copointed quantum heaps and the category of Hopf algebras. In this paper, we use the term herd for a coalgebra with an appropriate ternary operation. As far as possible we work with comonoids in a braided monoidal category V (admitting reﬂexive coequalizers preserved by X ⊗ −) rather than the special case of categories of modules over a commutative ring in which comonoids interpret as coalgebras over the ring. The algebra version is included by taking V to be a dual category of modules (plus some ﬂatness assumptions). When V is the category of sets, we obtain the classical notion of herd. The theory of torsors for a group in a topos E appears in [6]. Torsors give an interpretation of the ﬁrst cohomology group of the topos with coeﬃcients in the group. In Section 2 we review how, when working in a topos E , torsors are herds with chosen elements existing locally. Locally here means after applying a functor − × R : E → E / R where R → 1 is an epimorphism in E . Such a functor is conservative and, since it has both adjoints, is monadic. In Section 3 we deﬁne herds in a braided monoidal category and make explicit the codescent condition causing them to be torsors for a Hopf monoid. This generalizes the topos case and slightly extends aspects of recent work of Grunspan [7] and Schauenburg [15,16]. Finite dimensional representations of a Hopf algebra form an autonomous (= compact = rigid) monoidal category. In fact, as explained in [4], Hopf algebras and autonomous monoidal categories are structures of the same kind: autonomous pseudomonoids in different autonomous monoidal bicategories. In Section 4, we introduce a general structure in an autonomous monoidal bicategory we call a “ﬂock”. It generalizes herd in the same way that autonomous pseudomonoid generalizes Hopf algebra. However, by looking at the autonomous monoidal bicategory V Mod, in Section 4 this gives us the notion of enriched ﬂock, roughly described as an autonomous monoidal V category without a chosen unit for the tensor product. We believe our use of the term ﬂock is new. However, our use is close to the concept of “heapy category” in [14]. The comodules admitting a dual over a herd in V form a V ﬂock. In Section 5 we adapt Tannaka duality (as presented, for example, in [9]) to relate V ﬂocks and herds in V . In Section 3 we need to know that the existence of an antipode in a bimonoid is equivalent to the invertibility of the socalled fusion operator. For completeness, we include in an Appendix A a direct proof shown to us by Micah McCurdy at the generality required. It is a standard result for Hopf algebras over a ﬁeld. 2. Recollections on torsors for groups Let G be a monoid in a cartesian closed, ﬁnitely complete and ﬁnitely cocomplete category E . A Gtorsor [6] is an object A with a Gaction μ : G × A −→ A such that: (i) the unique ! : A −→ 1 is a regular epimorphism; and, (ii) the morphism (μ, pr2 ) : G × A −→ A × A is invertible. The inverse to (μ, pr2 ) must have the form ( , pr2 ) : A × A −→ G × A where : A × A −→ G has the property that the following two composites are equal to the ﬁrst projections. 348 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 1×δ μ×1 1×δ ×1 μ G × A −→ G × A × A −→ A × A −→ G , A × A −→ A × A × A −→ G × A −→ A . If the epimorphism A −→ 1 is a retraction with right inverse a : 1 −→ A then 1×a μ G −→ G × A −→ A is an invertible morphism of Gactions where G acts on itself by its own multiplication. It follows that the existence of such a torsor forces G to be a group. If A is a Gtorsor in E , we obtain a ternary operation on A as the composite ×1 μ q : A × A × A −→ G × A −→ A . The following properties hold: q×1×1 A×A×A×A×A A×A×A = 1×1×q A×A×A (2.1) q A q A×A×A 1×δ q = A×A (2.2) A pr1 A×A×A δ×1 q = A×A (2.3) A pr2 That is, A becomes a herd (“Schar” in German) in E ; references for this term are [13,1,8]. Conversely, given a herd A in E for which ! : A −→ 1 is a regular epimorphism, a group G is obtained as the coequalizer : A × A −→ G of the two morphisms A×A×A 1×1×δ A×A×A×A q×1 A×A A×A×A and 1×1×! A×A with multiplication induced by q×1 A × A × A × A −→ A × A . δ The unit for G is constructed using the composite A −→ A × A −→ G and the coequalizer A×A A −→ 1. Moreover, there is an action of G on A induced by q which causes A to be a Gtorsor. T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 349 Our purpose is to generalize this to the case of comonoids in a monoidal category V in place of the cartesian monoidal E and to examine a higherdimensional version. We relate the two concepts using the Tannakian adjunction; see Chapter 16 of [17]. 3. Noncommutative torsors Let V be a braided monoidal category with tensor product ⊗ having unit I , and with reﬂexive coequalizers preserved by each X ⊗ −. For a comonoid A = ( A , δ : A → A ⊗ A , ε : A → I ) in V , we write A ◦ for the opposite comonoid δ c A, A ( A , A −→ A ⊗ A −→ A ⊗ A , ε : A → I ). For comonoids A and B, there is a comonoid A ⊗ B = ( A ⊗ B, A ⊗ B δ⊗δ 1⊗c A , B ⊗1 A⊗A⊗B⊗B A ⊗ B ⊗ A ⊗ B, A ⊗ B ⊗ I ). Deﬁnition 1. A comonoid A in V is called a herd when it is equipped with a comonoid morphism q : A ⊗ A ◦ ⊗ A −→ A (3.1) for which the following conditions hold: A⊗A⊗A⊗A⊗A q ⊗1 ⊗1 A⊗A⊗A = 1⊗1⊗q A⊗A⊗A (3.2) q A q A⊗A⊗A 1⊗δ A⊗A q = (3.3) A 1 ⊗ A⊗A⊗A δ⊗1 A⊗A q = (3.4) A ⊗1 Such structures, stated dually, occur in [11] and [2], for example. For any comonoid A in V we have the category Cml A of left Acomodules δ : M −→ A ⊗ M. There is a monad T A on Cml A deﬁned by δ⊗1 δ T A ( M −→ A ⊗ M ) = ( A ⊗ M −→ A ⊗ A ⊗ M ), (3.5) with multiplication and unit for the monad having components 1 ⊗ε ⊗1 A⊗A⊗M− −−−−−→ A ⊗ M and δ M −→ A ⊗ M . (3.6) 350 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 There is a comparison functor K A : V −→ (Cml A ) T A into the category of Eilenberg–Moore algebras taking X to the Acomodule δ ⊗ 1 : A ⊗ X −→ A ⊗ A ⊗ X with T A action 1 ⊗ ε ⊗ 1 : A ⊗ A ⊗ X → A ⊗ X . We say that ε : A → I is a codescent morphism when K A is fully faithful. If K A is an equivalence of categories (that is, the right adjoint to the underlying functor Cml A −→ V is monadic) then we say ε : A → I is an effective codescent morphism. Deﬁnition 2. A torsor in V is a herd A for which the counit ε : A → I is a codescent morphism. σ and τ : A ⊗ A ⊗ A → A ⊗ A deﬁned Let A be a herd. We asymmetrically introduce morphisms by σ = (A ⊗ A ⊗ A 1⊗1⊗δ A⊗A⊗A⊗A τ = (A ⊗ A ⊗ A 1 ⊗1 ⊗ε q ⊗1 A ⊗ A) and A ⊗ A ). These form a reﬂexive pair using the common right inverse 1⊗δ A ⊗ A −→ A ⊗ A ⊗ A . Let : A ⊗ A −→ H be the coequalizer of σ and τ . It is easily seen that there is a unique comonoid structure on H such that : A ⊗ A ◦ −→ H becomes a comonoid morphism which means the following diagrams commutes (where c 1342 is the positive braid whose underlying permutation is 1342). A ⊗2 H ⊗ δ A ⊗2 H ⊗2 = = ⊗ δ⊗δ A ⊗4 A ⊗4 c 1342 I (3.7) H Since H is a reﬂexive coequalizer, we have the coequalizer A ⊗6 σ ⊗σ τ ⊗τ ⊗ A ⊗4 −→ H ⊗2 . (3.8) q ⊗1 A ⊗4 −→ A ⊗2 −→ H (3.9) It is readily checked that A ⊗6 commutes. So there exists a unique σ ⊗σ τ ⊗τ μ : H ⊗2 −→ H such that μ q ⊗1 ⊗ A ⊗4 −→ H ⊗2 −→ H = A ⊗4 −→ A ⊗2 −→ H . (3.10) T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 351 Proposition 1. This μ : H ⊗2 → H is associative and a comonoid morphism. μ is a comonoid morphism Proof. Associativity follows easily from Eqs. (3.2) and (3.10). To show that we need to prove the equations δ⊗δ μ⊗μ 1⊗c ⊗1 μ δ H ⊗2 −→ H ⊗4 −−−−−→ H ⊗4 −→ H ⊗2 = H ⊗2 −→ H −→ H ⊗2 , (3.11) μ ⊗ H ⊗2 −→ I = H ⊗2 −→ H −→ I . (3.12) The following diagram proves (3.11) while Eq. (3.12) follows easily from the second diagram of (3.7) and the fact that and q preserve counits. δ⊗δ H ⊗2 H ⊗4 1⊗c ⊗1 ⊗ ⊗ ⊗ H ⊗4 A ⊗8 ⊗ ⊗ ⊗ μ⊗μ ⊗ c 13425786 A ⊗4 δ⊗δ⊗δ⊗δ A ⊗8 c 12563478 A ⊗8 c 15623784 H ⊗2 q⊗1⊗q⊗1 c 14523678 c 12356748 A ⊗8 A ⊗4 ⊗ q ⊗1 ⊗ A ⊗2 c 1324 q⊗q⊗1⊗1 δ⊗δ A ⊗4 δ H ⊗2 H μ 2 Proposition 2. If A is a torsor then H is a Hopf monoid and A is a left H torsor. Proof. In order to construct a unit for the multiplication on H , we use the codescent condition on ε : A → I . We deﬁne η : I −→ H by providing an Eilenberg–Moore T A algebra morphism from K A I = ( A, A ⊗ A 1 ⊗ε A) to K A H = (A ⊗ H, A ⊗ A ⊗ H 1 ⊗ε ⊗1 A ⊗ H) 352 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 and using the assumption that K A is fully faithful. The morphism is c A, A δ 1⊗δ 1 ⊗ A −→ A ⊗ A −→ A ⊗ A −→ A ⊗ A ⊗ A −→ A ⊗ H . (3.13) It is an ( A ⊗ −)algebra morphism since (1 ⊗ ε ⊗ 1)(1 ⊗ 1 ⊗ )(1 ⊗ 1 ⊗ δ)(1 ⊗ c )(1 ⊗ δ) = (1 ⊗ )(1 ⊗ δ)(1 ⊗ ε ⊗ 1)(1 ⊗ c )(1 ⊗ δ) = (1 ⊗ )(1 ⊗ δ) = (1 ⊗ )(1 ⊗ 1 ⊗ 1 ⊗ ε )c 2341 (δ ⊗ δ) = (1 ⊗ )(1 ⊗ q ⊗ 1)(1 ⊗ δ ⊗ δ)c 231 (1 ⊗ δ) = (1 ⊗ )(1 ⊗ δ)c 21 δ(1 ⊗ ε ). So indeed we can deﬁne η by 1 A ⊗ η = (1 ⊗ )(1 ⊗ δ)c A , A δ. Here is the proof that η is a right unit: (1 ⊗ μ)(1 ⊗ 1 ⊗ η)(1 ⊗ ) = (1 ⊗ μ) c −1 ⊗ 1 (1 ⊗ 1 ⊗ η)c (1 ⊗ ) = (1 ⊗ μ) c −1 ⊗ 1 (1 ⊗ 1 ⊗ )(1 ⊗ 1 ⊗ δ)(1 ⊗ c δ)c (1 ⊗ ) = (1 ⊗ μ)(1 ⊗ ⊗ δ)(c 231 )−1 (1 ⊗ 1 ⊗ c δ)c 231 = (1 ⊗ )(1 ⊗ q ⊗ 1)(1 ⊗ 1 ⊗ δ)(c 231 )−1 (1 ⊗ 1 ⊗ c δ)c 231 = (1 ⊗ )(1 ⊗ 1 ⊗ 1 ⊗ ε )(c 231 )−1 (1 ⊗ 1 ⊗ c δ)c 231 = 1 ⊗ . Here is the proof that η is a left unit: (1 ⊗ μ)(1 ⊗ η ⊗ 1)(1 ⊗ ) = (1 ⊗ μ)(1 ⊗ ⊗ 1)(1 ⊗ δ ⊗ 1)(c δ ⊗ 1)(1 ⊗ ) = (1 ⊗ μ)(1 ⊗ ⊗ )(1 ⊗ δ ⊗ 1 ⊗ 1)(c δ ⊗ 1 ⊗ 1) = (1 ⊗ )(1 ⊗ q ⊗ 1)(1 ⊗ δ ⊗ 1 ⊗ 1)(c δ ⊗ 1 ⊗ 1) = (1 ⊗ )(1 ⊗ ε ⊗ 1 ⊗ 1)(c δ ⊗ 1 ⊗ 1) = 1 ⊗ . To complete the proof that H is a bimonoid, one easily checks the remaining properties: (3.14) T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 η 353 η ⊗η δ I −→ H −→ H ⊗ H = I −→ H ⊗ H , η ε (3.15) 1 I −→ H −→ I = I −→ I . (3.16) Using the coequalizer σ ⊗1 A ⊗4 τ ⊗1 ⊗1 A ⊗3 −→ H ⊗ A , (3.17) we can deﬁne a morphism μ : H ⊗ A −→ A (3.18) by the condition ⊗1 μ q A ⊗ A ⊗ A −→ H ⊗ A −→ A = A ⊗ A ⊗ A −→ A . (3.19) It is easy to see that this is a comonoid morphism and satisﬁes the two axioms for a left action of the bimonoid H on A. Moreover, the fusion morphism μ⊗1 1⊗δ v = ( H ⊗ A −→ H ⊗ A ⊗ A −→ A ⊗ A ) (3.20) has inverse 1⊗δ ⊗1 A ⊗ A −→ A ⊗ A ⊗ A −→ H ⊗ A . (3.21) In fact, H is a Hopf monoid. To see this, consider the fusion morphism μ⊗1 1⊗δ v = ( H ⊗ H −→ H ⊗ H ⊗ H −→ H ⊗ H ). (3.22) It suﬃces to prove this is invertible (see Appendix A for a proof that this implies the existence of an antipode). Again we appeal to the codescent property of ε : A → I ; we only require that V −→ Cml A is conservative (reﬂects invertibility). For, we have the fusion equation H⊗H⊗A 1⊗ v H⊗A⊗A v ⊗1 v ⊗1 = H⊗H⊗A A⊗A⊗A 1⊗ v (3.23) 1⊗ v H⊗A⊗A c ⊗1 A⊗H⊗A which holds in Cml A. All morphisms here are known to be invertible with the exception of H ⊗ H ⊗ v ⊗1 A −→ H ⊗ H ⊗ A. So that possible exception is also invertible. Alternatively, for the herd A, we can introduce morphisms by 2 σ and τ : A ⊗ A ⊗ A → A ⊗ A deﬁned 354 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 1⊗q δ⊗1⊗1 σ = ( A ⊗ A ⊗ A −−−−−→ A ⊗ A ⊗ A ⊗ A −−−−→ A ⊗ A ) and τ = (A ⊗ A ⊗ A ε ⊗1 ⊗1 A ⊗ A) . These form a reﬂexive pair using the common right inverse δ⊗1 A ⊗ A −→ A ⊗ A ⊗ A . Let : A ⊗ A −→ H be the coequalizer of σ and τ . Then there is a unique comonoid structure on H such that : A ◦ ⊗ A −→ H becomes a comonoid morphism. Symmetrically to H , we see that A becomes a right H torsor for the Hopf monoid H . Indeed, A is a torsor from H to H in the sense that the actions make A a left H , right H bimodule. 4. Flocks Flocks are a higherdimensional version of herds. Our use of the term may be in conﬂict with other uses in the literature (such as [3]). Deﬁnition 3. Let M denote a right autonomous monoidal bicategory [5]. So each object X has a bidual X ◦ with unit n : I −→ X ◦ ⊗ X and counit e : X ⊗ X ◦ −→ I . A left ﬂock in M is an object A equipped with a morphism q : A ⊗ A ◦ ⊗ A −→ A (4.1) and 2cells q ⊗1 ⊗1 A ⊗ A◦ ⊗ A ⊗ A◦ ⊗ A 1⊗1⊗q ∼ = φ A ⊗ A◦ ⊗ A A ⊗ A◦ ⊗ A (4.2) q A q A ⊗ A◦ ⊗ A 1⊗n A q α (4.3) A 1 e ⊗1 A ⊗ A◦ ⊗ A A β q n satisfying the following three conditions (where 1n = 1 ⊗ · · · ⊗ 1): (4.4) T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 355 q(12 ⊗ q)(q ⊗ 14 ) ∼ = q(q ⊗ 12 )(14 ⊗ q) φ(q⊗14 ) φ(14 ⊗q) q(q ⊗ 12 )(q ⊗ 14 ) q(12 ⊗ q)(14 ⊗ q)) = (4.5) q(12 ⊗φ) q(φ⊗12 ) q(q ⊗ 12 )(12 ⊗ q ⊗ 12 ) 1 q(12 ⊗ q)(12 ⊗ q ⊗ 12 ) φ(12 ⊗q⊗12 ) ∼ = (e ⊗ 1)(1 ⊗ n) β(1⊗n) q(1 ⊗ n) = α 1 1 q(12 ⊗ q)(11 ⊗ n ⊗ 12 ) (4.6) φ −1 (11 ⊗n⊗12 ) q(q ⊗ 12 )(11 ⊗ n ⊗ 12 ) = q(12 ⊗β)(11 ⊗n⊗12 ) q (α ⊗1 2 ) q(12 ⊗ e ⊗ 11 )(11 ⊗ n ⊗ 12 ) ∼ =q = q13 ∼ (4.7) q∼ = q13 1 Example 1. Suppose A is a left autonomous pseudomonoid in M in the sense of [4] and [12]. A pseudomonoid consists of an object A, morphisms p : A ⊗ A → A and j : I → A, 2cells φ : p ( p ⊗ 1) ⇒ p (1 ⊗ p ), λ : p ( j ⊗ 1) ⇒ 1 and ρ : p (1 ⊗ j ) ⇒ 1, satisfying coherence axioms. It is left autonomous when it is equipped with a left dualization morphism d : A ◦ → A having 2cells α : p (d ⊗ 1)n ⇒ j and β : je ⇒ p (1 ⊗ d), satisfying two axioms. Put q = ( A ⊗ A◦ ⊗ A 1⊗d⊗1 A⊗A⊗A p ⊗1 A⊗A p A ), φ : q(q ⊗ 12 ) = p ( p ⊗ 1)(1 ⊗ d ⊗ 1)( p ⊗ 12 )( p ⊗ 13 )(1 ⊗ d ⊗ 13 ) ∼ = p ( p ⊗ 1)( p ⊗ 12 )( p ⊗ 13 )(1 ⊗ d ⊗ 1 ⊗ d ⊗ 1) ∼ = p 5 (1 ⊗ d ⊗ 1 ⊗ d ⊗ 1) ∼ = p ( p ⊗ 1)(12 ⊗ p )(12 ⊗ p ⊗ 1)(1 ⊗ d ⊗ 1 ⊗ d ⊗ 1) ∼ = q(12 ⊗ q), α : q(1 ⊗ n) ∼ = p (1 ⊗ ( p (d ⊗ 1)n)) β :e⊗1∼ = p ( j ⊗ 1)(e ⊗ 1) ∼ = p (( je ) ⊗ 1) p (β⊗1) p (1 ⊗α ) p (1 ⊗ j ) ∼ = 1, p (( p (1 ⊗ d)) ⊗ 1) ∼ = q. The axioms for the ﬂock ( A , q, φ, α , β) follow from those on ( A , p , φ, λ, ρ , d, α , β) in [4]. Remark. As noted in [5], Baez–Dolan coined the term “microcosm principal” for the phenomenon whereby a concept ﬁnds its appropriate level of generality in a higherdimensional version of the concept. In particular, monoids in the category of sets generalize to monoids in monoidal categories. Similarly, monoidal categories generalize to pseudomonoids in monoidal bicategories. For 356 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 autonomous pseudomonoids the context is an autonomous monoidal bicategory. Example 1 shows that autonomous monoidal categories become ﬂocks. Although we shall not explicitly deﬁne a biﬂock, an autonomous monoidal bicategory would be an example. The general context for ﬂock would be biﬂock. Given a left ﬂock A, consider the mate q̂ : A ◦ ⊗ A −→ A ◦ ⊗ A of q under the biduality A b (4.8) A ◦ ; that is, q̂ is the composite 1⊗q n ⊗1 2 A ◦ ⊗ A −→ A ◦ ⊗ A ⊗ A ◦ ⊗ A −→ A ◦ ⊗ A . Deﬁne (4.9) μ : q̂q̂ ⇒ q̂ to be the composite (1 ⊗ q)(n ⊗ 12 )(1 ⊗ q)(n ⊗ 12 ) ∼ = (1 ⊗ q)(1 ⊗ q ⊗ 1)(1 ⊗ n ⊗ 1)(n ⊗ 12 ) and (1⊗q)(1⊗α ⊗1)(n⊗12 ) (1 ⊗ q)(n ⊗ 12 ) (4.10) η : 12 ⇒ q̂ to be the composite 12 ∼ = (1 ⊗ e ⊗ 1)(n ⊗ 12 ) (1⊗β)(n⊗12 ) (1 ⊗ q)(n ⊗ 12 ). (4.11) Proposition 3. (q̂, μ, η) is a monad on A ◦ ⊗ A. Proof. Conditions (4.5), (4.6) and (4.7) translate to the monad axioms. 2 Assume a Kleisli object K exists for the monad q̂. So we have a morphism h : A ◦ ⊗ A −→ K (4.12) χ : hq̂ ⇒ h (4.13) and a 2cell forming the universal Eilenberg–Moore algebra for the monad deﬁned by precomposition with q̂. It follows that h is a map in the bicategory M; that is, h has a right adjoint h∗ : K −→ A ◦ ⊗ A. Proposition 4. The morphism 1 ⊗ q : A ◦ ⊗ A ⊗ A ◦ ⊗ A −→ A ◦ ⊗ A induces a pseudoassociative multiplication p : K ⊗ K −→ K . T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 357 Proof. We obtain a monad opmorphism (1 ⊗ q, ψ) : ( A ◦ ⊗ A ⊗ A ◦ ⊗ A , q̂ ⊗ q̂) −→ ( A ◦ ⊗ A , q̂) where ψ is the composite 2cell (1 ⊗ q)(q̂ ⊗ q̂) = (1 ⊗ q)(13 ⊗ q)(1 ⊗ q ⊗ 14 )(14 ⊗ n ⊗ 12 )(n ⊗ 14 ) ∼ = (1 ⊗ q)(13 ⊗ q)(13 ⊗ q ⊗ 12 )(14 ⊗ n ⊗ 12 )(n ⊗ 14 ) ⇒ (1 ⊗ q)(13 ⊗ q)(n ⊗ 14 ) ∼ = (1 ⊗ q)(n ⊗ 12 )(1 ⊗ q) ∼ = q̂(1 ⊗ q). (4.14) Since the Kleisli object for the monad q̂ ⊗ q̂ is K ⊗ K , the morphism p : K ⊗ K −→ K is induced by (1 ⊗ q, ψ). The invertible 2cell φ of (4.2) induces an invertible φ : p ( p ⊗ 1) ⇒ p (1 ⊗ p ) satisfying the appropriate “pentagon” condition following from condition (4.5). 2 Proposition 5. The morphism q : A ⊗ A ◦ ⊗ A −→ A induces a pseudoaction q : A ⊗ K −→ A . If q is a map then so is q. Proof. The Kleisli object for the monad 1 ⊗ q̂ on A ⊗ A ◦ ⊗ A is A ⊗ K . So there exists a unique q : A ⊗ K −→ A such that q (1 ⊗χ ) q(1 ⊗ q̂) ∼ = q(1 ⊗ h)(1 ⊗ q̂) = q(1 ⊗ h) ∼ =q q(1 ⊗ q̂) = q(12 ⊗ q)(1 ⊗ n ⊗ 12 ) ∼ = q(q ⊗ 12 )(1 ⊗ n ⊗ 12 ) q (α ⊗1 2 ) q . 2 Proposition 6. The morphism h : A ◦ ⊗ A −→ K is a parametric left adjoint for q : A ⊗ K −→ A. That is, the mate 1⊗q n ⊗1 K −→ A ◦ ⊗ A ⊗ K −→ A ◦ ⊗ A (4.15) of q is right adjoint to h : A ◦ ⊗ A −→ K . 5. Enriched ﬂocks Suppose V is a base monoidal category of the kind considered in [10] and we will use the enriched category theory developed there. In particular recall that the end A T ( A , A ) of a V functor T : Aop ⊗ A → V is constructed as the equalizer T ( A, A) A V A( A , B ), T ( A , B ) . T ( A, A) A,B A For a V functor T : Aop ⊗ A → X , the end isomorphisms A T ( A , A ) and coend A T ( A , A ) are deﬁned by V natural 358 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 A A A X X X , T ( A , A ) and T ( A, A) ∼ = X X, T ( A , A ), X X T ( A , A ), X . ∼ = A Deﬁnition 4. A left V ﬂock is a ﬂock A in V Mod for which the structure module Q : A ⊗ Aop ⊗ A −→ A (5.1) of (4.1) is a V functor. More explicitly, we have a V category A and a V functor Q : A ⊗ Aop ⊗ A −→ A equipped with V natural transformations ∼ = φ : Q Q ( A , B , C ), D , E −→ Q A , B , Q (C , D , E ) , (5.2) α AB : Q ( A , B , B ) −→ A , (5.3) β BA (5.4) : B −→ Q ( A , A , B ), with φ invertible, such that the following three conditions hold: Q ( Q ( A , B , C ), D , Q ( E , F , G )) φ Q ( A , B , Q (C , D , Q ( E , F , G ))) φ = Q ( Q ( Q ( A , B , C ), D , E ), F , G ) Q (1,1,φ) (5.5) Q (φ,1,1) Q ( Q ( A , B , Q (C , D , E )), F , G ) φ Q ( A , B , Q ( Q (C , D , E ), F , G ) φ −1 Q ( A , B , Q ( B , B , C )) Q ( Q ( A , B , B ), B , C ) = Q (1,1,β) Q ( A, B , C ) Q (α ,1,1) (5.6) Q ( A, B , C ) 1 Q ( A, A, A) β A = 1 α (5.7) A The following observation characterizes the special case of Example 3 (below) where H = A. Proposition 7. Suppose A is a V ﬂock which has an object J for which α AK : Q ( A , J , J ) → A and β BK : B −→ Q ( J , J , B ) are invertible for all A and B. Then A becomes a left autonomous monoidal V category by deﬁning A ⊗ B = Q ( A, J , B ) and A ∗ = Q ( J , A , J ). (5.8) T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 The associativity and unit constraints are deﬁned by instances of φ , A ∗ A are α KA and β KA . 359 α and β , while the counit and unit for Now, given any V ﬂock, we have the Kleisli V category K and a V functor H : Aop ⊗ A −→ K (5.9) constructed as in (4.12). The objects of K are pairs ( A , B ) as for Aop ⊗ A and the homobjects are deﬁned by K ( A , B ), (C , D ) = A B , Q ( A , C , D ) . (5.10) Composition C ,D K (C , D ), ( E , F ) ⊗ K ( A , B ), (C , D ) −→ K ( A , B ), ( E , F ) (5.11) for K is deﬁned to be the morphism C ,D A D , Q (C , E , F ) ⊗ A B , Q ( A , C , D ) −→ A B , Q ( A , E , F ) equal to the composite of the canonical Yoneda isomorphism with the composite C A B , Q A , C , Q (C , E , F ) C A(1,φ −1 ) −−−−−−−−−→ C C A(1, Q (α ,1,1)) A B , Q Q ( A , C , C ), E , F −−−−−−−−−−−−→ A B , Q ( A , E , F ) . The V functor H of (5.9) is the identity on objects and its effect on homobjects H A(C , A ) ⊗ A( B , D ) −→ A B , Q ( A , C , D ) is the V natural family corresponding under the Yoneda Lemma to the composite jB A(1,β) I −→ A( B , B ) −→ A B , Q ( A , A , B ) . Example 2. Let X and Y be V categories for a suitable V . There is a V category Adj(X , Y ) of adjunctions between X and Y : the objects F = ( F , F ∗ , ε , η) consist of V functors F : X → Y , F ∗ : Y → X , and V natural transformations ε : F F ∗ ⇒ 1Y and η : 1X ⇒ F ∗ F which are the counit and unit for an adjunction F F ∗ ; the hom objects are deﬁned by X (F X, G X) ∼ = [Y , X ] G ∗ , F ∗ Adj(X , Y )( F , G ) = [X , Y ]( F , G ) = X where [X , Y ] is the V functor V category [10]. We obtain a V ﬂock A = Adj(X , Y ) by deﬁning Q ( F , G , H ) = H G ∗ F , F ∗ G H ∗ , ε, η 360 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 where ε and η come by composition from those for F F ∗ , G G ∗ and H H ∗ . In this case φ is an equality while α and β are induced by the appropriate counit ε and unit η . Notice that K ( F , G ), ( H , K ) = [Y , Y ] G F ∗ , K H ∗ . Example 3. Let H be a left autonomous monoidal V category for suitable V . For each X ∈ H, we have X ∗ ∈ H and ε : X ∗ ⊗ X −→ I and η : I −→ X ⊗ X ∗ inducing isomorphisms H X∗ ⊗ Y , Z ∼ = H(Y , X ⊗ Z ) and H( Y ⊗ X , Z ) ∼ = H Y , Z ⊗ X∗ . So X (5.12) (5.13) Z = Z ⊗ X ∗ acts as a left internal hom for H and X, Y = X∗ ⊗ Y (5.14) acts as a left internal cohom. Notice that X (Z ⊗ Y ) ∼ = Z ⊗ XY = Z ⊗ Y ⊗ X∗ ∼ and (5.15) X, Y ⊗ Z ∼ = X, Y ⊗ Z. = X∗ ⊗ Y ⊗ Z ∼ (5.16) Suppose A is a right Hactegory; that is, A is a V category equipped with a V functor ∗ : A ⊗ H −→ A (5.17) (A ∗ X) ∗ Y ∼ = A ∗ ( X ⊗ Y ) and (5.18) A∗I∼ =A (5.19) and V natural isomorphisms satisfying the obvious coherence conditions. Suppose further that each V functor A ∗ − : H −→ A (5.20) A , − : A −→ H. (5.21) −,− : Aop ⊗ A −→ H (5.22) H A, B , Z ∼ = A( B , A ∗ Z ). (5.23) has a left adjoint It follows that we have a V functor and a V natural isomorphism To encapsulate: Aop is a tensored Hop category. Observe that we have a canonical isomorphism A, B ∗ Z ∼ = A, B ⊗ Z . For, we have the natural isomorphisms (5.24) T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 361 H A , B ∗ Z , X A( B ∗ Z , A ∗ X ) ∼ = A B, A ∗ X ⊗ Z ∗ = A B, ( A ∗ X) ∗ Z ∗ ∼ ∼ = H A, B ⊗ Z , X . = H A, B , X ⊗ Z ∗ ∼ There are also canonical morphisms e : A , B −→ I and (5.25) d : B −→ A ∗ A , B (5.26) corresponding respectively under isomorphism (5.23) to A ∼ = A ∗ I and 1 : A , B −→ A , B . Finally, we come to our example of a left V ﬂock. The V category is A and the V functor Q : A ⊗ Aop ⊗ A −→ A of (5.1) is Q ( A, B , C ) = A ∗ B , C . (5.27) The isomorphism φ of (5.2) is derived from (5.24) as Q Q ( A , B , C ), D , E = A ∗ B , C ∗ D , E ∼ = A ∗ B, C ∗ D, E ∼ = Q A , B , Q (C , D , E ) . The natural transformations (5.28) α and β of (5.3) and (5.4) are 1∗e Q ( A , B , B ) = A ∗ B , B −→ A ∗ I ∼ =A and (5.29) d B −→ A ∗ A , B = Q ( A , A , B ). (5.30) In this case the Kleisli V category K of (5.9) is given by the V functor −,− of (5.22). Notice that K is closed under binary tensoring in H since, by (5.24), we have A, B ⊗ C , D ∼ = A, B ∗ C , D . However, K may not contain the I of H. Deﬁnition 5. Suppose F : A −→ X is a V functor between V ﬂocks A and X . We call F ﬂockular when it is equipped with a V natural family consisting of maps ρ A , B ,C : Q ( F A , F B , F C ) → F Q ( A , B , C ) such that Q ( F Q ( A , B , C ), F D , F E ) ρ Q (ρ ,1,1) Q ( Q ( F A , F B , F C ), F D , F E ) Fφ = F Q ( A , B , Q (C , D , E )) ρ φ Q ( F A , F B , Q ( F C , F D , F E )) F Q ( Q ( A , B , C ), D , E ) Q (1,1,ρ ) Q ( F A , F B , F Q (C , D , E )) 362 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 F Q ( A, B , B ) ρ Fα = Q ( F A, F B , F B ) (5.31) FA α Q ( F A, F A, F B ) β ρ = FB (5.32) F Q ( A, A, B ) Fβ (where we have suppressed the subscripts of is invertible. ρ ) commute. We call F strong ﬂockular when each ρ A , B ,C 6. Herd comodules Let A be a herd in V in the sense of Section 3. In the ﬁrst instance, A is a comonoid. Write Cmr f A for the V category of right Acomodules whose underlying objects in V have duals. Write V f for the full subcategory of V consisting of the objects with duals. We shall show that Cmr f A is a V ﬂock in the sense of Section 5 and that the forgetful V functor U : Cmr f A −→ V f is strong ﬂockular. The ﬂock structure on V f is Q ( L , M , N ) = L ⊗ M ∗ ⊗ N which is a special case of Example 3 with A = H = V f . We wish to lift this ﬂock structure on V f to Cmr f A. So, assuming L, M and N are right Acomodules with duals in V , we need to provide a right Acomodule structure on L ⊗ M ∗ ⊗ N. This is deﬁned as the composite L ⊗ M∗ ⊗ N 1 ⊗1 ⊗η ⊗1 M∗ ⊗ N ⊗ A L ⊗ M∗ ⊗ M ⊗ M∗ ⊗ N 1 ⊗1 ⊗ε ⊗1 ⊗1 ⊗1 ⊗1 L ⊗ M∗ ⊗ N ⊗ A ⊗ A ⊗ A δ⊗1⊗δ⊗1⊗δ L ⊗ A ⊗ M ∗ ⊗ M ⊗ A⊗ L ⊗ A ⊗ A ⊗ M∗ ⊗ N ⊗ A 1⊗1⊗1⊗q c 145236 L ⊗ M ∗ ⊗ N ⊗ A. (6.1) In terms of strings we can write this as (6.2) Theorem 8. For all right Acomodules L, M and N with duals in V , the composite (6.1) renders Q ( L , M , N ) = L ⊗ M ∗ ⊗ N a right Acomodule such that the canonical morphisms T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 363 φ : Q Q ( L , M , N ), R , S → Q L , M , Q ( N , R , S ) , α : Q ( L , M , M ) → L and β : M → Q ( L , L , M ) in V are right Acomodule morphisms. Further, Cmr f A is a ﬂock such that the forgetful functor U : Cmr f A → V f is strong ﬂockular. Proof. The main coaction axiom for the composite (6.1) follows by using a duality (“snake”) identity for M ∗ M, the coaction axioms for L, M and N, and that q is a comonoid morphism. The fact that φ , α and β are right comodule morphisms follow from duality identities, comonoid axioms, and Eqs. (3.2), (3.3) and (3.4). The calculation for φ is straight forward The one for α is 364 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 and the one for β is the forgetful functor Cmr f A → V f is faithful, the ﬂock axioms hold in Cmr f A because they do in V f , it follows that Cmr f A is a ﬂock. Clearly also the V functor U : Cmr f A −→ V f is strong ﬂockular. 2 7. Tannaka duality for ﬂocks and herds Given a strong ﬂockular V functor F : A −→ V f , we show that, when the coend A E = End∨ F = ( F A )∗ ⊗ F A , (7.1) T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 365 exists in V , it is a herd in V . To simplify notation, let us put e X = X ∗ ⊗ X for X ∈ V so that, for A ∈ A, we have a coprojection copr A : e F A −→ E . It is well known that E is a comonoid (see [17] for example). The comultiplication δ : E → E ⊗ E is deﬁned by commutativity of 1 ⊗η ⊗1 eF A eF A ⊗ eF A copr A ⊗copr A copr A E (7.2) E⊗E δ The counit ε : E → I restricts along the coprojection copr A : e F A → E to yield the counit for the duality F A ∗ F A. We have the following isomorphisms A E ⊗3 = B eF A ⊗ C eF B ⊗ eF C A , B ,C ∼ = eF A ⊗ eF B ⊗ eF C A , B ,C ∼ = ( F A ) ⊗ ( F B )∗ ⊗ F C ∗ ⊗ F A ⊗ ( F B )∗ ⊗ F C A , B ,C ∼ = Q ( F A , F B , F C )∗ ⊗ Q ( F A , F B , F C ) A , B ,C ∼ = ∗ F Q ( A, B , C ) ⊗ F Q ( A, B , C ) A , B ,C = e F Q ( A, B , C ) compatible with the coprojections. The morphism q : E ⊗ E ⊗ E → E is deﬁned by commutativity of e F Q ( A, B , C ) copr A , B ,C E ⊗3 q (7.3) copr Q ( A , B ,C ) E Theorem 9. For any strong ﬂockular V functor F : A → V f , the coend (7.1), and diagrams (7.2) and (7.3), deﬁne a herd E in V . 366 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 Proof. Once again we proceed by strings. As usual, morphisms in string diagrams are depicted as nodes shown as circles with the morphism’s name written inside. However, coprojections copr A : e A → E are labeled as A. We deﬁne the comonoid multiplication by and the counit by The q for the herd structure is There are also the α and β where, for example β= T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 β∗ = for which we have the following identiﬁcations: From now on we will drop the labels on the strings and take them to be understood. By Deﬁnition 1 we require q to be a comonoid morphism. We proceed as follows: 367 368 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 The map q deﬁned above is also associative since: 369 370 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 The calculation for comultiplying on the left is: T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 The one for comultiplying on the right is: 2 371 372 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 Appendix A. Invertible fusion implies antipode For completeness we provide a proof shown to us by Micah Blake McCurdy. This proof applies in any braided monoidal category. The result for Hopf algebras is classical. Proposition 10. A bimonoid A is a Hopf monoid if and only if the fusion morphism (3.22) is invertible. Proof. If A has an antipode v by ν : A → A denoted by a black node with one input and one output, deﬁne v= so that vv 1 A ⊗3 , vv 1 A ⊗3 ; v is invertible. Conversely, (M.B. McCurdy) suppose v has an inverse v denoted by a black node with two inputs and two outputs. Deﬁne ν by ν= We shall use (i) A ⊗3 1⊗ v μ⊗1 A ⊗2 A ⊗3 μ⊗1 v A ⊗2 (ii) A δ η ⊗1 A ⊗2 v A ⊗2 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 A ⊗2 (iii) 1⊗δ A ⊗3 A ⊗2 (iv) v ⊗1 v A ⊗2 1⊗δ 373 v (iii) A ⊗3 A (ii) (μ ⊗ 1)(1 ⊗ v ) = v (μ ⊗ 1), (1 ⊗ δ) v = ( v ⊗ 1)(1 ⊗ δ), (iv) which as strings are (i) (iii) Now so that ν is an antipode. 2 μ 1 ⊗ which follow from the more obvious (i) A ⊗2 (ii) (iv) δ = v (η ⊗ 1), μ = (1 ⊗ ) v , 374 T. Booker, R. Street / Journal of Algebra 330 (2011) 346–374 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] R. Baer, Zur Einführung des Scharbegriffs, J. Reine Angew. Math. 160 (1929) 199–207. T. Brzezinski, J. Vercruysse, Bimodule herds, J. Algebra 321 (2009) 2670–2704. W. Cherowitzo, T. Penttila, I. Pinneri, G.F. Royle, Flocks and ovals, Geom. Dedicata 60 (1996) 17–37. B. Day, P. McCrudden, R. Street, Dualizations and antipodes, Appl. Categ. Structures 11 (2003) 229–260. B. Day, R. Street, Monoidal bicategories and Hopf algebroids, Adv. Math. 129 (1997) 99–157. J. 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