Zoran Skoda
internal bialgebroid

This is an adaptation of a definition in

  • Gabriella Böhm, Internal bialgebroids, entwining structures and corings, AMS Contemp. Math. 376 (2005) 207-226, math.QA/0311244

Let C=(C,,1,τ)C = (C,\otimes,1,\tau) be a symmetric monoidal category with symmetry τ\tau. Assume that CC allows coequalizers of parallel pairs which commute with \otimes.

Let (R,m R,η R)(R,m_R,\eta_R) be a monoid in CC. An internal left RR-bialgebroid in CC consists of the following data

  • A monoid (H,m H,η H)(H,m_H,\eta_H) in CC equipped with two morphisms of monoids, α:RH\alpha: R\to H and β:R opH\beta: R^{op}\to H, such that m Hτ H,H(αβ)=m H(αβ)m_H\circ\tau_{H,H}\circ(\alpha\otimes\beta)=m_H\circ(\alpha\otimes\beta).

  • Consider HH as an internal RR-bimodule in CC via the left action m H(αH)m_H\circ (\alpha\otimes H) and the right action m Hτ H,H(Hβ)m_H\circ\tau_{H,H}\circ (H\otimes\beta). that HHH\otimes H is a monoid in CC and the coequalizer H RHH\otimes_R H. Denote by

    π:HHH RH\pi:H\otimes H\to H\otimes_R H

    the canonical map of the coequalizer.

HRHm Hτ H,H(Hβ)HHm H(αH)HHπH RH H\otimes R \otimes H\overset{H\otimes m_H\circ (\alpha\otimes H)}{\underset{m_H\circ\tau_{H,H}\circ (H\otimes\beta)\otimes H}{\Rightarrow}} H\otimes H\overset\pi\longrightarrow H\otimes_R H

The coequalizer is equipped with the unique right HHH\otimes H-action ρ\rho satisfying

ρ(πid Hid H)=πm HH \rho \circ (\pi \otimes \id_H \otimes \id_H) = \pi \circ m_{H\otimes H}

Require that

ρ(Δβη H)ρ(Δη Hα), \rho\circ (\Delta\otimes\beta\otimes\eta_H) \cong \rho\circ (\Delta\otimes\eta_H\otimes\alpha),

where we identified the domains HR1H\otimes R\otimes 1 and H1RH\otimes 1\otimes R. In the case of vector spaces, this is the condition h (1)β(r) Rh (2)=h (1) Rh (2)α(r) h_{(1)}\beta(r)\otimes_R h_{(2)}=h_{(1)}\otimes_R h_{(2)}\alpha(r).

The above condition implies that there exist (unique) left action λ:H(H RH)(H RH)\lambda : H\otimes (H\otimes_R H)\to (H\otimes_R H) such that

λ(Hπ)=ρ(ΔHH) \lambda\circ(H\otimes\pi) = \rho\circ(\Delta\otimes H\otimes H)

The fact that λ\lambda exists requires a long check in the categorical setup. It follows that H RHH\otimes_R H is an internal HH-HHH\otimes H-bimodule in CC.

  • We require that RH R{}_R H_R be equipped with a comonoid structure (H,Δ,ϵ)(H,\Delta,\epsilon) in the monoidal category of RR-bimodules in CC. Thus Δ:HH RH\Delta:H\to H\otimes_R H is coassociative a map of RR-bimodules, and ϵ:HR\epsilon : H\to R is a counit, also a map of RR-bimodules. Require
Δη H=π(η Hη H) \Delta\circ\eta_H = \pi\circ (\eta_H\otimes\eta_H)

In the case of vector spaces, this is the condition Δ(1)=1 R1\Delta(1) = 1\otimes_R 1.

Now the most subtle axiom:

Δm H=λ(HΔ) \Delta\circ m_H = \lambda\circ (H\otimes\Delta)

In the case when CC is the category of vector spaces, this is simply written Δ(hh)=h (1)h (1) Rh (2)h (2)\Delta(h h') = h_{(1)} h'_{(1)}\otimes_R h_{(2)}h'_{(2)} for all h,hHh,h'\in H, but the fact that the right-hand side is well defined requires the axioms above.

A couple of axioms on ϵ\epsilon are the remaining ones:

ϵη H=η R \epsilon\circ\eta_H = \eta_R

In the case of vector spaces, this is the condition ϵ(1 H)=1 R\epsilon(1_H) = 1_R (equivalently, the black action given by hr=ϵ(hα(r))h\blacktriangleright r = \epsilon(h\alpha(r)) is unital).

ϵm H(Hαϵ)=ϵm H=ϵm H(Hβϵ) \epsilon\circ m_H\circ(H\otimes\,\alpha\circ\epsilon) = \epsilon\circ m_H = \epsilon\circ m_H\circ(H\otimes\,\beta\circ\epsilon)

In the case of vector spaces, this is the condition ϵ(hα(ϵ(h))=ϵ(hh)=ϵ(hβ(ϵ(h))\epsilon(h\alpha(\epsilon(h')) = \epsilon(h h') = \epsilon(h\beta(\epsilon(h')) (or equivalently, the black action satisfies the action (associativity) axiom).

Last revised on March 25, 2015 at 17:43:45. See the history of this page for a list of all contributions to it.