nLab
cop

Idea

A coalgebra is half of the structure of a bialgebra. A bialgebra with antipode is a Hopf algebra, and morally a “Hopf algebra without counit” is essentially a quantum heap. So a cop should be “half way toward a quantum heap”.

Definitions

Let (A,,1)(A,\otimes,1), or AA for short, be a strict monoidal category with unit object 11. A cop CC in (A,,1)(A,\otimes, 1) is a pair (C,τ)(C,\tau), where CC is an object in AA and τ:CCCC\tau : C \rightarrow C \otimes C \otimes C is a morphism in AA satisfying the law

(1)(ididτ)τ=(τidid)τ. (\id \otimes \id \otimes \tau) \circ \tau = (\tau \otimes \id \otimes \id) \circ \tau.

Let (A,,1,σ)(A,\otimes,1,\sigma) be a strict symmetric monoidal category and CC a monoid (= algebra) object in that category, i.e. CC is equipped with a product μ:CCC\mu : C \otimes C \rightarrow C and a unit morphism η:1C\eta : 1 \rightarrow C satisfying the standard axioms. Then an opposite monoid C opC_{op} (lower index as it is on the monoid side and upper index is for the comonoid-side duality here) is the same object CC equipped with product σ C,Cμ\sigma_{C,C} \circ \mu and with the same unit map η\eta.

A symmetric cop monoid CC in a strict symmetric monoidal category (A,,1,σ)(A,\otimes,1,\sigma) is a monoid object CC with a morphism of monoids τ:CCC opC\tau : C \rightarrow C \otimes C_{op} \otimes C satisfying the law (eqref:law). Notice the passage to the opposite monoid in the second tensor factor (which does not make sense for usual cops in nonsymmetric monoidal categories). Here the tensor product has the usual tensor product structure of a monoid in a strict symmetric monoidal category (for two monoids CC and DD one takes (μμ)(idσ D,Cid)(\mu \otimes \mu) \circ (\id \otimes \sigma_{D,C} \otimes \id) as a product on CDC \otimes D).

A counit of a cop CC in AA is a morphism ϵ:C1\epsilon : C \rightarrow 1 such that

(2)(idϵϵ)τ=id=(ϵϵid)τ, (\id \otimes \epsilon \otimes \epsilon) \circ \tau = \id = (\epsilon \otimes \epsilon \otimes \id) \circ \tau ,

where the identification morphism 11CCC111 \otimes 1 \otimes C \equiv C \equiv C \otimes 1 \otimes 1 is used.

A typical interest is in the cops in the (symmetric) monoidal category of vector spaces Vec kVec_k or supervector spaces SVec kSVec_k over some fixed field kk.

A coheap monoid in a symmetric monoidal category is a symmetric cop monoid such that (idμ)τ=(μid)τ=id(\id \otimes \mu)\circ \tau = (\mu\otimes \id)\circ\tau = \id, where the identification C1=C=1CC\otimes 1 = C = 1\otimes C is implicitly used. A character of a monoid (C,μ,η)(C,\mu,\eta) in a strict monoidal category is a morphism ϵ:Cbf1\epsilon : C \rightarrow {\bf 1} such that ϵη=id 1\epsilon \circ \eta = \id_1 and (ϵϵ)=ϵμ(\epsilon \otimes \epsilon) = \epsilon \circ\mu.

A character of a symmetric cop monoid CC is any character of (C,η,μ)(C,\eta,\mu) in AA.

Proposition

A character of a coheap monoid is a automatically a counit of the underlying cop.

Proof

This is straightforward:

(idϵϵ)τ=(id(ϵμ))τ=(idϵ)(idμ)τ=(idϵ)(idη)=id, (ϵϵid)τ=((ϵμ)id)τ=(idϵ)(μid)τ=(ϵid)(ηid)=id,\array{ (\id \otimes \epsilon\otimes \epsilon)\tau = (\id\otimes (\epsilon\circ\mu))\tau = (\id\otimes\epsilon)(\id\otimes\mu)\tau = (\id\otimes\epsilon)(\id\otimes\eta) = \id, \\ (\epsilon\otimes\epsilon\otimes\id)\tau = ((\epsilon\circ\mu)\circ\id)\tau = (\id\otimes\epsilon)(\mu\otimes\id)\tau = (\epsilon\otimes\id)(\eta\otimes\id) = \id, }

where obvious identifications are implicitly used.

A copointed cop is a pair (C,ϵ)(C,\epsilon) of a cop CC and a counit ϵ\epsilon of CC. A copointed coheap monoid is a coheap monoid with a character ϵ\epsilon of CC. Warning: a copointed cop which is also a coheap is not necessarily a copointed coheap, as the counit does not need to be a character of a coheap.

Remarks

  • Clearly all this may be generalized to nonstrict monoidal categories.

  • About the language: the ‘co’ in ‘cop’ mimics the ‘co’ in ‘coalgebra’; furthermore, in the dialect of Kent, according to the OED, ‘cop’ means a small heap of hay or straw. This way it is kind of coalgebra-like entity half way toward a (quantum) heap as in the idea section above.

Literature

  • Zoran Škoda, Quantum heaps, cops and heapy categories, Mathematical Communications 12, No. 1, pp. 1-9 (2007); math.QA/0701749
Revised on July 30, 2009 19:03:16 by Toby Bartels (71.104.230.172)