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”.
Let , or for short, be a strict monoidal category with unit object . A cop in is a pair , where is an object in and is a morphism in satisfying the law
Let be a strict symmetric monoidal category and a monoid (= algebra) object in that category, i.e. is equipped with a product and a unit morphism satisfying the standard axioms. Then an opposite monoid (lower index as it is on the monoid side and upper index is for the comonoid-side duality here) is the same object equipped with product and with the same unit map .
A symmetric cop monoid in a strict symmetric monoidal category is a monoid object with a morphism of monoids 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 and one takes as a product on ).
A counit of a cop in is a morphism such that
where the identification morphism is used.
A coheap monoid in a symmetric monoidal category is a symmetric cop monoid such that , where the identification is implicitly used. A character of a monoid in a strict monoidal category is a morphism such that and .
A character of a symmetric cop monoid is any character of in .
A character of a coheap monoid is a automatically a counit of the underlying cop.
This is straightforward:
where obvious identifications are implicitly used.
A copointed cop is a pair of a cop and a counit of . A copointed coheap monoid is a coheap monoid with a character of . 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.
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.