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 $(A,\otimes ,1)$, or $A$ for short, be a strict monoidal category with unit object $1$. A cop $C$ in $(A,\otimes ,1)$ is a pair $(C,\tau )$, where $C$ is an object in $A$ and $\tau :C\to C\otimes C\otimes C$ is a morphism in $A$ satisfying the law

(1)$$(id\otimes id\otimes \tau )\circ \tau =(\tau \otimes id\otimes id)\circ \tau .$$(\id \otimes \id \otimes \tau) \circ \tau = (\tau \otimes \id \otimes \id) \circ \tau.

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

A symmetric cop monoid $C$ in a strict symmetric monoidal category $(A,\otimes ,1,\sigma )$ is a monoid object $C$ with a morphism of monoids $\tau :C\to C\otimes C_{\mathrm{op}}\otimes C$ 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 $C$ and $D$ one takes $(\mu \otimes \mu )\circ (id\otimes {\sigma }_{D,C}\otimes id)$ as a product on $C\otimes D$).

A counit of a cop $C$ in $A$ is a morphism $ϵ:C\to 1$ such that

(2)$$(id\otimes ϵ\otimes ϵ)\circ \tau =id=(ϵ\otimes ϵ\otimes id)\circ \tau ,$$(\id \otimes \epsilon \otimes \epsilon) \circ \tau = \id = (\epsilon \otimes \epsilon \otimes \id) \circ \tau ,

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

A typical interest is in the cops in the (symmetric) monoidal category of vector spaces ${\mathrm{Vec}}_k$ or supervector spaces ${\mathrm{SVec}}_k$ over some fixed field $k$.

A coheap monoid in a symmetric monoidal category is a symmetric cop monoid such that $(id\otimes \mu )\circ \tau =(\mu \otimes id)\circ \tau =id$, where the identification $C\otimes 1=C=1\otimes C$ is implicitly used. A character of a monoid $(C,\mu ,\eta )$ in a strict monoidal category is a morphism $ϵ:C\to bf1$ such that $ϵ\circ \eta ={id}_1$ and $(ϵ\otimes ϵ)=ϵ\circ \mu$.

A character of a symmetric cop monoid $C$ is any character of $(C,\eta ,\mu )$ in $A$.

Proposition

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

Proof

This is straightforward:

$$\begin{array}{c}(id\otimes ϵ\otimes ϵ)\tau =(id\otimes (ϵ\circ \mu ))\tau =(id\otimes ϵ)(id\otimes \mu )\tau =(id\otimes ϵ)(id\otimes \eta )=id,\\ (ϵ\otimes ϵ\otimes id)\tau =((ϵ\circ \mu )\circ id)\tau =(id\otimes ϵ)(\mu \otimes id)\tau =(ϵ\otimes id)(\eta \otimes id)=id,\end{array}$$\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 $(C,ϵ)$ of a cop $C$ and a counit $ϵ$ of $C$. A copointed coheap monoid is a coheap monoid with a character $ϵ$ of $C$. 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