This is a follow-up to The definition of a group object is wrong?. The basic setup is as follows. Let $C$ be a category with finite products, $S : C \to D$ a product-preserving faithful functor, and $G \in C$ be a monoid object. It sometimes happens that $G$ has inverses in the sense that there exists a morphism $S(i) : S(G) \to S(G)$ making $S(G)$ a group object in $D$, but that this morphism does not lift to a morphism in $C$.

(Not an example; see the comments.) **Example:** A Poisson-Lie group with nontrivial Poisson bracket is not a group object in the category of Poisson manifolds $C$. However, it is a group object in the category $D$ of smooth manifolds; the inverse map negates the Poisson bracket, so does not lift to a Poisson map.

In the above question, Ryan Reich suggested two suitably weak notions of group object (in which the inverse is required to be a morphism) which allow the above example. Here they are, with ad hoc names for the sake of discussion.

A

$D$-virtual group objectin $C$ is a monoid object $G \in C$ together with a morphism $S(i) : S(G) \to S(G)$ such that $S(G)$ is a group object in $D$ with $S(i)$ as the inverse.Let $F : C \to C$ be a functor such that $SF \cong S$. An

$(F, D)$-virtual group objectin $C$ is a monoid object $G \in C$ together with a morphism $i : G \to F(G)$ such that $S(G)$ is a group object in $D$ with $S(i)$ as the inverse.

The first definition is the most general one suggested by the basic setup, but the second definition suffices, at least, for the case of Poisson manifolds, where $F$ negates the Poisson bracket.

**Question 1:** Is every $D$-virtual group object actually an $(F, D)$-virtual group object for some $F$?

I expect the answer to be "no," but I don't know how I would go about constructing a counterexample. A basic observation is that $S(i)^2 = \text{id}_{S(G)}$, so the types of morphisms that can actually occur in $D$ as inverses are somewhat restricted: while they are "virtual morphisms" (Ben Webster suggested the term *heteromorphism*) in $C$, they must square to "real morphisms." A sufficiently good theory of heteromorphisms might make it possible to construct $F$ given $C, D, S$.

Various follow-up questions suggest themselves.

**Question 2:** Given $C, D, S$ as above, is there a unique $F$ such that $C$ admits $(F, D)$-virtual group objects which don't lift to group objects in $C$?

**Question 3:** Given a category $C$, how can I tell if it admits $D$-virtual group objects for some $D$ which don't lift to group objects in $C$?

**Question 4:** Can we describe heteromorphisms in a manner internal to $C$? (For example, if $C = \text{Cat}$, it seems a heteromorphism ought to be a contravariant functor. Here we can take $D$ to be the category of graphs, $S : C \to D$ the obvious forgetful functor, and $F$ the opposite category functor. Did we need to do this, or does the notion of contravariant functor naturally fall out of the structure of $C$ in some way?)

poissonif the two Poisson structures are "$f$-related", but the diagonal map $M \to M\times M$ relates $\varpi\oplus0\oplus\varpi$ with $2\pi$, not $\varpi$. – Theo Johnson-Freyd Jun 1 '11 at 17:54