For $k$ a field, a $k$-linear operad is defined to be a symmetric operad in the category of $\mathbb{Z}$-graded $k$-vector spaces (that is, a symmetric sequence of $k$-vector spaces such that certain conditions are satisfied). Morphisms between $k$-linear operads are sequences of $k$-linear equivariant maps that respect the operadic composition and preserve the operadic units.
The category $\mathsf{Oper}_k$ of $k$-linear operads can equivalently be defined as the category of monoids and monoid morphisms in the monoidal category $([\mathbb{P}^{op},\mathsf{vect}_k^{\mathbb{Z}}],\circ)$ of functors $\mathbb{P}^{op}\rightarrow \mathsf{vect}_k^{\mathbb{Z}}$. Here $\circ$ denotes the subsitution/composition product. Clearly the functor category $[\mathbb{P}^{op}, \mathsf{vect}_k^{\mathbb{Z}}]$ is abelian, since the category of $k$-vector spaces $\mathsf{vect}_k^{\mathbb{Z}}$ is.
- I am trying to disprove that also $\mathsf{Oper}_k$ is abelian. But where does abelianness go wrong?
$\mathsf{Oper}_k$ is pre-additive. Finite coproducts exist. (And products?) Every morphism has a kernel and cokernel. Is every monomorphism normal, every epimorphism conormal? Quick ways to see this?