Further interesting examples? Obtaining (co)monoids from dual objects

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1. Context

Obtaining (co)monoids from dual objects
Let $(C, \otimes, I, a, l,r)$ be a monoidal category. To simplify notation (and work with string diagrams) we assume that $C$ is strict. Let $V \in C$ be a right dualizable object, i.e. there exists an object $V^* \in C$ and morphisms $b_V: I \rightarrow V \otimes V^*$, $d_V: V^* \otimes V \rightarrow I$ that satisfy the zigzag-identities. It seems, this data alone induces the structure of a monoid object $(V \otimes V^*, \mu, \eta)$ where $\mu = (r_V \otimes id_{V^*})\circ (id_V \otimes d_V \otimes id_{V^*})$ and $\eta =b_V$. This can be verified by using the zigzag identities. Analogously, it seems we have the structure of a comonoid object $(V^* \otimes V, \Delta, \epsilon)$ where $\Delta:(id_{V^*} \otimes b_V \otimes id_V)\circ (r^{-1}_{V^*} \otimes id_V)$ and $\epsilon=d_V$.

Two motivating examples

  • The category of endofunctors $End(C)$ of any small category $C$. It becomes a monoidal category in the following way: Composition of functors is the monoidal product. The monoidal unit is given by the identity functor on $C$. As the composition of functors is associative this category is strict. A right dual to an object $F \in End(C)$ is a right adjoint functor to that functor $F$. (Co)monads are (co)monoid objects in the category of endofunctors. Hence, above construction shows how one can obtain a (co)monad from a pair of adjoint functors (i.e. by suitably composing the pair of adjoint functors, and defining the respective natural transformations as described above.)

  • Consider the monoidal category of finite dimensional vector spaces (over a field) with tensor product of vector spaces as the monoidal product. This category is rigid. (The dual vector space is precisely the right/left dual object. Evaluation and coevaluation are the morphisms $d$ and $b$ respectively.) Let $V$ be an object in that category. We then have the identification $End(V) \cong V \otimes V^*$. The above construction hence endows $End(V)$ with the structure of a unital, associative algebra.

2. Questions

  • This algebra structure is the same as the the algebra structure on $End(V)$ given by the composition of maps (multiplication) and $\eta (1_{\mathbb k})=id_V$ (unit). Correct?
  • By the above construction we can turn $V \otimes V^* \cong End(V)$ into a coalgebra. Is the induced coproduct $\Delta:End(V) \rightarrow End(V) \otimes End(V)$ simply the diagonal map $\Delta(f)=f \otimes f$? What is the counit specified on a basis of $End(V)$?
  • What are other (enlightening or interesting) examples of the above construction (obtaining (co)monoids from dual objects) in other monoidal categories from the ones mentioned?

Best Answer

  1. Yes, the algebra structure on $\text{End}(V)$ is the familiar one. I don't know a super clean way to see this off the top of my head but you can just work it out by picking a basis.

  2. No, the diagonal map isn't linear. The counit $\text{End}(V) \to k$ is the trace. The comultiplication $\Delta : \text{End}(V) \to \text{End}(V) \otimes \text{End}(V)$ is given by inserting the unit into the middle, so explicitly in a basis $e_i$ of $V$ and a dual basis $e_i^{\ast}$ of $V^{\ast}$ this means

$$\Delta \left( e_i \otimes e_j^{\ast} \right) = e_i \otimes \left( \sum_{k=1}^n e_k^{\ast} \otimes e_k \right) \otimes e_j^{\ast}.$$

  1. (cont) Honestly I don't know a super clean way of thinking about this other than as the dual of the algebra structure on $\text{End}(V^{\ast})$ (or $\text{End}(V)$ itself, I suppose). I suppose you can think of it as a "path coalgebra" structure, where if $e_i \otimes e_j^{\ast}$ denotes an edge between two vertices $i$ and $j$ in the complete multigraph on $n$ vertices (so including edges between each vertex and itself, and edges are directed) then the comultiplication sends it to a sum over all paths of length $2$ between $i$ and $j$, and repeated comultiplication is a sum over paths of longer length.

  2. I'm actually not aware of substantially different examples than these. Note that the monad / comonad construction is actually more general, when generalized to 2-categories; see this blog post for more. The string diagrams look almost exactly the same.

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