# Can every object in a monoidal category be a monoid?

Categories from the Working Mathematician says

A monoid $$c$$ in $$B$$ is an object $$c \in B$$ together with two arrows $$\mu:c \square c \to c$$ and $$\eta: e \to c$$ such that the diagrams are commutative. Can every object in a monoidal category be a monoid?

Specifically, for every object in a monoidal category, are there always two arrows $$\mu:c \square c \to c$$ and $$\eta: e \to c$$ such that the diagrams are commutative?

What decides whether an object in a monoidal category can be a monoid?

Thanks.

• Can you not just take the subcategory of monoidal objects in a monoidal category? For example the category of $k$-algebras as a subcategory of the monoidal category of $k$-vector spaces? – desiigner Jul 11 '19 at 19:13
• Oh just to clarify, I misinterpreted your first question - I took it to mean "do there exist monoidal categories in which every object is a monoid," which is true. But as Max stated below, given a monoidal category it is not always true that every object is a monoid. – desiigner Jul 11 '19 at 19:40

Why "obviously no" ? Well you can easily create artifical examples that show that it's not always possible. For instance take a monoid $$M$$ and view it as a monoidal category with objects elements of $$M$$, no nonidentity arrows and $$\otimes$$ the multiplication of $$M$$. The associator and unitors are the identity (they have to be, because there are no other arrows) and clearly all the axioms are satisfied.
Now in that category a monoid is an element $$m$$ of $$M$$ with a map $$1\to m$$ and a map $$mm\to m$$ satisfying certain equations. But if there is a map $$1\to m$$, then by definition $$1=m$$. So if $$M$$ is nontrivial, there are no other monoids than $$1$$ (which is easily seen to be a monoid).