Suppose we have a semi-monoidal category $(\mathcal{C}, \otimes, a)$, i.e., a category $\mathcal{C}$, a functor $\otimes:\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$ and an associative constraint satisfying the pentagonal axiom $a$. Let's consider two cancellable objects $1,1'\in\mathcal{C}$, i.e., the functors $1\otimes-\quad$, $\quad-\otimes1\quad$, $\quad1'\otimes-\quad$, $\quad-\otimes1'$ are full and faithful. We say that a morphism $\psi:1\rightarrow1'$ is left tensor cancellable if for every $X,Y\in\mathcal{C}$ the function given by $(f:X\rightarrow Y)\mapsto\psi\otimes f$ between the respective hom-sets is a bijection. We define similarly a right tensor cancellable morphism. Finally, we say that $\psi$ is tensor cancellable if it is left and right tensor cancellable. It is easy to see that every isomorphism between cancellable objects is tensor cancellable. My question is: is every tensor cancellable morphism between cancellable objects an isomorphism?
Is every tensor cancellable morphism an isomorphism?
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category-theory
monoidal-categories
1 Answers
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Consider the ordered set $\mathbb{N}$ viewed as a category. Equip it with a strict monoidal structure with unit $0$ and monoidal operation $+$.
Every object is cancellable, and every morphism is tensor cancellable, but the only isomorphisms in this category are the identity maps.