Skip to main content
MathOverflow

Return to Question

Small clarifications in notation and re-tag
Source Link
edit approved Jun 21, 2020 at 18:55
Ender Wiggins
edit approved Jun 21, 2020 at 18:55
Ender Wiggins
  • 718
  • 7
  • 21

Let $\mathcal{C}$ be a right rigid (not strict) monoidal category with associativity constraint $\Phi$. Let $J_{UV}: U^*\otimes V^*\to (V\otimes U)^*$$J_{U,V}: U^*\otimes V^*\to (V\otimes U)^*$ be the canonical isomorphism for every objects $U,V\in\mathcal{C}$ . I would like to show that the pair $((-)^*, J)$ is an anti-monoidal functor, i.e. for any three objects $U,V,W\in\mathcal{C}$ $$J_{U,(W\otimes V)}(1\otimes J_{VW})\Phi_{U^*V^*W^*}=\Phi_{WVU}^*J_{(V\otimes U), W}(J_{UV}\otimes 1)$$$$J_{U,(W\otimes V)} \circ (U^*\otimes J_{V,W}) \circ \Phi_{U^*,V^*,W^*}=\Phi_{W,V,U}^* \circ J_{(V\otimes U), W}\circ (J_{U,V}\otimes W^*)$$ It should be an easy exercise of diagram chasing, but... I am stuck.

Let $\mathcal{C}$ be a right rigid (not strict) monoidal category with associativity constraint $\Phi$. Let $J_{UV}: U^*\otimes V^*\to (V\otimes U)^*$ the canonical isomorphism for every objects $U,V\in\mathcal{C}$ . I would like to show that the pair $((-)^*, J)$ is an anti-monoidal functor, i.e. for any three objects $U,V,W\in\mathcal{C}$ $$J_{U,(W\otimes V)}(1\otimes J_{VW})\Phi_{U^*V^*W^*}=\Phi_{WVU}^*J_{(V\otimes U), W}(J_{UV}\otimes 1)$$ It should be an easy exercise of diagram chasing, but... I am stuck.

Let $\mathcal{C}$ be a right rigid (not strict) monoidal category with associativity constraint $\Phi$. Let $J_{U,V}: U^*\otimes V^*\to (V\otimes U)^*$ be the canonical isomorphism for every objects $U,V\in\mathcal{C}$ . I would like to show that the pair $((-)^*, J)$ is an anti-monoidal functor, i.e. for any three objects $U,V,W\in\mathcal{C}$ $$J_{U,(W\otimes V)} \circ (U^*\otimes J_{V,W}) \circ \Phi_{U^*,V^*,W^*}=\Phi_{W,V,U}^* \circ J_{(V\otimes U), W}\circ (J_{U,V}\otimes W^*)$$ It should be an easy exercise of diagram chasing, but... I am stuck.

detail added
Source Link
Andrea
Andrea
  • 41
  • 2

Let $\mathcal{C}$ be a right rigid (not strict) monoidal category with associativity constraint $\Phi$. Let $J_{UV}: U^*\otimes V^*\to (V\otimes U)^*$ the canonical isomorphism for every objects $U,V\in\mathcal{C}$ . I would like to show that the pair $((-)^*, J)$ is an anti-monoidal functor, i.e. for any three objects $U,V,W\in\mathcal{C}$ $$J_{U,(W\otimes V)}(1\otimes J_{VW})\Phi_{U^*V^*W^*}=\Phi_{WVU}^*J_{(V\otimes U), W}(J_{UV}\otimes 1)$$ It should be an easy exercise of diagram chasing, but... I am stuck.

Let $\mathcal{C}$ be a right rigid (not strict) monoidal category. Let $J_{UV}: U^*\otimes V^*\to (V\otimes U)^*$ the canonical isomorphism for every objects $U,V\in\mathcal{C}$ . I would like to show that the pair $((-)^*, J)$ is an anti-monoidal functor, i.e. for any three objects $U,V,W\in\mathcal{C}$ $$J_{U,(W\otimes V)}(1\otimes J_{VW})\Phi_{U^*V^*W^*}=\Phi_{WVU}^*J_{(V\otimes U), W}(J_{UV}\otimes 1)$$ It should be an easy exercise of diagram chasing, but... I am stuck.

Let $\mathcal{C}$ be a right rigid (not strict) monoidal category with associativity constraint $\Phi$. Let $J_{UV}: U^*\otimes V^*\to (V\otimes U)^*$ the canonical isomorphism for every objects $U,V\in\mathcal{C}$ . I would like to show that the pair $((-)^*, J)$ is an anti-monoidal functor, i.e. for any three objects $U,V,W\in\mathcal{C}$ $$J_{U,(W\otimes V)}(1\otimes J_{VW})\Phi_{U^*V^*W^*}=\Phi_{WVU}^*J_{(V\otimes U), W}(J_{UV}\otimes 1)$$ It should be an easy exercise of diagram chasing, but... I am stuck.

Source Link
Andrea
Andrea
  • 41
  • 2

Show that duality functor is anti-monoidal

Let $\mathcal{C}$ be a right rigid (not strict) monoidal category. Let $J_{UV}: U^*\otimes V^*\to (V\otimes U)^*$ the canonical isomorphism for every objects $U,V\in\mathcal{C}$ . I would like to show that the pair $((-)^*, J)$ is an anti-monoidal functor, i.e. for any three objects $U,V,W\in\mathcal{C}$ $$J_{U,(W\otimes V)}(1\otimes J_{VW})\Phi_{U^*V^*W^*}=\Phi_{WVU}^*J_{(V\otimes U), W}(J_{UV}\otimes 1)$$ It should be an easy exercise of diagram chasing, but... I am stuck.