Monoidal category

• triple $(𝐂,\otimes ,I)$

• a natural isomorphism called the associator with components of the form ${\alpha }_{x,y,z}:(x\otimes y)\otimes z\cong x\otimes (y\otimes z)$.

• two natural isomorphisms with components of the form ${\lambda }_x:I\otimes x\cong x$ and ${\rho }_x:x\otimes I\cong x$ called the left and right unitors, respectively.

coherence conditions:

the pentagon identity defines associativity:

(quiver)\begin{tikzcd} & {\left(w \otimes x\right) \otimes \left(y \otimes z\right)} \\ {\left(\left(w \otimes x\right) \otimes y\right) \otimes z} && {w \otimes \left(x \otimes \left(y \otimes z\right)\right)} \\ \\ {\left(w \otimes \left(x \otimes y\right)\right) \otimes z} && {w \otimes \left(\left(x \otimes y\right) \otimes z\right)} \arrow["{\alpha_{w,x,\left(y \otimes z\right)}}", from=1-2, to=2-3] \arrow["{\alpha_{(w\otimes x),y,z}}", from=2-1, to=1-2] \arrow["{\alpha_{w,x,y} \otimes \id{z}}"', from=2-1, to=4-1] \arrow["{\alpha_{w,\left(x\otimes y\right),z}}"', from=4-1, to=4-3] \arrow["{\id{w} \otimes \alpha_{x,y,z}}"', from=4-3, to=2-3] \end{tikzcd} ParseError: No such environment: tikzcd at position 7: \begin{̲t̲i̲k̲z̲c̲d̲}̲ & {\left(w \o…

the triangle identity defines identity:

(quiver)\begin{tikzcd} {x \otimes \left(I \otimes y\right)} && {\left(x \otimes I\right) \otimes y} \\ & {x \otimes y} \arrow["{\id{x} \otimes \lambda}"', from=1-1, to=2-2] \arrow["{\alpha_{x,I,y}}"', from=1-3, to=1-1] \arrow["{\rho \otimes \id{y}}", from=1-3, to=2-2] \end{tikzcd} ParseError: No such environment: tikzcd at position 7: \begin{̲t̲i̲k̲z̲c̲d̲}̲ {x \otimes \l…

References

• [cite:@nlab:monoidal_category]