Timeline for A result on symmetric closed monoidal categories
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Dec 13, 2023 at 15:48 | comment | added | Max Demirdilek | … More generally, post-composing the monoid unit of a monoid structure on the monoidal unit $I$ with a non-trivial automorphism of $I$ (and "left-conjugating" the monoid multiplication by this automorphism) gives a different (though isomorphic) monoid structure on $I$. | |
| Dec 13, 2023 at 15:47 | comment | added | Max Demirdilek | @MartinBrandenburg: The monoid structure on the monoidal unit is not strictly unique. For instance, let $k$ be a field with more than two elements and consider the category of $k$-vector spaces with its usual monoidal structure. Pick an invertible element $x\in k$ different from $1$. Then for the canonical monoid structure $(\mu,\eta)$ on the monoidal unit $k$, the pair $(x\cdot \mu, x^{-1}\cdot \eta)$ gives a different (though isomorphic) monoid structure… | |
| Dec 13, 2023 at 15:46 | comment | added | Max Demirdilek | … By the Eckmann-Hilton argument, the endomorphism monoid $\operatorname{End}(I)$ is commutative, thus we also have $\eta\circ \mu \circ {\rho_I}^{-1}=\operatorname{id}_I$ and $\eta$ is invertible. Since any monoid $(A,m,e)$ together with an isomorphism $f:A\rightarrow I$ induces a monoid structure on $I$ whose unit is given by $f\circ e$ (transport of structure), this finishes the proof. | |
| Dec 13, 2023 at 15:46 | comment | added | Max Demirdilek | An explicit proof: Let $(\mu,\eta)$ be a unital monoid structure on the monoidal unit $I$. Since for the unitors we have $\lambda_I=\rho_I$ (one can use Mac Lane's coherence theorem or give an elementary proof), unitality of the monoid implies $\mu\circ (\eta\otimes \operatorname{id}_I)\circ {\rho_I}^{-1}=\operatorname{id}_I$. By naturality of the right unitor, this is equivalent to $\mu \circ {\rho_I}^{-1}\circ \eta=\operatorname{id}_I$ … | |
| Nov 30, 2023 at 8:00 | comment | added | Maxime Ramzi | @MartinBrandenburg : yes the coherence theorem implies it in general - but you can also prove it directly in an arbitrary monoidal category :) | |
| Nov 29, 2023 at 23:23 | comment | added | Martin Brandenburg | Very nice interpretation. In other words, the unit object has a unique monoid structure. This is rather easy for strict monoidal categories (not sure if coherence thm then implies it in general, or if we only get uniqueness up to isomorphism), and these are the ones we need for applying it to monads. | |
| Nov 29, 2023 at 23:20 | history | edited | Martin Brandenburg | CC BY-SA 4.0 |
added 25 characters in body
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| Nov 29, 2023 at 17:40 | vote | accept | Max Demirdilek | ||
| Nov 29, 2023 at 15:40 | history | answered | Maxime Ramzi | CC BY-SA 4.0 |