Ken wrote:
> Def 12 on monoidal equivalence rubs me strangely, because FG->1 and GF->1 feel like they "want" to be a unit & counit, as from an adjunction, or like a compact category. Except it's a counit & pseudo-counit! What am I to make of this?
Your instincts are good, but there's nothing wrong here.
Just so everyone knows what we're talking about, it's this definition:
> **Definition 12.** If \\(C\\) and \\(D\\) are monoidal categories, a monoidal functor \\(F \colon C \to D\\) is a **monoidal equivalence** if there is a monoidal functor \\(G \colon D \to C\\) such that there exist monoidal natural isomorphisms \\(\alpha \colon FG \Rightarrow 1_C\\), \\(\beta \colon GF \Rightarrow 1_D\\).
Note that \\(\alpha\\) and \\(\beta\\) are required to be _invertible_ in this definition: they are isomorphisms. So, I could equally well have written the definition with either of them turned around, like this:
> **Definition 12\\({}^\prime\\).** If \\(C\\) and \\(D\\) are monoidal categories, a monoidal functor \\(F \colon C \to D\\) is a **monoidal equivalence** if there is a monoidal functor \\(G \colon D \to C\\) such that there exist monoidal natural isomorphisms \\(\alpha \colon 1_C \Rightarrow FG \\), \\(\beta \colon GF \Rightarrow 1_D\\).
This makes them look more like the 'unit' and 'counit' in a compact closed category, which we have been calling the 'cup' and 'cap'. And that's actually good! Maybe I should have written the definition this way - but I wasn't writing it for people who knew about compact closed categories, so I figured they'd be more puzzled by
\[ \alpha \colon 1_C \Rightarrow FG , \qquad \beta \colon GF \Rightarrow 1_D \]
than by the superficially more symmetrical
\[ \alpha \colon FG \Rightarrow 1_C, \qquad \beta \colon GF \Rightarrow 1_D. \]
Note also, Ken, that:
1) We could delete the word 'monoidal' everywhere and get a simpler and incredibly important concept, which alas we have not yet discussed in this course:
> **Definition.** If \\(C\\) and \\(D\\) are categories, a functor \\(F \colon C \to D\\) is an **equivalence** if there is a functor \\(G \colon D \to C\\) such that there exist natural isomorphisms \\(\alpha \colon 1_C \Rightarrow FG \\), \\(\beta \colon GF \Rightarrow 1_D\\).
2) In these definitions we are not imposing the snake equations (aka zig-zag equations) for \\(\alpha\\) and \\(\beta\\), unlike what we've done for cups and caps.
3) However, there's a wonderful theorem that if we have \\(\alpha\\) and \\(\beta\\) as above, we can always _improve_ them in a systematic way to get new ones that _do_ satisfy the snake equations! Then we say we have an **adjoint equivalence**, because then \\(F\\) and \\(G\\) are also adjoint functors.
4) This works equally well in the monoidal situation: every monoidal equivalence can be improved to give a monoidal adjoint equivalence, meaning one where \\(\alpha\\) and \\(\beta\\) obey the snake equations.
> Def 12 on monoidal equivalence rubs me strangely, because FG->1 and GF->1 feel like they "want" to be a unit & counit, as from an adjunction, or like a compact category. Except it's a counit & pseudo-counit! What am I to make of this?
Your instincts are good, but there's nothing wrong here.
Just so everyone knows what we're talking about, it's this definition:
> **Definition 12.** If \\(C\\) and \\(D\\) are monoidal categories, a monoidal functor \\(F \colon C \to D\\) is a **monoidal equivalence** if there is a monoidal functor \\(G \colon D \to C\\) such that there exist monoidal natural isomorphisms \\(\alpha \colon FG \Rightarrow 1_C\\), \\(\beta \colon GF \Rightarrow 1_D\\).
Note that \\(\alpha\\) and \\(\beta\\) are required to be _invertible_ in this definition: they are isomorphisms. So, I could equally well have written the definition with either of them turned around, like this:
> **Definition 12\\({}^\prime\\).** If \\(C\\) and \\(D\\) are monoidal categories, a monoidal functor \\(F \colon C \to D\\) is a **monoidal equivalence** if there is a monoidal functor \\(G \colon D \to C\\) such that there exist monoidal natural isomorphisms \\(\alpha \colon 1_C \Rightarrow FG \\), \\(\beta \colon GF \Rightarrow 1_D\\).
This makes them look more like the 'unit' and 'counit' in a compact closed category, which we have been calling the 'cup' and 'cap'. And that's actually good! Maybe I should have written the definition this way - but I wasn't writing it for people who knew about compact closed categories, so I figured they'd be more puzzled by
\[ \alpha \colon 1_C \Rightarrow FG , \qquad \beta \colon GF \Rightarrow 1_D \]
than by the superficially more symmetrical
\[ \alpha \colon FG \Rightarrow 1_C, \qquad \beta \colon GF \Rightarrow 1_D. \]
Note also, Ken, that:
1) We could delete the word 'monoidal' everywhere and get a simpler and incredibly important concept, which alas we have not yet discussed in this course:
> **Definition.** If \\(C\\) and \\(D\\) are categories, a functor \\(F \colon C \to D\\) is an **equivalence** if there is a functor \\(G \colon D \to C\\) such that there exist natural isomorphisms \\(\alpha \colon 1_C \Rightarrow FG \\), \\(\beta \colon GF \Rightarrow 1_D\\).
2) In these definitions we are not imposing the snake equations (aka zig-zag equations) for \\(\alpha\\) and \\(\beta\\), unlike what we've done for cups and caps.
3) However, there's a wonderful theorem that if we have \\(\alpha\\) and \\(\beta\\) as above, we can always _improve_ them in a systematic way to get new ones that _do_ satisfy the snake equations! Then we say we have an **adjoint equivalence**, because then \\(F\\) and \\(G\\) are also adjoint functors.
4) This works equally well in the monoidal situation: every monoidal equivalence can be improved to give a monoidal adjoint equivalence, meaning one where \\(\alpha\\) and \\(\beta\\) obey the snake equations.
Version 2.1.8p2
