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I am working through the book "Frobenius Algebras and 2D TQFTs" and am stuck on an exercise: Show that the Frobenius relations and the (co)unit relations imply the (co)associativity relations.

It's obvious topologically, but I have tried manipulating the diagrams and can't find a solution using only these relations.

For examples of the diagrams involved see this other paper, Lemmas 3.15, 3.16 and 3.18.

Frobbened
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1 Answers1

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Denote the product and co-unit by prod and counit respectively (my diagrams are from top to down unlike the ones in your reference). One of the Frobenius relations is

frob1 $=$ frob2.

Attaching prodpluscounit to the lower right leg, using the equalities

frob3 $=$ frob4

and

frob5 $=$ frob6

(where applications of the Frobenius law are marked in orange color) and applying the co-unit law we arrive at the desired associativity relation. A proof for co-associativity using the unit and co-product can be obtained by vertically reflecting the diagrams.

pregunton
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  • I think maybe I was interpreting this question too literally. I found this solution earlier but it still seems to implicitly involve some topological movement rather than explicitly matching the equations in the book. For instance there is no equation in the book to attach the bit you do in that orientation without rotating one arm of arms around. – Frobbened Aug 10 '19 at 00:24