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## String diagrams, duality, and trace

Previously we introduced string diagrams and saw that they were a convenient way to talk about tensor products, partial compositions of multilinear maps, and symmetries. But string diagrams really prove their use when augmented to talk about duality, which will be described topologically by bending input and output wires. In particular, we will be able to see topologically the sense in which the following four pieces of information are equivalent:

• A linear map $U \to V$,
• A linear map $U \otimes V^{\ast} \to 1$,
• A linear map $V^{\ast} \to U^{\ast}$,
• A linear map $1 \to U^{\ast} \otimes V^{\ast}$.

Using string diagrams we will also give a diagrammatic definition of the trace $\text{tr}(f)$ of an endomorphism $f : V \to V$ of a finite-dimensional vector space, as well as a diagrammatic proof of some of its basic properties.

Below all vector spaces are finite-dimensional and the composition convention from the previous post is still in effect.

Below the composition of a map $f : a \to b$ with a map $g : b \to c$ will be denoted $f \circ g : a \to c$ (rather than the more typical $g \circ f$). This will make it easier to translate between diagrams and non-diagrams. All diagrams were drawn in Paper.