I'm reading this article on Circuit Theory by John Baez https://ncatlab.org/johnbaez/show/Circuit+theory, and I'm having trouble in understanding how a map is defined. In the "Cochain Complexes From Circuits" section he defines a map $r$ which should be the canonical isomorphism between the Hilbert space of chains $C_i(\Gamma)$ of a circuit and its dual $C^i(\Gamma)$. He says that this map is given by the relation $$ a(\beta)=\langle r(a),\beta\rangle, $$ where $a\in C_i(\Gamma)$ and $\beta\in C^i(\Gamma)$.
What I don't understand is: What is $a(\beta)$? That is, how can I evaluate a chain in a cochain? Moreover, what is the angle bracket between two cochains, as in the second member?
So, what is this function $r$ that he is defining?
I think I've got it. With $a(\beta)$ he really meant $\beta(a)$, so the relation is $$ \beta(a)=\langle r(a), \beta\rangle,\quad\text{for each $a\in C_i(\Gamma)$ and $\beta\in C^i(\Gamma)$.} $$
This is equivalent to saying that $r(b)=\langle b,\cdot\rangle$ for each $b\in C_i(\Gamma)$, which is the standard way of defining an isomorphism given an inner product. In fact, for each $b\in C_i(\Gamma)$, $$ \eqalign{ (rb)(a)&=\langle b,a\rangle\cr &=\langle a,b\rangle\cr &=\langle r(a),r(b)\rangle. } $$ So, upon defining $\beta=rb$, we have $\beta(a)=\langle r(a),\beta\rangle$.