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Title: New reasoning techniques for monoidal algebra

### Aleks Kissinger

Algebraic structures in monoidal categories are best reasoned about
using string diagrams and diagram rewriting. While this is a powerful
tool for proving individual equalities between diagrams, the methods
for defining and proving properties about whole families of diagrams
are largely ad hoc, relying on human intuition to "fill in the blanks"
in repeated diagrams of arbitrary size. One might hope to solve this
problem by identifying a hierarchy of string diagram languages and
developing induction principles for those languages whenever possible.

In this talk, I will talk about the "first rung" on this ladder, which
we call "bang-box languages". These allow us to express string
diagrams where certain parts are allowed to fan-out to arbitrary size.
While the expressive power of these "bang-graphs" is quite poor, when
combined with sound notions of bang-graph rewriting and induction, we
can prove a surprisingly rich family of theorems. I will illustrate
this by giving a purely graphical proof of a recent representation
theorem for interacting bialgebras, and show this theorem in action in
the graphical reasoning tool Quantomatic.