We will begin with an exploration of colimits in the category of sets, a categorical notion characterizing compatible maps out of a diagram, and use a universal construction for them to relate colimits and equivalence relations. We'll then define the notion of a congruence, an equivalence relation compatible with further structure on those sets, e.g. group, ring, etc, and ask:
"from which shapes emerge such equivalence relations?"
The answer to this question will be sifted diagrams. We'll then examine other properties that sifted colimits have, and stumble upon, albeit in the direction counter to history, Lawvere's algebraic theories, the first way that universal algebra was instantiated category theorically.
This talk is intended to be accessible. Very little will be assumed as is hopefully conveyed by the abstract.
Of Colimits, Sifted and Not, and the Naturalness of Algebraic Theories