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Stream: learning: questions

Topic: equationally presenting monads

Fawzi Hreiki (Mar 04 2021 at 12:25):

Where can I go to read about equational presentations of monads (in a similar vein to how finitary monads on $\text{Set}$ can be presented equationally using finitary operations and equations between derived operations)? In particular, has anything been written about presenting monads from a 2-monadic POV? I've heard about monads with arities and the relationship to enriched Lawvere theories, but I can't really find any details on equationally presenting such monads/theories.

Fawzi Hreiki (Mar 04 2021 at 12:32):

For example, the path (i.e. free category) monad on the category of reflexive directed graphs can be presented equationally using a composition operation of arity $3 = (0 \rightarrow 1 \rightarrow 2)$.

Nathanael Arkor (Mar 04 2021 at 13:14):

Perhaps Section 3 of Bourke–Garner's Theories and monads is along the lines of what you're looking for?

Fawzi Hreiki (Mar 04 2021 at 13:27):

Just glancing at section 3, that seems perfect. Thanks!