# An initial solution to the monad problem, and then some more

This is the second of two talks about monads, based on the very good notes by Andrea Schalk and continuing the one I gave on the 30th of May. Recall that we are trying to solve the following problem:

given a monad $T = (T, \eta, \mu)$, find an adjunction $(F, G, \eta, \varepsilon)$ such that $T = GF$ and $\mu = G \varepsilon_F$

If the adjunction $(F, G, \eta, \varepsilon)$ solves the problem above, we say that it generates the monad $T$.

The first solution to this problem was given by the Swiss mathematician Heinrich Kleisli, and is based on an alternative way of defining monads, as it is the case with adjunctions. Continue reading

# When does an endofunctor derive from an adjunction?

This is the first of two talks based on Andrea Schalk’s very good introduction to monads, which can be retrieved HERE

In the following, if $\mathcal{C}$ is a category, we indicate by $|\mathcal{C}|$ the collection of objects of $\mathcal{C}$, and by $\mathcal{C}(A,B)$ the collection of morphisms in $\mathcal{C}$ from $A$ to $B$.

As we know, there are two basic ways of defining an adjunction: Continue reading

# If monads are about syntax then algebras are about semantics

Today was the seventh theory lunch and the last one before christmas. So far it has been a lot of fun.

“If monads are about syntax then algebras are about semantics” – that’s the message, in this post I will attempt to illustrate it via a simple example. Continue reading