The Monads Hurt My Head — But Not Anymore
Posted by John Baez
My friend the combinatorist Bill Schmitt breezed through Paris recently, taking a train back from Hopf-in-Lux with Paul-André Melliès, and spending a day here before going home to DC. We had some time to talk, and during the course of it I realized I’d become less scared of certain topics involving monads.
Monads seem to bother a lot of people. There’s even a YouTube video called The Monads Hurt My Head! It’s a long aimless self-indulgent rant, and it’s not exactly focused on monads in the mathematical sense, but you may enjoy listening to it anyway if you start at this particular second: 7:30. Shortly thereafter, the woman speaking exclaims:
What the heck?! How do you even explain what a monad is?
If you’re one of the people who still finds monads terrifying, I’ll just offer you two words of advice — not a complete explanation, just pointers toward what you need to learn:
Monads as generalized monoids
A monoid is a set $T$ with an associative multiplication
$m : T \times T \to T$
and unit
$i : 1 \to T$
But the idea generalizes from $Set$ to any category with products, or indeed any monoidal category, or indeed any 2-category: you can still write down the definition. And that’s a monad.
Monads as a tool for studying algebraic gadgets
On the other hand, a monad is a way of describing and studying algebraic gadgets. This is especially clear for monads in the 2-category Cat. A monad in Cat is a functor $T : C \to C$ going from some category $C$ to itself, equipped with an associative multiplication
$m: T T \Rightarrow T$
and unit
$i: 1_C \Rightarrow T$
An algebra for this monad is an object $a \in C$ together with a morphism $\alpha : T a \to a$ satisfying laws that mimic those of an action. A typical example is to let $C = Set$ and let $T a$ be the underlying set of the free group on the set $a$. The algebras for this particular monad are just groups! Groups are just one of zillions of algebraic gadgets that we can describe this way.
It’s the interplay between these viewpoints that makes monads fun. For lots more, see the Catsters videos: Monads, Adjunctions, and String diagrams, adjunctions and monads.
(It was, in fact, looking for these videos that led me to The Monads Hurt My Head! I’m not sure if the author of that video would be helped or further damaged by watching the Catsters.)
Anyway: here are some topics in monad theory that used to hurt my head, which I am now eager to understand:
- If we have a monad on a monoidal category there are various ways these structures can interact. The monad can have a tensorial strength, which is a natural transformation $a \otimes T b \to T(a \otimes b)$ making some diagrams commute. What’s the point of this? I don’t know, but I’m sure there is one. Someone tell me the fundamental theorem that justifies this notion! Do their algebras form a monoidal category, or something?
- We can also talk about monoidal monads, which are just monads in MonCat. This seems like a perfectly nice notion, but I’d still like to know the fundamental theorem about them, if there is one.
- We can also talk about lax monoidal monads, which are monads in LaxMonCat.
- If we have a monad on a cartesian category, we can ask for it to be a cartesian monad. I know this concept comes up in Leinster’s work on higher-dimensional algebra… but again, there should be some really fundamental theorem that justifies this notion.
- If we have two monads on the same category, say $S,T : C \to C$ they can be related by a distributive law, which is a natural transformation $T S \Rightarrow S T$ with just the right properties to make $S T$ into a monad. Distributive laws are a lot like braidings, and you can understand them nicely using string diagrams. I’m perfectly happy with this concept, since I just stated the fundamental theorem about it: a distributive law lets you make $S T$ into a monad. So, I add it to this list just for completeness. If you’re not happy with distributive laws yet, try my description in week257 of Cheng’s paper Iterated distributive laws.
- If we have two monads on different categories, they can be related by a ‘map of monads’ — see section 6.1 of Leinster’s book. I should like these just as much as distributive laws, but I don’t yet. At least I know the fundamental theorem about them: if we have a map of monads, we get a functor sending algebras of the first to algebras of the second.
I’ve linked to the Wikipedia for some definitions, and the $n$Lab for others. Not all the above concepts are defined on the $n$Lab yet. Of course they eventually will be. And if you can help me out here, that’ll be a good start.
Re: The Monads Hurt My Head — But Not Anymore
I was about to have a ‘go’ at you for not linking to nLab’s monad, but you come up with a good excuse.
It does seem a shame we can’t find a reliable server. Perhaps we need to earn some money to pay for one with some relevant advertising – degrees for sale, mathematics project writing for a fee, etc.