# Monads, monoidal categories

I'm interested in learning about monads and their relations to algebraic structures (as a generalization of universal algebra, if I understand well -correct me if not) .

In the process of learning about them, I came across this sentence "A monad on $\chi$ is a monoid in the functor category $\chi^\chi$". Now obviously this category need not (unless I'm -again- mistaken ?) have finite products, and so I wondered what it could possibly mean for it to be a monoid in this category.

A moment's thought made me realize that it was a $\circ$-monoid (as opposed to a $\times$-monoid), that is the diagrams should be drawn, not with the cartesian product but with the composition of functors, seen as a "tensor product".

Having heard about them, I thought that this must be because $(\chi^\chi, \circ)$ is a monoidal category, and that we consider monoids with respect to this monoidal structure.

Suddenly, the monad axioms became much easier for me to state and remember, so I hope this isn't a mistake either. Is it ? Is my understanding of the sentence correct ?

My main question is the following: since initially my interest was in monads and algebras, but since I'm also (obviously) interested in monoidal categories, should I learn about the latter before diving into the former ? Or should I just keep the monoidal aspect in a corner of my mind and maybe come back to it later, and as of now focus on monads ? Essentially, this breaks down to: is it interesting (in the naive sense + could it bring some perspectives etc.) to learn about monoidal categories before learning about monads ?

A more general question that I could ask (that you don't have to answer to properly answer this question, but it would be a bonus) is: what big steps in category theory (+ examples if useful) should I take before learning about monads, and what big steps should I take more generally to study category theory ? (knowing that I know the basics: categories, functors, natural transformations, adjoints -still a bit rusty on that one-, limits, I know a bit about abelian categories, CCC's and topoi)

• You are correct about a monad being a $\circ$-monoid. My advice is to know a little bit of 2-categories, as I find them quite more useful to understand the theory of monads than monoidal categories are. But don't study 2-categories first: study this topic in its main lines together with monad theory. Be sure to have extremely clear basic examples. Of course, build your own favorite examples. – Fosco Aug 28 '17 at 19:49
• You may want to glance at multicategories and their variations and how they relate to monoidal categories as it puts a different lens on what's going on with monoidal categories. From a multicategory perspective, $\mu:(T,T)\to T$ and $\eta:()\to T$. This perspective, at the very least, makes some results about absolute coequalizers that are important for monadicity much more obvious. So, as Fosco Loregian suggested, you should explore these things in parallel. – Derek Elkins Aug 28 '17 at 22:41
• @FoscoLoregian : and, not for further study, but purely to help study monads, should I be interested only in strict $2$-categories, or also in bicategories ? (of course, later I will probably study bicategories because I assume they're interesting as well, but I'm wondering if they're useful to study monads) – Max Aug 30 '17 at 10:04
• This confusion arises because “a monoid in a category” is not “a monoid in set theory”. Hence I call “a monoid in a category” “a monoid object”. A monoid object should be defined relative to a monoidal category (which may be a product or a composition of endofunctors). Maybe we need to look at how a monoid is defined in your textbook. The operations of the monoidal category are implicit in the definition of a monad from your textbook. This creates confusion. – beroal Oct 13 '17 at 9:09