Monad in a \( 2 \)-Category
Table of Contents
1. Algebraic Motivation
We can motivation monads as a generalization of idempotents. In a \( 1 \)-category an indempotent is a morphism \( e \colon X \rightarrow X \) with \( e^2 = e \). Given an idempotent \( e \), we can ask if \( e \) splits, ie are there morphisms \( d \colon X \rightarrow Y \), \( i \colon Y \rightarrow X \) such that \( i \circ d = e \) and \( d \circ i = 1_Y \). In words we might say “is there an object of \( e \)-fix points?”. We can apply this to idempotent endofunctors of a category \( \mathcal C \). But note that \( \Cat \) is a strict \( 2 \)-category, hence it is natural to replace the equalities by natural transformations. Therefore the notion of an idempotent in a \( 2 \)-category generalizes to the notions of monads and comonads and the notion of a split generalizes to the notion of an adjunction. Well, the problem with this motivation is that it is unclear why a monad should contain a unit (a \( 2 \)-morphism \( 1_x \Rightarrow t \), see below).
Second try: let’s think of adjunctions as “Free-Forgetful” adjunctions. The left adjoint takes an object and equips it with additional structure (in a “free” sense, ie without adding additional constraints). The right adjoint takes an object with additional structure and just forgets the additional structure. If we can distinguish between the two categories (the starting category of objects, for example \( \Set \), and the category of objects with additional structure, for example the category of monoids), then we get an adjunction. But suppose we don’t distinguish between the two categories, then we are left with a monad on the category of objects. The functor \( T \colon \mathcal C \rightarrow \mathcal C \) tells us how to construct the underlying object \( TX \) of the free construction generated by \( X \) (think “\( T = \text{Forget } \circ \text{Free} \)”). The unit tells us how to map \( X \) into the free construction \( TX \). The multiplication tells us how to operate in \( TX \) (how the multiplication in \( TX \) works).
Third try: if we ignore the relation between monads and adjunctions, then a monad is just a \( 2 \)-categorification of a monoid.
2. Definition of a Monad
Like with adjunction we can define what a monad is not only in \( \Cat \) but in any \( 2 \)-category.
Let \( \mathcal C \) be a \( 2 \)-category. A monad \( t \) in \( \mathcal C \) consists of
- an object \( X \),
- a \( 1 \)-morphism \( t \colon X \rightarrow X \),
- a \( 2 \)-morphism \( \eta \colon 1_X \Rightarrow t \) (unit or return operation),
- a \( 2 \)-morphism \( \mu \colon t \circ t \Rightarrow t \) (multiplication, join operation or evaluation)
such that
unital axiom:
\begin{equation*} \xymatrix{ t \ar@{=>}[r] ^{1_t \circ \eta } \ar@{=>}[dr] & tt \ar@{=>}[d] ^{\mu } & t \ar@{=>}[l] _{\eta \circ 1_t} \ar@{=>}[dl] \\ & t & } \end{equation*}commutes (the \( 2 \)-morphisms are composed via vertical composition).
associativity:
\begin{equation*} \xymatrix{ (tt)t \ar@{=>}[rr] ^{\cong } \ar@{=>}[d] _{\mu \circ 1_t} & & t(tt) \ar@{=>}[d] ^{1_t \circ \mu }\\ tt \ar@{=>}[r] ^{\mu } & t & tt \ar@{=>}[l] _{\mu }} \end{equation*}commutes (\( 2 \)-morphisms are composed via vertical composition).
In words: a monad over \( X \) is a monoid object in (the \( 1 \)-category) \( \End(X) \).
3. Definition of a Module over a Monad
We define left modules (also called algebras) over a monad.
Let \( \mathcal C \) be a \( 2 \)-category. Let \( t \colon X \rightarrow X \) be a monad in \( \mathcal C \). A left module over \( t \) consists of
- a \( 1 \)-morphism \( m \colon Y \rightarrow X \),
- a \( 2 \)-morphism \( \lambda \colon mt \Rightarrow t \)
such that
unital axiom:
\begin{equation*} \xymatrix{ m \ar@{=>}[r] ^{\eta \triangleleft x} \ar@{=>}[dr] _{1} & tm \ar@{=>}[d] ^{\lambda } \\ & m } \end{equation*}commutes (can you find the omitted morphisms? Hint: \( \mathcal C \) is a \( 2 \)-category (bicategory).
associativity:
\begin{equation*} \xymatrix{ ttm \ar@{=>}[r] ^{\mu \triangleleft x} \ar@{=>}[d] _{t \triangleright \lambda } & tm \ar@{=>}[d] ^{\lambda } \\ tm \ar@{=>}[r] _{\lambda } & m } \end{equation*}commutes.
Note that \( t \) is a module over \( t \) !
Let \( t \colon \mathcal C \rightarrow \mathcal C \) be a monad in \( \Cat \). In this special case by an algebra over \( t \) (or \( t \)-algebra or left \( t \)-module) we mean a left module as above where \( m : * \rightarrow \mathcal C \) corresponds to just an object in \( \mathcal C \) in most cases.
4. Eilenberg-Moore Object
The Eilenberg-Moore object of a monad is (if it exists) the universal left module over the monad.
Let \( \mathcal C \) be a \( 2 \)-category. Let \( t \colon X \rightarrow X \) be a monad in \( \mathcal C \). An Eilenberg-Moore object \( X^t \) of \( t \) consists of
- an object \( X^t \),