### Context

#### 2-category theory

2-category theory

# Contents

## Idea

A lax-idempotent 2-monad encodes a certain kind of property-like structure that a category, or more generally an object of a 2-category, can carry.

The archetypal examples are given by 2-monads $T$ on Cat that take a category $C$ to the free cocompletion $TC$ of $C$ under a given class of colimits – then an algebra $TC\to C$ is a category $C$ with all such colimits, which are of course essentially unique. Moreover, given thus-cocomplete categories $C$ and $D$, a functor $F:C\to D$, and a diagram $S$ in $C$, there is a unique arrow $\mathrm{colim}TFS\to F\left(\mathrm{colim}S\right)$ given by the universal property of the colimit. It is this property that lax-idempotence generalizes.

## Definition

A 2-monad $T$ on a 2-category $K$ is called lax-idempotent if given any two (strict) $T$-algebras $a:TA\to A$, $b:TB\to B$ and a morphism $f:A\to B$, there exists a unique 2-cell $\overline{f}:b\circ Tf⇒f\circ a$ making $\left(f,\overline{f}\right)$ a lax morphism of $T$-algebras:

(1)$\begin{array}{ccc}TA& \stackrel{Tf}{\to }& TB\\ a↓& ⇙\overline{f}& ↓b\\ A& \underset{f}{\to }& B\end{array}$\array{ T A & \overset{T f}{\to} & T B \\ a \downarrow & \swArrow \bar f & \downarrow b \\ A & \underset{f}{\to} & B }

Dually, a 2-monad $T$ is called colax-idempotent if $f:A\to B$ gives rise to a colax $T$-morphism $\left(f,\stackrel{˜}{f}\right)$:

(2)$\begin{array}{ccc}TA& \stackrel{Tf}{\to }& TB\\ a↓& ⇗\stackrel{˜}{f}& ↓b\\ A& \underset{f}{\to }& B\end{array}$\array{ T A & \overset{T f}{\to} & T B \\ a \downarrow & \neArrow \tilde f & \downarrow b \\ A & \underset{f}{\to} & B }

### Equivalent conditions

A 2-monad $T$ as above is lax-idempotent if and only if for any $T$-algebra $a:TA\to A$ there is a 2-cell ${\theta }_{a}:1⇒\eta A\circ a$ such that $\left({\theta }_{a},{1}_{{1}_{A}}\right)$ are the unit and counit of an adjunction $a⊣{\eta }_{A}$.

Proof (Adapted from Kelly–Lack). The multiplication ${\mu }_{A}:{T}^{2}A\to TA$ is a $T$-algebra on $TA$, and ${\eta }_{A}:A\to TA$ is a morphism from the underlying object of $a$ to that of ${\mu }_{A}$. So there is a unique ${\overline{\eta }}_{A}:{\mu }_{A}\circ T{\eta }_{A}={1}_{TA}⇒{\eta }_{A}\circ a$ making ${\eta }_{A}$ into a lax $T$-morphism. Set ${\theta }_{a}={\overline{\eta }}_{A}$. The triangle equalities then require that:

1. $a{\overline{\eta }}_{A}:a⇒a\circ {\eta }_{A}\circ a=a$ is equal to ${1}_{a}$. The composite $a\circ {\overline{\eta }}_{A}$ makes $a\circ {\eta }_{A}$ a lax $T$-morphism from $a$ to $a$ (paste ${\overline{\eta }}_{A}$ with the identity square $a\circ {\mu }_{A}=a\circ Ta$). But $a\circ {\eta }_{A}={1}_{A}$, and ${1}_{a}$ also makes this into a lax $T$-morphism, so by uniqueness $a{\overline{\eta }}_{A}={1}_{a}$.

2. ${\overline{\eta }}_{A}{\eta }_{A}:{\eta }_{A}⇒{\eta }_{A}\circ a\circ {\eta }_{A}={\eta }_{A}$ is equal to ${1}_{{\eta }_{A}}$. But this follows directly from the unit coherence condition for the lax $T$-morphism ${\overline{\eta }}_{A}$.

Conversely, suppose ${\theta }_{a}$, algebras $a,b$ on $A,B$ and $f:A\to B$ are given. Take $\overline{f}$ to be the mate of ${1}_{f}:b\circ Tf\circ \eta A=f⇒f$ with respect to the adjunctions $a⊣{\eta }_{A}$ and $1⊣1$, which is given in this case by pasting with ${\theta }_{a}$, so we have that $\overline{f}=b\circ Tf\circ {\theta }_{a}$. The mate of $\overline{f}$ in turn is given by $\overline{f}\circ {\eta }_{A}$, which because mates correspond bijectively is equal to ${1}_{f}$. So $\overline{f}$ satisfies the unit condition.

Consider the diagrams expressing the multiplication condition: because $a\circ {\mu }_{A}=a\circ Ta$ (and the same for $b$), their boundaries are equal, so we have 2-cells $\alpha ,\beta :b\circ Tb\circ {T}^{2}f⇒f\circ a\circ Ta$. Their mates under the adjunction $\left(T{\theta }_{a},1\right):Ta⊣T{\eta }_{A}$ are given by pasting with $T{\eta }_{A}$. One is $\overline{f}$ pasted with $T\overline{f}\circ T{\eta }_{A}=T\left(f\circ {\eta }_{A}\right)=T{1}_{f}={1}_{Tf}$, and the other is given by composing $T{\eta }_{A}$ with the identity ${\mu }_{B}\circ {T}^{2}f=Tf\circ {\mu }_{A}$ (and then pasting with $\overline{f}$), but because ${\mu }_{A}\circ T{\eta }_{A}={1}_{TA}$ this is also equal to ${1}_{Tf}$. The two original 2-cells are hence equal, because their mates are equal, and so $\overline{f}$ is indeed a lax $T$-morphism.

Since $T$’s multiplication $\mu$ makes $T$ itself into a (generalized) $T$-algebra, the above implies (and in fact is implied by) the requirement that there exist a modification $\ell :{1}_{{T}^{2}}\to \eta T\circ \mu$ making $\left(\ell ,1\right):\mu ⊣\eta T$. Conversely, given an algebra $a:TA\to A$, the 2-cell ${\theta }_{a}$ is given by $Ta\circ {\ell }_{A}\circ T{\eta }_{A}$.

A different but equivalent condition is that there be a modification $d:T\eta \to \eta T$ such that $d\eta =1$ and $\mu d=1$; and given $\ell$ as above, $d$ is given by $\ell \circ T\eta$.

Dually, for $T$ to be colax-idempotent, it is necessary and sufficient that:

• For any $T$-algebra $a:TA\to A$ there is a 2-cell ${\zeta }_{a}:{\eta }_{A}\circ a⇒1$ such that $\left(1,{\zeta }_{a}\right):{\eta }_{A}⊣a$.

• There is a modification $m:\mu \circ \eta T\to 1$ making $\left(1,m\right):\eta T⊣\mu$.

• There is a modification $e:\eta T\to T\eta$ such that $e\eta =1$ and $\mu e=1$.

## Examples

As mentioned above, the standard examples of lax-idempotent 2-monads are those on $\mathrm{Cat}$ whose algebras are categories with all colimits of a specified class. Dually, there are colax-idempotent 2-monads which adjoin limits of a specified class.

A converse is given by Power et. al., who show that a 2-monad is a monad for free cocompletions if and only if it is lax-idempotent and the unit $\eta$ is dense (plus a coherence condition).

An important example of a colax-idempotent monad is the monad on $\mathrm{Cat}/B$ that takes $p:E\to B$ to the projection $B/p\to p$ out of the comma category. The algebras for this monad are Grothendieck fibrations over $B$; see also fibration in a 2-category. The monad $p↦p/B$ is lax-idempotent, and its algebras are opfibrations.

This latter is actually a special case of a general situation. If $T$ is a (2-)monad relative to which one can define generalized multicategories, then often it induces a lax-idempotent 2-monad $\stackrel{˜}{T}$ on the 2-category of such generalized multicategories (aka “virtual $T$-algebras”), such that (pseudo) $\stackrel{˜}{T}$-algebras are equivalent to (pseudo) $T$-algebras. When $T$ is the 2-monad whose algebras are strict 2-functors $B\to \mathrm{Cat}$ and whose pseudo algebras are pseudofunctors $B\to \mathrm{Cat}$, then a virtual $T$-algebra is a category over $B$, and it is a pseudo $\stackrel{˜}{T}$-algebra just when it is an opfibration. Similarly, there is a lax-idempotent 2-monad on the 2-category of multicategories whose pseudo algebras are monoidal categories, and so on.

## References

• Max Kelly, Steve Lack, On property-like structures, TAC 3(9), 1997.
• Kock, Monads for which structures are adjoint to units, JPAA 104:41–59, 1995.