nLab
lax-idempotent 2-monad

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 $\mathrm{colim}TFS\to F(\mathrm{colim}S)$ 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 $\overline{f}:b\circ Tf\Rightarrow f\circ a$ making $(f,\overline{f})$ a lax morphism of $T$-algebras:

(1)$$\begin{array}{ccc}TA& \stackrel{Tf}{\to }& TB\\ a\downarrow & ⇙\overline{f}& \downarrow b\\ A& \underset{f}{\to }& B\end{array}$$\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 $(f,\tilde{f})$:

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

Lax-idempotent monads are also called Kock–Zöberlein or KZ monads.

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 ${\theta }_a:1\Rightarrow \eta A\circ a$ such that $({\theta }_a,1_{1_A})$ 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 ${\overline{\eta }}_A:{\mu }_A\circ T{\eta }_A=1_{TA}\Rightarrow {\eta }_A\circ a$ making ${\eta }_A$ into a lax $T$-morphism. Set ${\theta }_a={\overline{\eta }}_A$. The triangle equalities then require that:

1. $a{\overline{\eta }}_A:a\Rightarrow a\circ {\eta }_A\circ a=a$ 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. ${\overline{\eta }}_A{\eta }_A:{\eta }_A\Rightarrow {\eta }_A\circ a\circ {\eta }_A={\eta }_A$ 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 $1_f:b\circ Tf\circ \eta A=f\Rightarrow f$ 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 $\alpha ,\beta :b\circ Tb\circ T^{2}f\Rightarrow f\circ a\circ Ta$. Their mates under the adjunction $(T{\theta }_a,1):Ta⊣T{\eta }_A$ are given by pasting with $T{\eta }_A$. One is $\overline{f}$ pasted with $T\overline{f}\circ T{\eta }_A=T(f\circ {\eta }_A)=T{1}_f=1_{Tf}$, 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 $(\ell ,1):\mu ⊣\eta T$. 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 ${\zeta }_a:{\eta }_A\circ a\Rightarrow 1$ such that $(1,{\zeta }_a):{\eta }_A⊣a$.

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

• 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 $p\mapsto p/B$ 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 $\tilde{T}$ on the 2-category of such generalized multicategories (aka “virtual $T$-algebras”), such that (pseudo) $\tilde{T}$-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 $\tilde{T}$-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.
• Power–Cattani–Winskel, A representation result for free cocompletions, JPAA 151:273–286, 2000.
• Marmolejo–Wood, Kan extensions and lax idempotent pseudomonads, TAC 26, p. 1–29 (2011)