# Lax algebras for a $2$-monad

## Idea

If $T$ is a 2-monad on a 2-category $\mathcal{K}$, then in addition to strict (if $T$ and $\mathcal{K}$ are strict) $T$-algebras, which satisfy their laws strictly, and pseudo $T$-algebras, which satisfy laws up to isomorphism, one can consider also lax and colax algebras, which satisfy laws only up to a transformation in one direction or the other.

## Definition

If $T$ is a 2-monad on $\mathcal{K}$, a lax $T$-algebra in $\mathcal{K}$ consists of an object $A$, a morphism $T A \overset{a}{\to} A$, and (not necessarily invertible) 2-cells

$\array{T^2A & \overset{T a}{\to} & T A\\ ^m\downarrow & \Downarrow & \downarrow^a\\ T A& \underset{a}{\to} & A}$

and

$\array{ A & & \overset{id}{\to} & & A\\ & _i \searrow & \Downarrow & \nearrow_a \\ & & T A}$

satisfying suitable axioms. (Here $m$ is the multiplication of $T$ and $i$ is the unit.)

Of course, in a colax $T$-algebra the transformations go the other way. The official way to remember which is lax and which is colax is that a lax $T$-algebra structure on $A$ is a lax M-morphism $T \to \langle A,A\rangle$, where $M$ is the 2-monad on the 2-category $[\mathcal{K},\mathcal{K}]$ of (some) endofunctors of $\mathcal{K}$ whose algebras are 2-monads, and $\langle A,A\rangle$ is the codensity monad of $A$, i.e. the right Kan extension of $1\overset{A}{\to} \mathcal{K}$ along itself.

Revised on September 6, 2011 22:34:18 by Mike Shulman (71.136.238.9)