nLab lax algebra 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 (also called an oplax $T$-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 $T$ 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.

If the transformations are invertible, then $A$ is a pseudo-algebra.

Related pages

• pseudoalgebra for a 2-monad

• lax monoidal category