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lax algebra for a 2-monad

Idea

If $T$ is a 2-monad on a 2-category $𝒦$, then in addition to strict (if $T$ and $𝒦$ 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 $𝒦$, a lax $T$-algebra in $𝒦$ consists of an object $A$, a morphism $TA\stackrel{a}{\to }A$, and (not necessarily invertible) 2-cells

$$\begin{array}{ccc}{T}^{2}A& \stackrel{Ta}{\to }& TA\\ {}^m\downarrow & \Downarrow & {\downarrow }^a\\ TA& \underset{a}{\to }& A\end{array}$$\array{T^2A & \overset{T a}{\to} & T A\\ ^m\downarrow & \Downarrow & \downarrow^a\\ T A& \underset{a}{\to} & A}

and

$$\begin{array}{ccccc}A& & \stackrel{\mathrm{id}}{\to }& & A\\ & {}_i\searrow & \Downarrow & {\nearrow }_a\\ & & TA\end{array}$$\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 $T\to ⟨A,A⟩$, where $M$ is the 2-monad on the 2-category $[𝒦,𝒦]$ of (some) endofunctors of $𝒦$ whose algebras are 2-monads, and $⟨A,A⟩$ is the codensity monad of $A$, i.e. the right Kan extension of $1\stackrel{A}{\to }𝒦$ along itself.

Related pages

• pseudoalgebra for a 2-monad