nLab pseudoalgebra for a 2-monad

Idea

If $T$ is a 2-monad on a 2-category $K$, then a pseudoalgebra for $T$ is a 2-dimensional version of an algebra over a monad which satisfies the laws only up to coherent isomorphism.

Definition

A pseudo $T$-algebra is the same as a lax algebra whose constraint 2-cells are invertible.

Examples

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Normalization

A pseudoalgebra is said to be normal or normalized if its unit constraint isomorphism is an identity.

While making a pseudoalgebra strict is quite difficult, usually making it normal is quite easy, and many pseudoalgebras arising naturally are normal. For instance, for the strict 2-monad $T$ whose strict algebras are strict monoidal categories and whose pseudoalgebras are unbiased non-strict monoidal categories, the unit constraint says that the “1-ary tensor product” $\otimes(x)$ is isomorphic to $x$ itself. Clearly in most cases it is most sensible to define the 1-ary tensor product to be $x$, so that the pseudoalgebra is normal.

This situation is fairly general: if $T$ is a strict 2-monad for which the components of the unit $\eta_X \colon X\to T X$ are isocofibrations, then any pseudoalgebra structure can be modified to a normalized one on the same underlying object.

Coherence theorems

One way to state a coherence theorem is to say that every pseudoalgebra for a given 2-monad is equivalent to a strict one, perhaps in a structured way. See coherence theorems.