pseudoalgebra for a 2-monad

**2-category theory**
## Definitions
* 2-category
* strict 2-category
* bicategory
* enriched bicategory
## Transfors between 2-categories
* 2-functor
* pseudofunctor
* lax functor
* equivalence of 2-categories
* 2-natural transformation?
* lax natural transformation
* icon
* modification
* Yoneda lemma for bicategories
## Morphisms in 2-categories
* fully faithful morphism
* faithful morphism
* conservative morphism
* pseudomonic morphism
* discrete morphism
* eso morphism
## Structures in 2-categories
* adjunction
* mate
* monad
* cartesian object
* fibration in a 2-category
* codiscrete cofibration
## Limits in 2-categories
* 2-limit
* 2-pullback
* comma object
* inserter
* inverter
* equifier
## Structures on 2-categories
* 2-monad
* lax-idempotent 2-monad
* pseudomonad
* pseudoalgebra for a 2-monad
* monoidal 2-category
* cartesian bicategory
* Gray tensor product
* proarrow equipment

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.

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

…

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.

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.

Revised on November 3, 2010 23:12:45
by Mike Shulman
(128.54.59.26)