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pseudoalgebra for a 2-monad

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

Definition

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

Examples

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 TT 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” (x)\otimes(x) is isomorphic to xx itself. Clearly in most cases it is most sensible to define the 1-ary tensor product to be xx, so that the pseudoalgebra is normal.

This situation is fairly general: if TT is a strict 2-monad for which the components of the unit η X:XTX\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.

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