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lax algebra for a 2-monad

Lax algebras for a 22-monad

Idea

If TT is a 2-monad on a 2-category 𝒦\mathcal{K}, then in addition to strict (if TT and 𝒦\mathcal{K} are strict) TT-algebras, which satisfy their laws strictly, and pseudo TT-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 TT is a 2-monad on 𝒦\mathcal{K}, a lax TT-algebra in 𝒦\mathcal{K} consists of an object AA, a morphism TAaAT A \overset{a}{\to} A, and (not necessarily invertible) 2-cells

T 2A Ta TA m a TA a A\array{T^2A & \overset{T a}{\to} & T A\\ ^m\downarrow & \Downarrow & \downarrow^a\\ T A& \underset{a}{\to} & A}

and

A id A i a TA\array{ A & & \overset{id}{\to} & & A\\ & _i \searrow & \Downarrow & \nearrow_a \\ & & T A}

satisfying suitable axioms. (Here mm is the multiplication of TT and ii is the unit.)

Of course, in a colax TT-algebra the transformations go the other way. The official way to remember which is lax and which is colax is that a lax TT-algebra structure on AA is a lax M-morphism TA,AT \to \langle A,A\rangle, where MM is the 2-monad on the 2-category [𝒦,𝒦][\mathcal{K},\mathcal{K}] of (some) endofunctors of 𝒦\mathcal{K} whose algebras are 2-monads, and A,A\langle A,A\rangle is the codensity monad of AA, i.e. the right Kan extension of 1A𝒦1\overset{A}{\to} \mathcal{K} along itself.

Revised on September 6, 2011 22:34:18 by Mike Shulman (71.136.238.9)