In general, if and are categories (or, more generally, any category-like things, such as objects of some 2-category) equipped with algebraic structure, a lax morphism is one which “preserves” the algebraic structure only up to a not-necessarily invertible transformation.
Of course, this transformation goes in one particular direction; a colax morphism is one where the transformation goes in the other direction. The case of 2-monads, below, provides an almost universally applicable way to decide which direction is “lax” and which is “colax”.
satisfying some axioms.
If the 2-cell goes in the other direction, then we say is a colax -morphism (or oplax -morphism). Equivalently, a colax -morphism is a lax -morphism, where is the induced 2-monad on the 2-cell dual (see opposite 2-category).
If the 2-cell is invertible, we call a pseudo or strong -morphism.
Let be a 2-comonad on , i.e. a 2-monad on the 1-cell dual , and let and be -coalgebras. A lax -morphism is a morphism in together with a 2-cell
satisfying some axioms.
Note that a lax morphism of algebras for the 2-comonad is a colax morphism of algebras for the 2-monad . The reason we choose to call this direction for coalgebras “lax” is that if is a 2-monad with a right adjoint , then automatically becomes a 2-comonad such that -coalgebras are the same as -algebras, and with the above definition, lax -morphisms coincide with lax -morphisms.
A lax natural transformation between 2-functors is a lax morphism for the 2-monad on whose algebras are 2-functors (which exists if is cocomplete and is small). Similarly, an oplax natural transformation is a colax morphism for this 2-monad. If is also complete, then this 2-monad has a right adjoint, which then as usual becomes a 2-comonad whose coalgebras are also 2-functors. The above conventions for lax morphisms between coalgebras mean that a lax natural transformation is unambiguously “lax” rather than “colax”, whether we regard the 2-functors as algebras for a 2-monad or coalgebras for a 2-comonad.
Some authors have tried to change the traditional meanings of “lax” and “colax” in this case, but the general framework of 2-monads gives a good argument for keeping it this way (even if in this particular case, oplax transformations are more common or useful).
A lax functor between 2-categories is a lax morphism for the 2-monad on Cat-graphs whose algebras are 2-categories.
If is a lax-idempotent 2-monad, then (by definition) every morphism in the underlying 2-category between (the objects underlying) -algebras has a unique structure of lax -morphism. For instance, every functor between categories with (some class of) colimits is a lax morphism for the 2-monad which assigns those colimits; the unique lax structure map is the canonical comparison . Such a morphism is strong/pseudo exactly when it preserves the colimits in question.
For any 2-monad , there are a 2-categories:
We have obvious 2-functors
Therefore, we can also assemble a number of F-categories of -algebras and any suitable pair of types of -morphism: strict+pseudo, strict+lax, strict+colax, pseudo+lax, or pseudo+colax.
If we want to consider both lax and colax -morphisms together, the natural structure is a double category: there is a straightforward definition of the squares in a double category whose vertical arrows are colax -morphisms and whose horizontal arrows are lax ones. We could then, if we wish, add some “F-ness” to incorporate pseudo and/or strict morphisms as well.
The 2-category is fairly well-behaved; for strict , it admits all strict PIE-limits (if the base 2-category does), and therefore all 2-limits (i.e. bilimits). When is accessible, admits all 2-colimits as well (but not, in general, many strict 2-colimits).
However, the 2-categories and are not so well-behaved; they do not have many limits or colimits. But once we enhance them to F-categories, they admit all rigged limits. All three 2-categories also admit morphism classifiers; that is, the inclusions have left 2-adjoints.