Many definitions of $2$-categories are given as categories equipped with some extra structure encoded by some functors and some natural (or extranatural) transformations between these functors (or with $Id$ functors, etc.), satisfying some commutativity conditions between them.

Sometimes, one of those "equipments" are of natural isomorphism, and sometimes one wishes to relax to a more general $2$-category where this is no longer a natural isomorphism, but only one direction is given, while retain some of the original theory (having a left adjoint to the inclusion of the $2$-categoires can give quite a lot of the original structure).


Is there a cannonial process to convert the axioms of the theory with the natural isomorphism to a more general theory, where we require only one direction of the isomorphism - but with a description of the same form: A list of commutativity of some diagrams given by the "extra structure"?

The generalized $2$-category given by this description should probably be definied via some universal property, I presume.

My motivation comes from a very specific example: I am working with Street's Skew closed categories, but it seems that for one construction to work, that one of the commutative diagrams in the definition of a skew-closed category be reversed (and this is due to the non-invertibility of $i$ which is a natural isomorphism in more restrictive $2$-category). So I am wondering, if such a process exists, and Skew closed categories are "the right generalization", then this means my construction is "not the right one".

  • $\begingroup$ I’d settle for a canonical way to know whether to call the resulting definition “lax” or “oplax.” $\endgroup$ – Noah Snyder Sep 10 '18 at 13:00
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    $\begingroup$ @NoahSnyder There is one which works in nearly all cases: represent it using algebras or morphisms for a 2-monad and apply the conventions of 2-monad theory. The only cases I'm aware of where this doesn't really work are "biased" ones like skew-monoidal categories (which is probably why Street didn't choose to call those either "lax" or "oplax" -- although the nlab page ncatlab.org/nlab/show/lax+monoidal+category points out that there is some relationship at least). $\endgroup$ – Mike Shulman Sep 10 '18 at 17:05

One situation in which this can be done if the original structure can be described as a pseudo-algebra structure for some 2-monad. In this case, to make the constraints noninvertible one can consider instead lax algebras or colax algebras for the same 2-monad. However, this process tends to produce only unbiased structures, whereas skew-things are generally biased (see for instance here), so it seems unlikely to solve your specific problem.


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