2
$\begingroup$

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).

Question:

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".

$\endgroup$
  • $\begingroup$ 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
  • 2
    $\begingroup$ @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
3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.