This question is a straight-up reference request, but of course I will be grateful for an answer in the event that no references are readily available. Consider a strict $2$-category $\mathbf{C}$ and assume the existence of a $1$-morphism $f:x \to y$ in $\mathbf{C}$ from some object $x$ to some object $y$ and moreover that there are no morphisms from $y$ back to $x$.

That is to say, $f$ is an object in the category $\mathbf{C}(x,y)$ of morphisms from $x$ to $y$ while the category $\mathbf{C}(y,x)$ is empty. To avoid set-theoretic issues, one can safely assume that $\mathbf{C}$ is hopelessly finite: there are finitely many objects, finitely many $1$-morphisms between any pair of objects, and finitely many $2$-morphisms between any pair of $1$-morphisms. Here's the question:

Is there a clean description of the $2$-cells in the localization $f^{-1}\mathbf{C}$ of $\mathbf{C}$ at the single morphism $f$?

I know that the objects are the same as those of $\mathbf{C}$, and it is clear that the $1$-morphisms in $f^{-1}\mathbf{C}$ from $w$ to $z$ are given by formally augmenting the old morphisms $\mathbf{C}(w,z)$ with "new ones" of the form $gf^{-1}h$ where $g:w \to y$ and $h:x \to z$. But what are the $2$-cells? I can describe them in an unweildy case-by-case manner (i.e. old-to-new, new-to-new,...) but was hoping for something nicer that can also be used as a reference.

I'd be happy with a reference that handles the more general case of localizing about a larger collection $W$ of morphisms, but what I am not looking for is the case when $W$ forms a multiplicatively closed set.

  • $\begingroup$ In mathoverflow.net/questions/160061/… I refer to a three author paper. This paper announced another paper about a 2-category constructed from a given one by adjoining adjoints. This would be related, but nothing was published. But perhaps adjoining inverses rather than adjoints is a simpler task. $\endgroup$ – Dimitri Chikhladze Jun 6 '14 at 1:16
  • $\begingroup$ Good question! I've thought about localisations of 2-categories, but only in a special setup disjoint from this case. $\endgroup$ – David Roberts Jun 10 '14 at 4:21
  • $\begingroup$ What exactly is the universal property you would require from your localization..? $\endgroup$ – Piotr Pstrągowski Jun 10 '14 at 8:49
  • $\begingroup$ @PiotrPstrągowski I'm even happy with the strict one: namely, there should be a strict 2-functor from the base category to the localization at f so that any other functor which sends f to an isomorphism factors through it. $\endgroup$ – Vidit Nanda Jun 10 '14 at 10:49
  • $\begingroup$ @DavidRoberts Thanks! Actually I did look at your paper (1402.7108) before posting this question but yes, your setup there is essentially disjoint from mine. $\endgroup$ – Vidit Nanda Jun 10 '14 at 14:14

Well, if you are happy with everything maximally strict and finite, then you are really looking for a description of the coinverter (a weighted colimit) in the 1-category of finite 2-categories enriched over the cartesian monoidal category of finite 2-categories. See section 2.1 of John Bourke' thesis, in particular pages 13-14 (and work backwards for definitions).

In your setup you want the coinverter of the 2-arrow $$ f\colon x \Rightarrow y \colon \ast \to \mathbf{C} $$ where $x$ and $y$ are the constant functors at the named objects, and the natural transformation has as only component the 1-arrow $f$. As Bourke mentions, this can be constructed from coinserters and coequifiers (these are described on previous pages). Since you are dealing with a particularly small class of arrows this may be not as tricky as it first sounds.

Note that this is not original to Bourke, and goes back to Max Kelly.

  • $\begingroup$ Thanks! I'm taking a look at Bourke's thesis as well as Kelly's book. $\endgroup$ – Vidit Nanda Jun 12 '14 at 19:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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