# Localizing 2-categories about a single morphism

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.

• 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. – Dimitri Chikhladze Jun 6 '14 at 1:16
• Good question! I've thought about localisations of 2-categories, but only in a special setup disjoint from this case. – David Roberts Jun 10 '14 at 4:21
• What exactly is the universal property you would require from your localization..? – Piotr Pstrągowski Jun 10 '14 at 8:49
• @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. – Vidit Nanda Jun 10 '14 at 10:49
• @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. – Vidit Nanda Jun 10 '14 at 14:14

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.