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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.

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  • $\begingroup$ 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
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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.

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  • $\begingroup$ 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

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