nLab inverter

Idea

An inverter is a particular kind of 2-limit in a 2-category, which universally renders a 2-morphism invertible.

Definition

Let $f,g\colon A\rightrightarrows B$ be a pair of parallel 1-morphisms in a 2-category and $\alpha\colon f\to g$ a 2-morphism. The inverter of $\alpha$ is a universal object $V$ equipped with a morphism $v\colon V\to A$ such that $\alpha v$ is invertible.

More precisely, universality means that for any object $X$, the induced functor

$$Hom(X, v) : Hom(X,V) \to Hom(X,A)$$

is fully faithful (so that $v$ is representably fully faithful), and its replete image consists precisely of those morphisms $u\colon X\to A$ such that $\alpha u$ is invertible. If the above functor is additionally an isomorphism of categories onto the exact subcategory of such $u$, then we say that $V\xrightarrow{v} A$ is a strict inverter.

Inverters and strict inverters can be described as a certain sort of weighted 2-limit, where the diagram shape is the walking 2-morphism

$T=$

category: svg

and the weight $T\to Cat$ is the diagram

$$\array{ & \to \\ 1 & \Downarrow & Inv\\ & \to }$$

where $1$ is the terminal category and $Inv$ is the walking isomorphism. Since $Inv$ is equivalent to $1$, this weight is equivalent to the terminal weight, and thus an inverter (but not a strict inverter) can also be defined simply as a conical 2-limit over a diagram of shape $T$.

An inverter in $K^{op}$ (see opposite 2-category) is called a coinverter in $K$. Coinverters in Cat are also called localizations.

Properties

• The above description makes it clear that any inverter is in particular a fully faithful morphism.

• Any strict inverter is, in particular, an inverter. (This is not true for all strict 2-limits.)

• Inverters can be constructed in a straightforward way from inserters and equifiers: first we insert a 2-morphism $\beta$ going in the opposite direction from $\alpha$, then we equify $\beta\alpha$ and $\alpha\beta$ with identities. In particular, it follows that strict inverters are PIE-limits.

References

In

• Peter Johnstone, Elephant

inverters appear in B1.1.4. The inverter of a transformation between geometric morphisms of toposes is constructed in the proof of corollary 4.1.7 in section B4.1. Coinverters are discussed in section B4.5 there.