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

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.

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.

Last revised on June 1, 2024 at 21:16:32.
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