nLab
inverter

Context

2-Category theory

Limits and colimits

Contents

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

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

Hom(X,V)Hom(X,A)Hom(X,V) \to Hom(X,A)

is fully faithful, and its replete image consists precisely of those morphisms u:XAu\colon X\to A such that αu\alpha u is invertible. If the above functor is additionally an isomorphism of categories onto the exact subcategory of such uu, then we say that VvAV\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=T= Layer 1

and the weight TCatT\to Cat is the diagram

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

where 11 is the terminal category and InvInv is the walking isomorphism. Since InvInv is equivalent to 11, 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 TT.

An inverter in K opK^{op} (see opposite 2-category) is called a coinverter in KK. 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 β\beta going in the opposite direction from α\alpha, then we equify βα\beta\alpha and αβ\alpha\beta with identities. In particular, it follows that strict inverters are PIE-limits.

References

In

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.

Revised on December 2, 2010 21:41:34 by Mike Shulman (128.54.60.251)