Let be a pair of parallel 1-morphisms in a 2-category and a 2-morphism. The inverter of is a universal object equipped with a morphism such that is invertible.
More precisely, universality means that for any object , the induced functor
is fully faithful, and its replete image consists precisely of those morphisms such that is invertible. If the above functor is additionally an isomorphism of categories onto the exact subcategory of such , then we say that 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
and the weight is the diagram
where is the terminal category and is the walking isomorphism. Since is equivalent to , 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 .
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 going in the opposite direction from , then we equify and 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.
Revised on December 2, 2010 21:41:34
by Mike Shulman