Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
An inserter is a particular kind of 2-limit in a 2-category, which universally inserts a 2-morphism between a pair of parallel 1-morphisms.
Let be a pair of parallel 1-morphisms in a 2-category. The inserter of and is a universal object equipped with a morphism and a 2-morphism .
More precisely, universality means that for any object , the induced functor
Hom(X,V) \to Ins(Hom(X,f),Hom(X,g))
is an equivalence, where denotes the category whose objects are pairs where is a morphism and is a 2-morphism. If this functor is an isomorphism of categories, then we say that is a strict inserter.
Inserters and strict inserters can be described as a certain sort of weighted 2-limit, where the diagram shape is the walking parallel pair and the weight is the diagram
1 \;\rightrightarrows\; I
where is the terminal category and is the interval category. Note that inserters are not equivalent to any sort of conical 2-limit.
An inserter in (see opposite 2-category) is called a coinserter in .
Revised on December 14, 2010 06:06:01
by Mike Shulman