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
A wide pullback in a category is a product (of arbitrary cardinality) in a slice category . In terms of , this can be expressed as a limit over a category obtained from a discrete category by adjoining a terminal object.
Yet more explicitly, the wide pullback of a family of coterminal morphisms is an object equipped with projection such that is independent of , and which is universal with this property.
Binary wide pullbacks are the same as ordinary pullbacks, a.k.a. fiber products.
Of course, a wide pushout is a wide pullback in the opposite category.
On the other hand, together with a terminal object, wide pullbacks generate all limits:
To build up arbitrary products in , take the wide pullback of the family . Then to build equalizers of diagrams , construct the pullback of the diagram
From products and equalizers, we can get arbitrary limits.
- Robert Paré, Simply connected limits. Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746, CMS
Revised on January 4, 2013 21:16:50
by Mike Shulman