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
The kernel pair of a morphism in a category is the fiber product of the morphism with itself.
The dual notion is that of cokernel pair.
The kernel pair of a morphism in a category is a pair of morphisms which form a limit of the diagram
We can think of this as the fiber product of with itself over , or as the pullback of along itself.
The kernel pair is always a congruence on ; informally, is the subobject of consisting of pairs of elements which have the same value under (sometimes called the ‘kernel’ of a function in ).
The coequalizer of the kernel pair, if it exists, is supposed to be the “object of equivalence classes” of the internal equivalence relation . In other words, it is the quotient object of in which generalized elements are identified if they are mapped by to equal values in . In a regular category (at least), this can be identified with a subobject of called the image of .
If a morphism has a kernel pair and is a coequalizer, then it is the coequalizer of its kernel pair. This is a special case of the correspondence of generalized kernels in enriched categories.
See also: regular epimorphism, regular category, exact category