Exact Sequences and Regular Functors
In a regular category, a diagram of the form is said to be an exact sequence if it is both a coequalizer and a kernel pair. The terminology is a generalization of exact sequences in homological algebra: in an abelian category, a diagram
is exact in this sense if and only if is a short exact sequence in the usual sense.
A functor between regular categories is called regular, if it preserves finite limits and coequalizers of kernel pairs. A functor is regular if and only if it preserves finite limits and exact sequences. For this reason, regular functors are sometimes called exact functors. Functors that preserve finite limits are often said to be left exact.
Read more about this topic: Regular Category
Famous quotes containing the words exact and/or regular:
“If we define a sign as an exact reference, it must include symbol because a symbol is an exact reference too. The difference seems to be that a sign is an exact reference to something definite and a symbol an exact reference to something indefinite.”
—William York Tindall (1903–1981)
“They were regular in being gay, they learned little things that are things in being gay, they learned many little things that are things in being gay, they were gay every day, they were regular, they were gay, they were gay the same length of time every day, they were gay, they were quite regularly gay.”
—Gertrude Stein (1874–1946)