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:
“Men are qualified for civil liberty in exact proportion to their disposition to put moral chains upon their own appetites; in proportion as their love to justice is above their rapacity; in proportion as their soundness and sobriety of understanding is above their vanity and presumption; in proportion as they are more disposed to listen to the counsels of the wise and good, in preference to the flattery of knaves.”
—Edmund Burke (1729–1797)
“This is the frost coming out of the ground; this is Spring. It precedes the green and flowery spring, as mythology precedes regular poetry. I know of nothing more purgative of winter fumes and indigestions. It convinces me that Earth is still in her swaddling-clothes, and stretches forth baby fingers on every side.”
—Henry David Thoreau (1817–1862)