Marco Benini

Monomorphisms as limits

Sometimes it is useful to think to monomorphisms as limits of a category.

Let f\colon A \to B be a monomorhism, i.e., for every pair of arrows g,h\colon C \to A, if f \circ g = f \circ h, then g = h.

This is equivalent to say that the square

is a pullback. In fact, the universal property says that, if f \circ g = f \circ h, then there is a unique arrow !\colon C \to A such that g = 1_A \circ ! and h = 1_a \circ !, so ! = g = h, that is, f is a monomorphism. Conversely, if f is a monomorphism, then the square always commutes, and it possesses the universal property since the (necessarily unique) arrow g = h is such that 1_A \circ g = g and 1_A \circ h = h.

Dually, one characterizes epimorphisms as colimits, specifically pushouts.

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This entry was posted on April 4, 2011 by in Category Theory, Mathematics.
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