# 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.