Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

What properties of a category have to be fulfilled such that every diagonal $\Delta:X\to X\times X$ is a monomorphism? Is this true for ''reasonable'' categories?

share|cite|improve this question
Quote from : The diagonal morphism is always a regular monomorphism, since it is the equaliser of the two projection maps $X^2\to X$. – Martin Sleziak Nov 30 '11 at 13:35
up vote 10 down vote accepted

Lemma: In general, if $f\circ g$ is a monomorphism, then $g$ is a monomorphism.

Proof: If $g\circ h_1=g\circ h_2$ then $(f\circ g)\circ h_1=(f\circ g)\circ h_2$, and hence, since $f\circ g$ is a monomorphism, $h_1=h_2$. So $g$ is a monomorphism.

Now, the category theory definition of $\Delta$ is that it is the unique function such that $\pi_1\circ \Delta = \pi_2\circ\Delta = 1_X$. And $1_X$ is a monomorphism, so necessarily $\Delta$ is a monomorphism.

So nothing additional is needed, other than the existence of $X\times X$ as a category-theoretic product.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.