POTW In a Category with Fibered Products

[Euge]

Let $\mathscr{C}$ be a category in which fibered products exist. If $f : X \to Y$ is a morphism in $\mathscr{C}$, prove that $f$ is a monomorphism if and only if the diagonal morphism $\Delta_f : X \to X\times_Y X$ is an isomorphism.

 

[julian]

I have actually done some category theory, so I might have a go at the problem.

Spoiler

"Only if" part:

We assume $f$ is monomorphic.

(i) Let $p_1 : X \times_Y X \rightarrow X$ and $p_2 : X \times_Y X \rightarrow X$ be the canonical projection morphisms. See figure A. We have $f \circ p_1 = f \circ p_2$. As $f$ is monomorphic we have $p_1 = p_2 = p$. Thus there is a unique $p$ such that $p \circ \Delta_f = \text{id}_X$.

Figure A

(ii) See figure B. By definition of a fibered product, $\Delta_f \circ p$ is uniquely determined. By (i) we have $p \circ \Delta_f \circ p = p$, meaning that by the uniqueness of $\Delta_f \circ p$, we have $\Delta_f \circ p = \text{id}_{X \times_Y X}$.

Figure B

"If" part:

We assume that $\Delta_f$ is an isomorphism.

We wish to prove that if $f \circ \alpha = f \circ \beta$ (i.e. figure C) then $\alpha = \beta$, when $\Delta_f$ is an isomorphism.

Figure C

First note: that as $\Delta_f$ is an isomorphism it has a unique inverse morphism, and as such $p_1 = p_2 = p$.

See figure D. By definition of a fibered product, there is a unique morphism $\xi : W \rightarrow X \times_Y X$ such that $\alpha = p_1 \circ \xi$ and $\beta = p_2 \circ \xi$. As $p_1 = p_2 = p$, we have $\alpha = p \circ \xi = \beta$.

Figure D

 

[Euge]

Thanks @julian for participating. Your solution is correct! So the postgraduate problems are not all beyond you.

 

[julian]

Thanks @julian for participating. Your solution is correct! So the postgraduate problems are not all beyond you.

I guess I have read some graduate maths in my time. But reading graduate maths and doing problems are two different things. This gives me some confidence in at least trying the problems. I may not always solve them but I would probably learn some more maths in attempting doing so, which I guess is one of the main reasons for the problems!

This has given me incentive to go back and better understand category theory!

 

[malawi_glenn]

Off topic. But any ideas for beginner books on cathegory theory and reasonable requirnents?

 

[Euge]

You can start with Steve Awodey's Category Theory and/or Emily Riehl's Category Theory in Context.

It would help to have some background knowledge in abstract algebra, at least at the undergraduate level. A topology background is also useful, but in my view not required.