intuition regarding monomorphic functions

Question

Instead of talking about the categorical way of defining monomorphism. I wanted to see if I have the correct intuition regarding monomorphic function. A function $f : A \rightarrow B$ is said to be monomorphic if for all sets $Z$ and for all functions such that $\alpha_1,\alpha_2 : Z \rightarrow A$ such that $f \circ \alpha_1 = f \circ \alpha_2 \implies \alpha_1 = \alpha_2$. The way I imagine what this is doing is that functions sends "spaces" to "spaces" in a one-to-one fashion. I proved that being injective and monomorphic is really the same. However, I would like to get an intuition that carries to categories as well.

Answer 1

Injectivity is good as a first intuition but you want to keep in mind that morphisms in a general category need not be functions. So I don't know if there is a "non-algebraic way" to think about them.

You should really think of monomorphisms as morphisms that can be "left-canceled". In particular, if a morphism has a left inverse then it can be left canceled implying that it is a monomorphism.

By the way, a monomorphism with a left inverse is called split. But be careful because, in a general category, not all monomorphisms are split.

If the morphisms are functions then it makes sense to talk about injectivity and injective morphisms are monomorphisms.

Standard example to keep in mind: the category of division groups. The projection $\pi: \mathbf{Q} \rightarrow \mathbf{Q}/\mathbf{Z}$ is not injective but it is a monomorphism. (can you see why?)

Answer 2

There is a generalized notions of element for which "monomorphism" again turns out to mean the same thing as "injective".

Morphisms with codomain $X$ turn out to have a good interpretation as being "elements" of $X$ — when doing so, we call them "generalized elements".

The definition of monomorphism turns out to be exactly the result of taking the usual definition of "injective" and replacing 'element' with 'generalized element'.

This even manifests in terms that are about elements in the ordinary sense:

• $f : X \to Y$ is monic if and only if $f_* : \hom(-, X) \to \hom(-, Y)$ is a monic natural transformation if and only if $f_* : \hom(Z,X) \to \hom(Z,Y)$ is an injective function for all $Z$
• If $\mathbf{C}$ is a small category, there is a faithful functor $\mathbf{C} \to \mathbf{Set}$ that sends each object $X$ to the set of all arrows with codomain $X$. (and $f : X \to Y$ to the function given by composition with $f$)

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Another useful characterization of monic, that can maybe also be interpreted along these lines, is that $f : X \to Y$ is monic if and only if the following is a pullback diagram: $$\begin{matrix} X &\xrightarrow{1_X}& X \\ \ \ \ \ \downarrow{\small 1_X} & & \ \ \downarrow{\small f} \\ X &\xrightarrow{f}& Y \end{matrix} $$

The pullback can be reasonably interpreted as all pairs $(x,y)$ with $f(x) = f(y)$, and the assertion that the above diagram is a pullback says that set is the same thing as the diagonal: i.e. it consists only of the pairs $(x,x)$