Take the 2-minute tour ×
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.

Is it possible to describe the property "being a monomorphism" as a universal property (with appropriate category/ies and functor)?

share|improve this question

1 Answer 1

Well, it really depends on what you mean by "described as a universal property". Maybe you find this description acceptable:

  • A morphism $f : A \to B$ is a monomorphism if and only if the diagram $$\begin{array}{rcl} A & \overset{\textrm{id}}{\rightarrow} & A \\ {\scriptstyle \textrm{id}} \downarrow & & \downarrow {\scriptstyle f} \\ A & \underset{f}{\rightarrow} & B \end{array}$$ is a pullback square.

    Indeed, suppose $g, h : X \to A$ are morphisms such that $f \circ g = f \circ h$; if the diagram is a pullback, then there is a unique $k : X \to A$ such that $g = h = k$, and so $f$ is a monomorphism; and if $f$ is a monomorphism then $g = h$ so there indeed a unique morphism completing the obvious diagram.

This description is certainly useful. For example, it implies:

  • Any functor that preserves pullbacks (or even just kernel pairs) also preserves monomorphisms. In particular, right adjoints preserve monomorphisms.

  • Any functor that reflects pullbacks (or kernel pairs) and isomorphisms must also reflect monomorphisms. In particular, monadic functors reflect monomorphisms.

Or perhaps you would prefer something in terms of hom-sets:

  • A morphism $f : A \to B$ in a (locally small) category is a monomorphism if and only if $f_* : \textrm{Hom}(X, A) \to \textrm{Hom}(X, B)$ is injective for all $X$.

If you think about it, this is just the definition of ‘monomorphism’. Amusingly this can be derived in a roundabout way by noting that $\textrm{Hom}(X, -)$ is a functor that preserves pullbacks, and the collection of all such functors jointly reflects pullbacks and isomorphisms.

share|improve this answer
    
Thank you Zhen Lin, I was aware of these descriptions, but I was hoping to get something along the lines of the general definition of universal arrow (as initial/terminal object in an appropriate slice category, given an appropriate functor). –  magma Dec 29 '12 at 18:30
    
There's no such definition that isn't tautological. For example, I could say that $f$ is a monomorphism if and only if it is equal to its own image... but the image of a morphism is defined to be the smallest monomorphism through which it factors. –  Zhen Lin Dec 30 '12 at 0:27

Your Answer

 
discard

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.