added pointer to:

- Saunders MacLane, §I.5 of:
*Categories for the Working Mathematician*, Graduate Texts in Mathematics**5**Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

below the statement that monomorphisms are preserved under pullback I added a quick pointer to *adhesive category* for the statement under pushout

Trivial fact about monos in preorders

]]>adding disambiguation message at top of page for the word “monic”.

Anonymous

]]>Gave the (counter-)examples of monos that are epi but not iso their own sub-section (here) and added mentioning of the example of dense subspace inclusions in Hausdorff spaces

]]>Fixed the errors noted by Hurkyl.

]]>Also, if I haven’t gotten turned around, the characterization of epimorphisms and monomorphisms via yoneda lemma is backwards. I.e. the monomorphism page has the version characterizing epis, and the epimorphism page has the version characterizing monos.

]]>Since you still have the lock… you forgot to change “pushout” to “pullback” in proposition 4.1.

]]>Added an extra property of monos (now the first one) to match the page on epis.

]]>That works well now. There are lots of places in papers written in English (and sometimes with native English speakers as authors!) where a very slight change in wording / word order can make a sentence much easier to read, even to parse for its intended meaning. There are questions of personal preference here, even of ’taste’, but, for example, starting a sentence with ’Then’ rarely works well in my view.

]]>Ah, I see. I think about this point (about adjacent formulas) a lot too. Let me rearrange. (Done.)

]]>I am doing this intentionally to avoid two mathematical formulas appearing in a sentence in the role of two consecutive words. (Even with a comma separating them, this is awkward). I do seem to recall that another native English speaker once active here used to do this, too. But if you tell me that I must be misremembering, then I’ll believe you.

]]>I agree with you, Zoran.

]]>Entry says

Since injective functions are precisely the monomorphisms in Set (example \ref{MonomorphismsInSet} below) this may be stated as saying that $f$ is a monomorphism if for all objects $Z$ then $Hom(Z,f)$ is a monomorphism.

The final “then” feels awkward to my feeling of English, but I am not going to correct it as many native speakers work around. Even logically we can do the analysis. I mean why implication between the quantifier and the clause. “If” is complemented by the preceeding clause “$f$ is a monomorphism”.

]]>Oh, I didn’t see that, sorry. But it doesn’t hurt to say it in the Idea-section already.

]]>I had added something to that effect later, there and at epimorphism:

A morphism $f \colon X \to Y$ in some category is called a

monomorphism(sometimes abbrieviated tomono), or described as beingmonic, if …

But maybe better earlier where you have it.

]]>In the spirit of the challenge in #1, I did add a line explaining the jargon to the Idea-sections at *monomorphism*, *epimorphism* and *isomorphism*.

Ok, I think both entries are consistent now.

]]>Yes, it should be ’monos’, and I say this even if I was the one who wrote that.

]]>So, Todd, would you have ’monos’ instead here, or is there ellipsis for ’monic arrows’?

]]>Frequently, regular and strong monics coincide.

I use ’monic’ and ’epic’ all the time as adjectives, and ’mono’ and ’epi’ as nouns. I thought the practice was widespread.

]]>David, please add a line explaining the jargon.

]]>While proof-reading monomorphism, I see we find ’monic’ used unannounced. I could just replace them all by ’monomorphism’, but how widely used is ’monic’?

Also epimorphism has ’epic’ used (once) unexplained. And ’epi’. Strangely, both seemed to be used as adjectives, where all ’monic’s are nouns.

]]>I agree, good idea. There is *so* much work to do here.

I find “dual of epimorphism” pretty funny. If there’s an order to these concepts, it’s arguably in the other direction: limit notions should come *before* colimit notions. What I mean is that $f: A \to B$ is epic iff $\hom(f, X)$ is monic in $Set$ for all $X$, and similarly all limits and colimits in categories reduce to limits in $Set$. Second, for the classical large categories of structures, epimorphisms are *harder* to understand than monomorphisms.