This is a purely syntax question. If $A$ is a structure and $\pi$ is a homomorphism sending $A$ to something.

We say that $\pi$ is a homomorphism of $A$.

Similarly we might want to say $A$ is a ____ of $\pi$.

What is that word? I found myself writing "$A$ is a structure for which $\pi$ is a homomorphism" but that really was verbose and convoluted.

What I had considered:

$A$ is a domain, but this doesn't carry the importance that $\pi$ is a homomorphism of $A$, it only says that $\pi$ is a function which maps $A$ to something.

  • 1
    $\begingroup$ We say that $\pi:A \rightarrow B$ is a homomorphism. So $B$ is a homomorphic image of $A$. I am not sure if you would say that $A$ is a homomorphic preimage? I don't agree with "$\pi$ is a homomorphism of $A$". It is only a homomorphism from $A$ to $B$. $\endgroup$ – Dietrich Burde Dec 30 '20 at 20:14
  • $\begingroup$ Categorically, $A$ is an object and $\pi$ is morphism, and $\pi$ is a member of $\text{Hom}(A,Z)$ for some object $Z$. $\endgroup$ – Ty Jensen Dec 30 '20 at 20:15
  • $\begingroup$ I usually say "$\pi$ is a homomorphism out of $A$". $\endgroup$ – Kaj Hansen Dec 30 '20 at 20:24
  • $\begingroup$ If $\pi:A\to B$ is a map, then it has no knowledge about the structure on $A$ or on $B$. Thus, $A$ can only be a domain of $\pi$. Similarly to a letter $\pi$ in the post office - you can only have a "sender" $A$ and "recipient" $B$ of the letter, which may be "business partners" (if the letter is a business letter) or "sweethearts" (if the letter is a love letter) but you would never say "$A$ is a business partner of $\pi$" or "$A$ is a sweetheart of $\pi$. (That information is encoded outside of the letter itself. The letter may only say "Yes" - which one is it then?) $\endgroup$ – Stinking Bishop Dec 30 '20 at 20:55

I'm not aware of any context where we have a fixed map $\pi$ but consider different structures on the domain of $\pi$ with respect to which $\pi$ remains a homomorphism, and this is the only sort of context I can think of where "$A$ is the domain of $\pi$" would seem inappropriate to me. So I don't think such a term exists.

(Contrast this with "$\pi$ is a homomorphism from $A$" - this language is useful since we do often consider multiple homomorphisms out of the same structure.)

EDIT: you clarify that this is in fact the context you're in. I stand by my claim that this context is rare enough that there probably isn't a standard term in the literature. That said, the term which fits the bill most cleanly for me is "compatible," and I would suggest the phrase "$A$ is a compatible domain for $\pi$," or - if you only ever talk about varying domains and not codomains - "$A$ is compatible with $\pi$." But of course unless I'm wrong and there is after all a standard term this should be defined explicitly in your text.

Alternatively, a phrase like "$A$ is a domain object for $\pi$" has a categorical flavor which may be desired (this is motivated by Ty Jensen's comment above). Personally I think this sort of thing is clunkier, but I could see it being preferred.

  • $\begingroup$ thats exactly the type of situation i'm considering $\endgroup$ – frogeyedpeas Dec 30 '20 at 20:16
  • $\begingroup$ @frogeyedpeas Oh interesting, can you elaborate? $\endgroup$ – Noah Schweber Dec 30 '20 at 20:18
  • $\begingroup$ @frogeyedpeas I still think there's no standard term but I've given a couple suggestions. $\endgroup$ – Noah Schweber Dec 30 '20 at 20:24
  • $\begingroup$ The situation arises when you ask a question of the type "what is this a homomorphism of". My particular use case was extremely specific: I started with a associative trinary function $f(x,y,z)=xy+yz+xz$ and observed $f(x,y,a+b) \ne f(x,y,a) + f(x,y,b)$ so that got me wondering what operation does the aforementioned actually distribute over. So now i thought to ask the question geometrically as, what closed surface with a binary operation * on it, is such that f(x,y,z)=xy+yz+xz is a homomorphism of the surface. $\endgroup$ – frogeyedpeas Dec 30 '20 at 20:24
  • $\begingroup$ at this point I wanted to be more direct with my verbiage so i wanted to say something akin to what are the _____ of xy+yz+xz, perhaps saying "what are the domain objects of xy+yz+xz" is a way to put it. $\endgroup$ – frogeyedpeas Dec 30 '20 at 20:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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