Jump to content

Wikipedia:Reference desk/Archives/Mathematics/2023 July 19

From Wikipedia, the free encyclopedia
Mathematics desk
< July 18 << Jun | July | Aug >> July 20 >
Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.


July 19

[edit]

By an "embedding" of a first given function in a second given function or a "monomorphism" from a first given function to a second given function I mean, as expected, a one-to-one mapping from the image of to the domain of satisfying

Question:

Given a first function "embedded" in a second given function and given a third function satisfying that the composition has a one-to-one mapping from the image of this composition to the domain of the second composition can the first composition always be embedded in the second composition

2A06:C701:7471:3000:39AA:1A85:25C2:975B (talk) 19:59, 19 July 2023 (UTC)[reply]

Your definition of "embedding" doesn't really make sense for functions with different domains and codomains. If and , then with your definition, an embedding would be a one-to-one map such that (where is just but with a codomain of ), and by comparing domains and codomains, that would imply that and .
Even if the domain of were required to be the entire codomain of instead of just the image, the definition would still be too restrictive (though it would work if one were dealing only with endofunctions).
Instead, an embedding should be a pair of one-to-one maps and such that , i.e., a morphism in the arrow category of the category of sets (which is a special case of a comma category).
GeoffreyT2000 (talk) 21:01, 19 July 2023 (UTC)[reply]
Correct. I've just corrected my question in the following thread. See below. 2A06:C701:7471:3000:39AA:1A85:25C2:975B (talk) 17:03, 20 July 2023 (UTC)[reply]