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

= July 19 =

Embedding, monomorphism
By an "embedding" of a first given function $$f$$ in a second given function $$ g\text{ (}$$or a "monomorphism" from a first given function $$f$$ to a second given function $$g),$$ I mean, as expected, a one-to-one mapping $$\phi$$ from the image of $$f$$ to the domain of $$g,$$ satisfying$$:\text{ }g\circ \phi=\phi\circ f.$$

Question:

Given a first function $$f$$ "embedded" in a second given function $$g,$$ and given a third function $$h$$ satisfying that the composition $$ h\circ f$$ has a one-to-one mapping from the image of this composition to the domain of the second composition $$h\circ g,$$ can the first composition $$h\circ f$$ always be embedded in the second composition $$h\circ g?$$ 2A06:C701:7471:3000:39AA:1A85:25C2:975B (talk) 19:59, 19 July 2023 (UTC)


 * Your definition of "embedding" doesn't really make sense for functions with different domains and codomains. If $$f:A \to B$$ and $$g:C \to D$$, then with your definition, an embedding would be a one-to-one map $$\phi:f(A) \to C$$ such that $$g \circ \phi=\phi \circ \bar{f}$$ (where $$\bar{f}$$ is just $$f$$ but with a codomain of $$f(A)$$), and by comparing domains and codomains, that would imply that $$f(A)=A$$ and $$C=D$$.
 * Even if the domain of $$\phi$$ were required to be the entire codomain of $$f$$ 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 $$\phi_1:A \to C$$ and $$\phi_2:B \to D$$ such that $$g \circ \phi_1=\phi_2 \circ f$$, 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)
 * 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)