Disjoint Union – Is meaningful to write $A \subseteq A \sqcup B$?

Question

I have a very basic question concerning disjoint union:

Given two arbitrary sets $A$, and $B$, is meaningful to write $A \subseteq A \sqcup B$?

Intuitively, I would say it does not, because the objects in $A$ are different from the objects in $A \sqcup B$.

[The point is that the elements of $A \sqcup B$ look like $(x, i)$, for an arbitrary $x \in A \cup B$, and an arbitrary index $i$ that can take two values.]

Thus, I am wondering:

1. If I am correct;
2. If the mathematical convention is different from my intuition (which is correct), and people actually write $A \subseteq A \sqcup B$ even if it is not completely correct.
   (I actually found such a statement in a book, and it left me puzzled).

Looking forward to any feedback.
Thank you for your time.

Answer 1

Yes, you are correct. Writing $A\subseteq A \sqcup B$ is an abuse of notation and I personally would not say it is "good abuse".

The intended meaning is something along the lines of:

There is a canonical monomorphism $\operatorname{copr}_1 : A \to A \sqcup B, a\mapsto (a,0)$

so you can view $A$ as being "embedded" into $A\sqcup B$. But the map $\operatorname{copr}_1$ really is necessary to specify how it is embedded.

PS: $\operatorname{copr}$ stands for "coprojection". Some people call this thing an "injection"; I am not a fan of that because (e.g.) it is not just any old injection.