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In his book Algebraic Topology (p.16), Spanier writes:

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So from this, I understand that its objects are the injective morphisms of $\mathscr{C}$ and its morphisms with domain $i_{1}:A_{1}\rightarrow B_{1}$ and range $i_{2}:A_{2}\rightarrow B_{2}$ are pairs of morphisms of $\mathscr{C}$ , $(g:A_{1}\rightarrow A_{2},h:B_{1}\rightarrow B_{2})$ , such that $h\circ i_{1}=i_{2}\circ g$ . Composition is define componentwise. So far, I understand everything.

But this is when I get confused. On the next page, he writes:

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So now he talks about pairs of the objects of $\mathscr{C}$ object consisting of a set, a subset of it, and the inclusion mapping. He says that this is an object of the category of pairs and he says that a morphism $f:(X,A)\rightarrow (Y,B)$ whose image is mapped into $B$ is a morphism in the category of pairs. Here are my questions:

  1. Are these two definitions different?

  2. In the second definition, he only considers inclusion mappings. While in the first definition, he talks about injections in general. Why does he do this?

  3. Could it be that Spanier is definining Categories of pairs in a horrible way? If so, could someone provide me with a clear definition?

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  1. Not fundamentally. Clearly the category from the "second definition" injects into the category from the "first definition", using the inclusion mapping and pairing a morphism with its restriction to the subset. It is not difficult to show that this is an equivalence. (What is the pseudo-inverse?) Note that we generally only care about categories up to equivalence, so it does not matter which one you take to be the "actual" definition, and which one a merely equivalent category.

  2. Sometimes it is useful to have multiple definitions of the "same" thing, to use whichever is more convenient or makes the notation easier.

  3. Not really. Either of the definitions works, and it is not very difficult to see that they give equivalent categories.

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