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This category is Example 3.5 of Chapter 1 from Algebra, Chapter 0 by Aluffi.

Let $\mathcal C$ be a category and fix an object $A$ in the category.

Define a new category $\mathcal C_A$ by declaring the objects to be all morphisms from any object in $\mathcal C$ to $A$, and define the morphisms as follows:

Given two morphisms $f_1: Z_1 \to A$ and $f_2: Z_2 \to A$ in $\mathcal C_A$, a morphism between $f_1$ and $f_2$ in $\mathcal C_A$ is represented by a morphism $\sigma: Z_1 \to Z_2$ in $\mathcal C$ such that $f_1=f_2\circ \sigma$.


Is this really the only "sensible" way to define morphisms in this category? Are there other sensible ways to define morphisms in this category?

(I have a bit of a gripe since the author recommends the reader try to define the morphisms by themselves before reading on, and I couldn't figure it out.)

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The morphism you might define must of course

  • have a link with the morphisms of $\mathcal C$
  • use the information there is in the domain and the codomain
  • be compatible with anything you can (you often want compatibility everywhere, which can be intuitively acceptable)

So, naively, you have two maps $X\to A$ and $Y\to A$. To define a map bewteen those two, you need to use the information you already have, i.e. the morphisms of $\mathcal C$. This is logic to ask for a map $X\to Y$ in $\mathcal C$. Now, you "want every diagram you could imagine to commute". So the only possibility is the one you suggested.

Now there is also a less naive way of finding the morphisms. For that you can make links with other fields of maths where you encounter objects like that. This is the case in covering theory, where the category you are interested in (covering spaces of $X$) is a subcategory of $\mathrm{Top}_X$, consisting of pairs $Y,X$ with a covering map. Then, you can remember that the maps used in this context are exactly the maps of the category $\mathcal C_A$ that we were talking about. This category ($\mathcal C_A$) is called the category of objects over $A$, and the best way to think of its objects is to consider that the very objects are the domains of the maps, and that they have an additional structure which is a kind of a "projection" to another object which must be preserved.

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