Does this fail to be a category?

Question

I've been skimming over Aluffi's Chapter 0 the past few days to get back to where I once was and recall everything -- and I got curious over something in Section I.3.

On pages 22-23, he introduces a category defined the following way:

Let $\textsf{C}$ be a category. Define a new category $\textsf{C}_A$ as follows

• $\text{Obj}(\textsf{C}_A)$ are morphisms $f\in \text{Hom}_\textsf{C}(Z,A)$, for any $Z \in \text{Obj}(\textsf{C})$.

• Morphisms $f_1\to f_2$ are commutative diagrams corresponding to morphisms $\sigma: Z_1 \to Z_2$ such that $f_1 = f_2 \sigma$.

He then mostly verifies this a category (leaving some exercises for the reader). When introducing this category, he said that the way morphisms are defined here are the "most natural choice". I'm curious perhaps then if this would work as a category as well:

• $\text{Obj}(\textsf{C}^*_A)$ are morphisms $f\in \text{Hom}_\textsf{C}(Z,A)$, for any $Z \in \text{Obj}(\textsf{C})$.

• Morphisms $f_1\to f_2$ correspond to morphisms $\sigma: A \to A$ (so the diagram looks like a "U")

I don't really have a paper and pencil on me to meticulously look into this -- but it seems this satisifies almost every property of a category:

1. Every object has an identity, namely $1_A$.
2. You can compose morphisms, as they're just identities and $\textsf{C}$ is a category. (Also, wouldn't composition be commutative here?)
3. Identities respect composition.
4. Composition is associative, follows from $\textsf{C}$ being a category.

What makes me believe this isn't a category though is it fails this fifth requirement:

$$\text{Hom}_{\textsf{C}^*_A}(A,B)\cap \text{Hom}_{\textsf{C}^*_A}(C,D)= \emptyset$$ unless $A=C$ and $B=D$.

Since in this category essentially every set of morphisms between objects is the set $\text{End}_\textsf{C}(A)$. So does this mean $\textsf{C}^*_A$ doesn't form a category?

Answer 1

There's something not written but implicit in your definition: in order for $\sigma f_1$ and $f_2$ to be equal, they must have the same domain, i.e. morphisms $f_1\to f_2$ only exist in your (prospective) category when $f_1$ and $f_2$ are both maps $Z\to A$ for the same $Z$.

That is, an element of $\mathrm{Hom}_{\mathsf{C}^*_A}(f_1,f_2)$ is a commutative triangle made up of the maps $f_1: Z\to A$, $f_2: Z\to A$ and $\sigma: A\to A$ (I don't know how to draw commutative diagrams on here, so you can do it yourself!). In particular, elements of the set $\mathrm{Hom}_{\mathsf{C}^*_A}(f_1,f_2)$ "remember" their domain and codomain $f_1$ and $f_2$ by definition: these elements aren't just maps $\sigma$, but maps $\sigma$ along with domain $f_1$ and codomain $f_2$. The domain and codomain are built into these commutative triangles.

In practice, people will usually write these maps as $\sigma\in \mathrm{Hom}_{\mathsf{C}^*_A}(f_1,f_2)$, but strictly speaking that's not quite true. A better notation, capturing all of the information in the commutative triangle, would be $(\sigma, f_1, f_2)\in\mathrm{Hom}_{\mathsf{C}^*_A}(f_1,f_2)$.

Long story short: if $\mathrm{Hom}_{\mathsf{C}^*_A}(f_1,f_2)$ and $\mathrm{Hom}_{\mathsf{C}^*_A}(g_1,g_2)$ intersect - say, the element $(\sigma,f_1,f_2)\in \mathrm{Hom}_{\mathsf{C}^*_A}(f_1,f_2)$ can also be written as $(\tau,g_1,g_2)\in \mathrm{Hom}_{\mathsf{C}^*_A}(g_1,g_2)$ - then the whole commutative triangle this element represents must be the same. That is, $\sigma = \tau, f_1 = g_1, f_2 = g_2$.

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To address something you said about "naturality": there's nothing wrong with your category at all. I wouldn't call it "unnatural". It's just that it doesn't crop up all that often in mathematics. Aluffi's category $\mathsf{C}_A$ does crop up a lot, though; it's even got its own special name, the slice category. It's a particular way of viewing how objects in $\mathsf{C}$ behave "relative to" $A$. In this case, the notion of "being relative to $A$" is a part of the structure of an object $Z\to A$, and if you want to study objects with a certain standing relative to $A$, then it's natural to ask for the morphisms to preserve that structure. I think this was the sense in which Aluffi used the word.

Answer 2

Requiring that the hom-sets be disjoint is an irrelevant technical condition; category theory is essentially the same the same whether or not you impose this requirement. Requiring it allows some slightly simpler approaches to the subject, but makes other approaches slightly less convenient.

If you do require hom-sets to be disjoint, there's a systematic to always guarantee that when constructing a category: you simply define $\hom(A,B)$ to be the set of all tuples of the form $(A,f,B)$ where $f$ is the kind of object you want to consider as a morphism from $A$ to $B$.

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In the category you define (improving your definition as indicated above, if necessary), every object of $\textsf{C}^*_A$ is going to be isomorphic to every other object. In fact, there is a canonical choice of isomorphism: the one corresponding to the identity map $1_A : A \to A$.

So, $\textsf{C}^*_A$ is equivalent to the full subcategory of $\textsf{C}$ consisting of the object $A$ (and its endomorphisms), meaning that it's not a particularly interesting construction.