# In category theory, should morphisms apply to specific objects or to any objects in the category?

I am not sure to correctly understand the notion of morphism in category theory. To try to better understand, let's take a very simple example. Let's say that we have a category $$\mathcal{C}$$:

• whose objects are three singleton sets of natural numbers $$S_{0} = \{0\}, S_{1} = \{1\}, S_{2} = \{2\}$$
• whose morphisms are $$f: \forall x \in S, f\left(x\right) \rightarrow x + 1$$ and $$g: \forall x \in S, g\left(x\right) \rightarrow x + 2$$

When specified in that way, I am trying to understand which option is the correct one:

• Option A: $$f$$ is a morphism
• or Option B: $$f$$ is just a nice way to call two different morphisms: $$f_{0}$$ (whose source object is $$S_{0}$$ and target object is $$S_{1}$$) and $$f_{1}$$ (whose source object is $$S_{1}$$ and target object is $$S{2}$$) (in that case what is the correct mathematical notion corresponding to $$f$$ since $$f_{0}$$ and $$f_{1}$$ are the morphisms ?)

But that triggers another question. If the correct option is A, then I guess $$\mathcal{C}$$ is not a category because if we apply $$f$$ to $$S_{2}$$ the codomain ($$S_{3} = \{3\}$$) is not in $$\mathcal{C}$$ as it should be. If the correct option is B then if I understand correctly every morphism is specific to a single source object and to a single target object in the category.

So which option is the correct one? A clarification (with simple illustrative examples if necessary) would be very welcomed.

Bonus question: If the correct option is B, is there a way to call a category that would be "closed under its families of morphisms (if we call $$f$$ and $$g$$ families of morphisms), meaning all the domains and codomains for all possible compositions of morphisms would be in $$\mathcal{C}$$" (it's probably very handwavy but I hope you'll get what I mean).

• A morphism $f: A \to B$ has a single source ($A$) and a single target ($B$). Hence the notation. Apr 21, 2022 at 17:29
• your Option B is correct, there isn't a standardized way to refer to $f$. Apr 21, 2022 at 17:44

To start with, you did not define a category, since there should be an identity morphism for each object. Next, neither A nor B is correct. You need to define precisely, for each pair of objects $$(S_i, S_j)$$ the morphisms from $$S_i$$ to $$S_j$$. If I understand your idea, you want to define $$f_0: S_0 \to S_1$$ by $$f_0(0) = 1$$, $$f_1: S_1 \to S_2$$ by $$f_1(1) = 2$$ and $$g = f_1 \circ f_0: S_0 \to S_2$$ by $$g(0) = 2$$.

• Just to elaborate, for all we know, we could have $f \in Hom(S_2, S_0)$. There need not be any connection between the “intrinsic nature” of $f$ and the role it plays in the category. Apr 21, 2022 at 17:57