category theory - what is the difference between morphism and arrow?

The Arrow Category confuses my understanding between morphism and arrow. I can understand a morphism is a mapping that preserves the context structure.

is Arrow just an alias of morphism or to some-extent mapping ?

Axioms:
1. for each arrow f, target * source (f) = source(f),


isn't source of f just the data ?, then target of a data is just meaningless.

• In the context of category theory, both an "arrow" and a "morphism" are synonymous. The difference here is that you can either specify a category by giving a collection of objects $\mathrm{Ob}(\mathbf{C})$, and for each pair of objects $X$ and $Y$ a collection of "arrows"/morphisms $\mathbf{C}(X,Y)$ (and usually assume that collections corresponding to distinct ordered pairs are disjoint). In the "arrow theory" you instead specify only a collection of arrows, and operations that identify the "domain" and "codomain" of each arrow. They are different ways of specifying the same information. Commented Mar 25, 2021 at 15:38
• P.S.: It's not "Arrow category" (that is, it is not a category of arrows). It is "the arrow point of view to define a category". Commented Mar 25, 2021 at 15:43
• @ArturoMagidin what does the first Axiom mean ? why there is still target on the source of an arrow? Commented Mar 25, 2021 at 15:48
• $s(f)$ is both the “source of $f$” (the domain object) and the identity function of the source of $f$. $t(f)$ is both the “terminal of $f$” (the codomain) and the identity function for the terminal of $f$. It’s just another way of providing you the information that makes up a category. When it says “data”, it is not describing the objects. It is describing the information they are providing you. You are completely and utterly misunderstanding this,. Commented Mar 27, 2021 at 13:56
• P.S. Questions should be self contained. Relying on screenshots is bad practice. They are not searchable, screen readers cannot handle them. Commented Mar 27, 2021 at 13:58

2 Answers

Ironically, the full snapshot answers your question...

The “Arrow Theoretic” view of category theory is not a specific category, as your “Arrow Category” nomenclature seems to suggest. Instead, it is a point of view of how to describe a category, one in which the morphisms are not just front-and-center, but they are front-and-only.

Just as there are multiple ways of defining a topological space (by giving the open sets, the close sets, the closure operator, the interior operator, the system of neighborhood families, etc), there are also multiple ways of defining a category.

Here are a few, going from the “classical” view of objects-and-maps, all the way to the “arrow point of view.”

1. A category $$\mathbf{C}$$ consists of a collection of “objects”, $$\mathrm{Ob}(\mathbf{C})$$, and for each ordered pair $$(X,Y)$$ of objects, a collection of “morphisms from $$X$$ to $$Y$$”, $$\mathbf{C}(X,Y)$$. We assume that if $$(X,Y)\neq(X’,Y’)$$, then $$\mathbf{C}(X,Y)\cap\mathbf{C}(X’,Y’)=\varnothing$$. In addition, there is a (family) of operations called composition, defined for each ordered triple $$(X,Y,Z)$$ of objects, $$\circ\colon\mathbf{C}(X,Y)\times\mathbf{C}(Y,Z)\to\mathbf{C}(X,Z)$$, which is associative; and distinguished morphisms $$\mathrm{id}_X\in\mathbf{C}(X,X)$$ for each $$X\in\mathrm{Ob}(\mathbf{C})$$ such that for all $$f\in\mathbf{C}(X,Y)$$, $$\mathrm{id}_Y\circ f = f$$ and $$f\circ\mathrm{id}_X=f$$.

2. A category $$\mathbf{C}$$ consists of a collection of objects $$\mathrm{Ob}(\mathbf{C})$$, and a collection of morphisms or “arrows”, $$\mathrm{Ar}(\mathbf{C})$$, together with two functions $$s,t\colon\mathrm{Ar}(\mathbf{C})\to\mathrm{Ob}(\mathbf{C})$$ (informally, the “source” and “target” functions). And an associative partial binary operation on $$\mathrm{Ar}(\mathbf{C})$$, $$\circ\colon (f,g)\mapsto g\circ f$$, defined whenever $$t(f)=s(g)$$. In addition, there is a distinguished family of morphisms, $$(\mathrm{id}_X)_{X\in\mathrm{Ob}(\mathbf{C})}$$ such that for all morphisms $$f$$ with $$s(f)=X$$ and $$t(f)=Y$$, $$\mathrm{id}_Y\circ f = f = f\circ\mathrm{id}_X$$.

3. A category $$\mathbf{C}$$ consists of a collection of arrows $$\mathrm{Ar}(\mathbf{C})$$, functions $$s,t\colon\mathrm{Ar}(\mathbf{C})\to\mathrm{Ar}(\mathbf{C})$$, informally the “source” and “target”; and a “composition” associative binary partial operation defined by $$\circ\colon (f,g)\mapsto g\circ f$$, defined whenever $$t(f)=s(g)$$. The functions $$s$$ and $$t$$ satisfy the conditions: $$s(s(f))=t(s(f))=s(f)$$, $$t(t(f))=s(t(f))=t(f)$$, $$f\circ s(f)=f$$, $$t(f)\circ f = f$$.

It is fairly clear that (1) and (2) are essentially the same thing; the main difference is that in (1) we segregate the morphisms and associate them with their source and target from the start, whereas in (2) we have all morphisms collected into a single grouping, and use $$s$$ and $$t$$ to identify the source and targets.

The point, however, is that what is important in both (1) and (2) are the morphisms. The objects are essentially nothing but index sets used to keep track of the morphisms, the identity maps, and when we can compose maps. But we can drop all of that if we just identify each object with its identity function, and use the identity functions as the way to keep track of “source” and “target”. That is what definition (3) does. The value of $$s(f)$$ is what we want to think of as “the identity function of the source of $$f$$”, while $$t(f)$$ is “the identity function of the target of $$f$$”. The “axioms” are just a way of making sure that these have the properties we want of the identity functions.

When the slide says “data”, they are describing what information they are giving you to define/describe the category. So, to describe a category with definition (3), I need to give you the collection of arrows, the function $$s$$, the function $$t$$ and the relation/partial operation $$\circ$$ that will play the role of “composition”. That will determine the category. The theorem on the right is showing that this way of describing a category is equivalent to (1) or (2), by using $$s(f)$$ to define the family of objects and the family of identities, etc.

• went through this again today. I guess all this expresses the idea in programming language "function should be treated as first class citizen" Commented May 12, 2021 at 15:15

In my understanding, the Arrow Theory is not meant to be a category, so you should not understand "data" as the objects of some category. It is rather another way of expressing the same information that is given by a category. Category theory and Arrow theory are two points of view of the same theory, and the point of the Arrow theory's point of view is that it is much clearer that what category theory focus on are morphisms, and not objects (category theory is about relationship between objects and not the objects themselves).

About the distinction between arrow and morphism, at least in a categorical context, these terms are usually used interchangeably.