What defines arrows equality

Question

I have decided to learn basics of category theory, but have stumbled upon the very first exercise: given a category C, prove that identity arrow is unique among arrows with domain of X and codomain of X, where X is from the objects of the given category C. But I fail to see or find any definition of arrows equality or inequality. In other words, given 2 arrows: $f:X\to Y $ and $g:X\to Y$, how can I say if they are same or not?

Answer 1

Say you have two identity arrows $\operatorname{Id}_1, \operatorname{Id}_2:X\to X$. By the defining property of identity arrow, we have $$ \operatorname{Id}_1 = \operatorname{Id}_1\circ \operatorname{Id}_2 = \operatorname{Id}_2 $$ and thus the two are equal.

Basically, the axioms and definitions of your theory will tell you when two things are equal. In this case, an identity arrow $\operatorname{Id}:X\to X$ is defined by the following: for any $f:X\to Y$ and any $g:Z\to X$, we have $f = f\circ{\operatorname{Id}}$ and $g = {\operatorname{Id}}\circ g$. One can deduce general results (usually called theorems) which will assist you in less simple cases so you don't have to appeal directly to the axioms all the time, but in the end, all equalities are proven from whatever equalities your axioms and definitions give you.

Exactly how you should prove that $f$ and $g$ in your question are equal will depend greatly on how they are defined, and what you know about the category in which you are working. Some categories only have one arrow for each (ordered) pair of objects, and in that case, they will automatically be equal. Other categories are more complicated. In most common categories, like the categories of groups (abelian or general), topological spaces, and so on, equality of arrows is not commonly shown on a category theoretical level, although some specific cases can benefit greatly from a category theoretical formulation.

Answer 2

This question has bugged me for a long time. Honestly, the fact that there is not a straightforward, satisfactory answer here tends to make me think that there are only like 100 people in the entire world who do category theory. Anyways, I finally found what I guess is a satisfactory answer to the question, but it's not quite as easy as I wish it were...

There is a nice text called FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES AND SOME ALGEBRAIC PROBLEMS IN THE CONTEXT OF FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES by Lawvere. I haven't digested all the material or even finished it - but I could say that about every single thing ever written about category theory. I guess this dude is old school enough to throw in a few logical symbols when going through the basic definitions, axioms, etc.

The following formula appears on page 27:

$$ \forall \mathbb{A}, \forall \mathbb{B}, \forall f, \forall g, \Big[\, \mathbb{A} \xrightarrow{\;\; f\;\;} \mathbb{B} \,\land\, \mathbb{A} \xrightarrow{\;\; g\;\;} \mathbb{B} \,\land\, \forall u \,[ 2 \xrightarrow{\;\; u\;\;} \mathbb{A} \Rightarrow uf = ug ] \Rightarrow f = g \,\Big] $$

Here, $2$ is the category $\cdot\to \cdot$

Roughly speaking, two arrows are equal if no other arrow could distinguish them. I'm not sure if this is truly an axiom of category theory, or if it follows from some more fundamental definitions, but I have no problem taking it as an axiom given that this seemingly important issue seems to have been disregarded in general. Like most things in category theory, I'm not sure if this will help you calculate anything, but it's at least something.

EDIT Well, for what it's worth, just take $u = \mathrm{Id}_{\mathbb{A}}$.