What is a morphism, broadly speaking?

I'm trying to learn category and set theory, and homomorphisms are mentioned right off the bat, and I'm having trouble understanding them. I know this question has been posited time after time on this website, but the answers and questions usually already have some weight of knowledge on this subject I haven't even attained yet. I'm completely new to anything related to this kind of abstract mathematics, and as such even the most basic forms of jargon on this subject is something I struggle with.

Basic examples homomorphisms I come across seem oddly unremarkably complex in what they're essentially stating but I can't piece together what it all means. Let me put this into an example. Here's an excerpt from Wikipedia on an example of a homomorphism:

Here's what I'm interpreting:

• The set of real numbers is a ring. The set of all 2x2 matrices is a ring.

• We're defining a function that is in a matrix and thus in the set of all 2x2 matrices but it uses elements of the set of real numbers as inputs. Thus, it is a function using two sets, both rings.

• If a set preserves addition and multiplication in its operations, it can be called a ring.

• Since, in this operation, which takes the set of all 2x2 matrices and the set of all natural numbers, applies $f(r)$ and maps it to a new set (let's say $\gamma$), and $f(r+s) = f(r)+f(s)$, this still preserves addition and multiplication (although I'm only showing the addition bit here but it's shown in the graphic).

• Since this mapped elements of two rings to create another ring, this function is a homomorphism of rings.

• Since this function preserves the algebraic structures of the sets being used, it is a homomorphism.

I assume some bits of my interpretation are incorrect, and I beseech any logical fallacies to be fixed for my understanding.

If this is true, then what implies that the elements of set $\gamma$ have preservation of addition and multiplication?

Finally, how can I relate this to my general understanding of morphisms if most of what I'm saying is true? In other words: in a general sense then, what are morphisms? What happened if preservation of addition and multiplication wasn't preserved? I've honestly never seen that happen before in my experience.

• There is a lot of confusion here. I think the best thing for you to do right now is to forget about category theory for a while, and grab an elementary abstract algebra textbook. In order of difficulty, or so-called "mathematical maturity", I'd suggest Pinter, Artin, and then Dummit & Foote. – Alex Provost Jun 29 '17 at 18:04
• To answer your main question pedantically: broadly speaking, a morphism is an arrow between two objects. One will usually require morphisms to preserve some structure present in the objects of the ambient category. In the category of sets, a morphism is just a function. In the category of groups, a morphism is a group homomorphism. In the category of topological spaces, a morphism is a continuous function. And so on. – Alex Provost Jun 29 '17 at 18:09
• Although, note that the idea of "preserving certain structure that the objects have" only makes sense in categories where the morphisms are functions (or function-like things) and the objects have properties other than simply being an object (e.g. the property of being a group). – Hurkyl Jun 29 '17 at 18:11
• That is correct. As an example of a category which falls out of the class of examples mentioned above, one might extract from a given partially ordered set a category whose objects are the poset elements and whose morphisms are dictated by the partial order. – Alex Provost Jun 29 '17 at 18:13
• Another non-function-like example: "math made difficult" defines the natural numbers as a specific category. It has a single object, and a single non-identity morphism from that object to itself (plus compositions). It exists by virtue of "As an axiom on which to base the positive numbers and the integers, which have in the past produced much harmless amusement and are still widely accepted as useful by most mathematicians, some such proposition as the following is sometimes considered as being pleasant, elegant, or at least handy: AXIOM: Equalisers exist in the category of categories." – Arthur Jun 29 '17 at 18:38