Jacob wrote:

> Binary relations apply to specific elements while morphisms are like functions in that they apply to categories and not specific elements. Is that correct?

Not quite. Let's start by saying what a binary relation is. Given a set \\(S\\), a **binary relation** is something that can be either true or false for any pair of elements of \\(S\\). An example is equality: for any pair of elements of \\(S\\) the statement \\(x = y\\) is either true or false.

I suppose you could summarize this by saying "binary relations apply to specific elements of a set".

Next for something much harder: a taste of category theory. A category has a set of "objects", and for any pair of objects \\(x\\) and \\(y\\), there is a set of "morphisms from \\(x\\) to \\(y\\)". There's more to a category than this - see my lecture above for a small taste - but this is the first part of the story.

So, I hope you see why it's wrong to say "morphisms are like functions in that they apply to categories and not specific elements".

If this still doesn't make sense, please ignore all my remarks about categories until I get around to actually explaining this subject!

> Binary relations apply to specific elements while morphisms are like functions in that they apply to categories and not specific elements. Is that correct?

Not quite. Let's start by saying what a binary relation is. Given a set \\(S\\), a **binary relation** is something that can be either true or false for any pair of elements of \\(S\\). An example is equality: for any pair of elements of \\(S\\) the statement \\(x = y\\) is either true or false.

I suppose you could summarize this by saying "binary relations apply to specific elements of a set".

Next for something much harder: a taste of category theory. A category has a set of "objects", and for any pair of objects \\(x\\) and \\(y\\), there is a set of "morphisms from \\(x\\) to \\(y\\)". There's more to a category than this - see my lecture above for a small taste - but this is the first part of the story.

So, I hope you see why it's wrong to say "morphisms are like functions in that they apply to categories and not specific elements".

If this still doesn't make sense, please ignore all my remarks about categories until I get around to actually explaining this subject!