[Jonatan wrote](https://forum.azimuthproject.org/discussion/comment/16179/#Comment_16179):

> What is the difference between an element and an object?

I hope you know a bit about [sets](https://en.wikipedia.org/wiki/Set_(mathematics)): if you feel shaky about them, click the link and read! A set \\(S\\) has **elements**, and we write \\(x \in S\\) when \\(x\\) is an element of \\(S\\).

A category \\(C\\) consists of two sets: a set \\(\mathrm{Ob}(C)\\) of **objects**, and a set \\(\mathrm{Mor}(C)\\) of **morphisms**. There's also more to a category (re-read my lecture for some of this), but that's where we start.

I emphasize that 'element', 'object' and 'morphism' are undefined terms: that is, they don't really mean anything except insofar as we demand that they obey certain rules.

Summarizing my answer to your first question: an object of a category \\(C\\) is an element of the set \\(\mathrm{Ob}(C)\\).

Note that you're forcing me to talk about categories, which is not at all what I want to be doing now!. I'm trying to explain preorders. My remark about categories was just a whispered comment to those who already understand categories - to explain why I'm starting a course on categories by explaining preorders!

Focus on preorders: objects and morphisms can wait.

> What is the difference between an element and an object?

I hope you know a bit about [sets](https://en.wikipedia.org/wiki/Set_(mathematics)): if you feel shaky about them, click the link and read! A set \\(S\\) has **elements**, and we write \\(x \in S\\) when \\(x\\) is an element of \\(S\\).

A category \\(C\\) consists of two sets: a set \\(\mathrm{Ob}(C)\\) of **objects**, and a set \\(\mathrm{Mor}(C)\\) of **morphisms**. There's also more to a category (re-read my lecture for some of this), but that's where we start.

I emphasize that 'element', 'object' and 'morphism' are undefined terms: that is, they don't really mean anything except insofar as we demand that they obey certain rules.

Summarizing my answer to your first question: an object of a category \\(C\\) is an element of the set \\(\mathrm{Ob}(C)\\).

Note that you're forcing me to talk about categories, which is not at all what I want to be doing now!. I'm trying to explain preorders. My remark about categories was just a whispered comment to those who already understand categories - to explain why I'm starting a course on categories by explaining preorders!

Focus on preorders: objects and morphisms can wait.