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My instructor introduced the definition of a category where the objects and morphisms are said to be in "collections" instead of "sets". I am new to Category Theory so I am not really sure what is the difference between these concepts? Does this have anything to do with having sets as categories (with collection of sets and functions between sets)?
And later on in class there are definitions like:

Definition: A category $\mathscr{C}$ is small if $\text{Mor}(\mathscr{C})$ is a set (which implies $\text{Object}(\mathscr{C})$ is a set).

I also have trouble understanding why a category being small implies the objects also form a set?
Thanks in advance!

SummerAtlas
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  • Quote: "You might wonder why the framers of the notion of category bothered to use two difficult words class and set rather than only one, in their definition. One could, after all, simply require that there be a set of objects of the category, rather than bring in the airy word class. In fact, people do that, at times: it is standard to call a category whose objects form a set a small category, and these small categories do have their uses. ... – Eric Nathan Stucky Apr 01 '21 at 08:00
  • ... Indeed, if you are worried about foundational issues, it is hardly a burden to restrict attention to small categories. But I think the reason that the notion of class is invoked has to do with the high ambition we have for categories: categories are meant to offer a fluid vocabulary for whole ‘fields of mathematics’ like group theory or topology, with a Fregean desire for freedom from the contingency implicit in subjective choices."

    — Barry Mazur, When is one thing equal some other thing? (page 10).

     – Eric Nathan Stucky Apr 01 '21 at 08:00
  • Check this question out https://math.stackexchange.com/questions/1601545/whats-the-definition-of-a-collection – Noel Lundström Apr 01 '21 at 12:52

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I also define categories using the notion of "collections". The reason is that I do not want to deal with set-theoretic problems (at least at first). Therefore I refrain from specifying whether something is a set or class and just call them collections. This is very common and most likely the reason why your instructor also did that.

Regarding your second question: For each object $X$ of a category $\mathcal{C}$ we have (per definition) one morphisms $X \rightarrow X$, namely the identity morphism. Therefore, the size of $\text{Mor}(\mathcal{C})$ is bigger or equal than the size of $\text{Ob}(\mathcal{C})$. If now $\text{Mor}(\mathcal{C})$ is small enough to be set, then so is $\text{Ob}(\mathcal{C})$.

Con
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  • Thank you, I wonder if you can elaborate on some of the ideas you mentioned? 1) what would be a key difference between "sets" and "collections"? Is it just because sets follow ZFC, etc? (My instructor actually talked about Russell's paradox and stuff to support the idea of using "collections", but I still don't see a key difference.) 2) When you say Morphism being "small enough to be a set", what does that actually mean? – SummerAtlas Apr 01 '21 at 07:48
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    The most satisfying definition I have seen of a class is "A class is a well-formed formula". It avoids the "class of all class" problem because it does not contain anything (this is not quite true; a careful justification is more subtle, but it can be done). The formula does not even refer to anything, necessarily, it is just an arrangement of symbols that is allowed in your logic scheme. A class $F$ with a free variable is "small enough to be a set" if ${x: F(x)}$ is a well-defined set, where $F(x)$ means that by instantiating the free variable as $x$, the resulting statement is true. – Eric Nathan Stucky Apr 01 '21 at 07:57
  • @EricNathanStucky Oh I see, thank you for these comments! – SummerAtlas Apr 01 '21 at 08:07