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A category $\mathsf C$ consists of the following three mathematical entities:

  • A class $\operatorname{ob}(\mathsf{C})$, whose elements are called objects;

  • A class $\hom(\mathsf{C})$, whose elements are called morphisms or maps or arrows. Each morphism $f$ has a source object $a$ and target object $b$.

  • A binary operation $\circ$, called composition of morphisms, such that for any three objects $a$, $b$, and $c$, we have $\hom(b, c) \times \hom(a, b) \to \hom(a, c)$. The composition of $f : a \to b$ and $g : b \to c$ is written as $g \circ f$ or $gf$, governed by two axioms: [...]

What the exact meaning of 'consist of' in the first sentence? Of course, I know the usual meaning. However, since it is not a mathematical term, I don't know the mathematical meaning of 'consists of'.

Community
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Evzone
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  • It just means that a category is the data of : a class of objects, etc. It could be formalized as a n-uple if the category were small, but it general you have to be careful. – Captain Lama Mar 07 '16 at 16:11
  • I understood that in some cases, consists of means n-tuple. But what's the definition of 'data of' ? – Evzone Mar 07 '16 at 16:23

2 Answers2

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To be formal, you can say that a category is a triple $(Ob(C), Hom(C), \circ)$ such that, etc ...

The notion of triple is perfectly and formaly defined in set theory.

Of course, I use the definition of category which states that $Ob(C)$ and $Hom(C)$ must be sets. To work with this definition, one usually uses Grothendieck Universes.

C. Dubussy
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    You have to be careful if your category is not small. For instance, you may want to fix a Grothendieck universe or something. In any case, there is something to be said. – Captain Lama Mar 07 '16 at 16:12
  • I add this precision. Thank you. – C. Dubussy Mar 07 '16 at 16:12
  • Ok, I understood that in some cases, consists of means n-tuple. But, including other cases, what is the meaing of 'consists of' that can apply to all situation? – Evzone Mar 07 '16 at 16:31
  • @Evzone you want an interpretation for "consists of" that covers every possible mathematical usage? It's just a phrase of language! The tuple interpretation works for all such definitions of algebraic objects. – Kevin Carlson Mar 07 '16 at 18:56
  • @KevinCarlson No. I want an interpretation for 'consists of' just for this matter. They said tuple interpretation works well when ob(C) is a set. I understood it. But I want more general interpretation that can cover the case when ob(C) is not a set. – Evzone Mar 07 '16 at 21:22
  • But you can assume that Ob(C) is always a set. I don't know what is your reference book for category theory but all the theory can be obtained in ZFC + axiom of universes. – C. Dubussy Mar 07 '16 at 21:35
  • @C.Dubussy How about class of all sets? It is clearly not a set. It is a well-known example of a proper class. – Evzone Mar 08 '16 at 18:26
  • @Evzone : Of course but instead of working with the class of all sets, you will fix a universe $U$ and work with the category off all sets contained in $U$. This is still a set. We call this category $U-Set$ or simply $Set$ if the context is clear. You could think that it is "restrictive" but it is not, all the theory can be done like this. I can recommand the following excellent book : "Categories and Sheaves" from Shapira and Kashiwara, this is well explained at the beginning. – C. Dubussy Mar 08 '16 at 19:51
  • @C.Dubussy Ok. I understood your mention. 'all sets contained in U' is a set and therefore there's no problem to 'all sets contained in U' be the first coordinate of the tuple. And adopting universe U or something like that is a common way. Thank you for all your comments and I also thanks to your recommendation of a good book. – Evzone Mar 08 '16 at 20:40
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To say an object "consists of" followed by a list of entities means that entities are the data types you have to specify to describe the object.

"Consists of" in the case of the definition of category means that to define a category, you have to specify three mathematical objects: Two classes and a binary operation, and they must satisfy the requirements in the definition.

The definition using an ordered triple also works, but it has the minor problem that the three entities are specified in order, which adds a superfluous fact that is not necessary to the definition. No one thinks that the "first part" of a category is its class of objects. It is not wrong to define a category as a triple, but it certainly adds unnecessary structure and is therefore inelegant.

SixWingedSeraph
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  • Thanks. I understood your answer. I understood that 'consists of' just means category needs following 3 data. Also, the way of formalizing category as a concrete thing might be different to all people. Tuple is just one way among many possible ways. – Evzone Mar 08 '16 at 20:48
  • As you said, Tuple might work well. But it adds unnecessary structure and is therefore inelegant. Then what do you think that it is a good way to formalize category? What do you recommend instead of a tuple? – Evzone Mar 08 '16 at 20:48
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    The wording given in the original question is a formal definition of category in my opinion. It is clear (once you understand what "consists of" means!) and gives a precise specification of what is required to be a category. But maybe you are asking to put it into a formal language. The Homotopy Type Theory book defines "category" in type theory on pages 298ff. Note that "set", "function" and other words used in the definition are previously defined in the HoTT book. I am quite sure you could define "category" in first order logic as well, but I don't know of a reference. – SixWingedSeraph Mar 10 '16 at 01:49