What exactly is the semantic difference between category and set?

Question

In this question, I asked what the difference is between set and type. These answers have been really clarifying (e.g. @AndrejBauer), so in my thirst for knowledge, I submit to the temptation of asking the same about categories:

Every time I read about category theory (which admittedly is rather informal), I can't really understand how it differs from set theory, concretely.

So in the most concrete way possible, what exactly does it imply about $x$ to say that it is in the category $C$, compared to saying that $x\in S$? (e.g. what is the difference between saying $x$ is a group, versus saying that $x$ is in the Category $\mathrm {Grp} $?).

(You may pick any category and set that makes the comparison most clarifying).

Answer 1

In brief, set theory is about membership while category theory is about structure-preserving transformations – but only about the relationships between those transformations.

Set theory is only about membership (i.e. being an element) and what can be expressed in terms of that (e.g. being a subset). It does not concern itself with any other properties of elements or sets.

Category theory is a way to talk about how mathematical structures of a given type¹ can be transformed into one another² by functions that preserve some aspect of their structure; it provides a uniform language for speaking of a great range of types¹ of mathematical structure (groups, automata, vector spaces, sets, topological spaces, … and even categories themselves!) and the mappings within those types¹. Although it formalises the properties of mappings between structures (really: between the sets on which the structure is imposed), it only deals with abstract properties of maps and structures, calling them morphisms (or arrows) and objects; the elements of such structured sets are not the concern of category theory, and nor are the structures on those sets. You ask “what is it a theory of”; it is a theory of structure-preserving mappings of mathematical objects of an arbitrary type¹.

The theory of Abstract categories³, however, as just stated, totally ignores the sets, operations, relations and axioms that specify the structure of the objects in question; it just provides a language in which to talk about how mappings that do preserve some such structure behave: without knowing what structure is preserved, we know that the combination of two such maps also preserves structure. For that reason, the axioms of category theory require that there be an associative composition law on morphisms and, similarly, that there be an identity morphism from each object to itself. But it does not assume that morphisms actually are functions between sets, just that they behave like them.

In Concrete categories, however, the objects really do correspond to sets and the morphisms to mappings of those sets, or they correspond to objects and morphisms of some other ‘base category’ than $\mathsf{Set}$. Concrete categories model the idea of adding structure to the objects of the base category, although we still ignore that actual structure: we consider only the morphisms of the base category and the abstract behaviour of the morphisms of the concrete category. For example $\mathsf{Grp}$, the category of groups, may be made into a concrete category (over $\mathsf{Set}$), by specifying the homomorphisms as mappings of the underlying sets of the groups; however, this tells us nothing about the group multiplication operations themselves.

As for the implications of your formulations, saying that “$G$ is a group”, that “$G$ is an element of the set of groups” (actually a proper class) or that “$G$ is (an object) in $\mathsf{Grp}$” (or a “$\mathsf{Grp}$-object”) mean the same thing logically, but talking about the category suggests you are interested in group homomorphisms (the morphisms in $\mathsf{Grp}$) and perhaps in what they have in common with other morphisms. On the other hand, saying $G$ is a group might suggest you are interested in the structure of the group (its multiplication operation) itself or perhaps in how the group acts on some other mathematical object. You would be unlikely to talk about $G$ belonging to the set of groups, though you could easily write $ G ∈ S $ for some particular set $ S $ of groups you are interested in.

See also

• https://math.stackexchange.com/questions/tagged/category-theory?sort=votes
• In particular https://math.stackexchange.com/questions/272875/concrete-category-and-abstract-category#1774821
• Abstract and Concrete Categories: The Joy of Cats by Adámek, Herrlich and Strecker
• https://en.wikipedia.org/wiki/Category_theory
• https://en.wikipedia.org/wiki/Concrete_category

¹ _{Here and throughout I do not refer to “type” in the sense of type theory, but rather a set of properties required of the mathematical objects/structures, i.e. a set of axioms they satisfy. Normally these describe the behaviour of some operations or relations on elements of the sets considered to carry the structure, though in the case of sets themselves ($\mathsf{Set}$) there is no structure beyond the sets themselves. In any case, as said above, category theory ignores the details of this structure.}

² _{I should perhaps say “into all or part of one another”: one allows the homomorphism from $ \mathbb Z $ (integers) into $ \mathbb Q $ (rationals) given by $ n \mapsto \frac n 2 $ .}

³ _{Without qualification, ‘category’ normally means ‘abstract category’, introduced, as far as I can see, in 1945 and developed in the 1960’s while Concrete categories seem to appear in the 1970’s.}

Answer 2

Category theory is in some sense a generalization of set theory: the category $C$ could be the category of sets, or it could be something else. So, you learn less if you learn that $x$ is an object in some unspecified category than if you learn that $x$ is a set (since in the latter case it follows that $x$ is an object in specifically the category of sets). If you learn that $x$ is an object in a particular specified category (other than the category of sets), what you learn is different from learning that $x$ is a set (i.e., an object in the category of sets); neither implies the other.

There's no difference between saying that $x$ is a group vs saying that $x$ is an object in the category Grp. Those two statements are equivalent.

Note: we don't say that $x$ is in the category Grp; we say that $x$ is an object in the category Grp. A category has both objects and arrows. You need to specify which you are talking about.

Answer 3

A further point on D.W.'s explanation

There's no difference between saying that $x$ is a group vs saying that $x$ is an object in the category $\mathsf{Grp}$. Those two statements are equivalent.

I'd like to make a stronger statement:

A concept is defined by its category

Think of it from the pespective of an inventor wanting to explain his concept. Suppose your new concept is called $M$. First, you might have to specify how many variations of instances of things that are $M$ can there be. Let's call that collection of instances $M_0$.

Now since you said that there are many things that are $M$, you have to explain each of them compares/relate to each other. You explain why do you think they are different instances of $M$. There might even be multiple ways in which $A \in M_0$ could be compared to $B \in M_0$ each other. Or in some cases, there might be no way to compare them at all. Let's denote that collection of ways to compare $A$ to $B$ as $M(A,B)$.

You probably notice already that $M_0$ forms the collection of objects and $M(A,B)$ is the homset of a category. The laws of category theory then lays out the expected behaviour of 'comparison'.

Once you have that, the category gives you many default property of the concept. Examples range from

• "which instances are essentially the same --- isomorphism",
• "which of these two instance is more and which is less --- section-retraction pair",
• "How many of the basic elements are inside this instance? --- homset from terminal object"

and so on.

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As for the question you ask in comment

What mathematical process/structure is category theory a theory of?

You now know the drill. Want to know what a concept really is? Look at its category. In this case, $\mathsf{Cat}$ the category of small categories and functors between them.

Answer 4

Sets

Basic concept. membership relation $$x\in A$$

Other concepts. Function is explained in terms of membership relation as a set $f$ of ordered pairs with $$(x,y)\in f\text{ and }(x,z)\in f \Rightarrow y=z$$

Philosophy. Sets have an inner structure - they are completely determined by their elements.

Remark. An axiomatic system widely used by set theorists is ZFC. Its strength is the simplicity: there are only sets and a membership relation. On the other hand many mathematicians feel that this leads to a set concept that diverges from their understanding and usage of sets (compare below Leinster). In fact tha vast majority of mathematicians (except set theorists) seems not to use the ZFC axioms. However, sets do not necessarily refer to ZFC (see below categories and ETCS).

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Categories

Basic concept. function (arrow, morphism) $$A\rightarrow B$$

Other concepts. Membership relation $x\in A$ is explained in terms of an arrow from a terminal object (in Set a terminal object is a singleton set $\{y\})$ $$x:1\rightarrow A$$

Philosophy. Objects of a category have a priori no inner structure. They are just characterized by their relations (morphisms) to other objects.

Remark. The basic concept of categories is function and this coincides with the usage of sets by the vast majority of mathematicians. Therefore you might see categories as a conceptual generalization of the way that (most) mathematicians from very different fields use sets in their daily work. Apart from categories (and toposes) as a generalization you might have a look at the axiomatic system ETCS that is axiomatizing sets (compare below Leinster and Lawvere).

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Question. What is the difference between saying x is a group, versus saying that x is in the Category Grp?

One might say the difference is not in the object $x$ itself but rather how you intend to deal with $x$.

(1) Do you ask for the inner structure of $x$? In this case it might appear natural to regard $x$ as a set.

(2) Do you ask how $x$ relates to other objects of same kind (via morphisms) or of other kind (via functors)? In this case you might tend to see $x$ as an object in the category of groups.

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Critics

In case of ZFC and ETCS these approaches can be translated into each other, although ETCS is weaker than ZFC but (seemingly) covers most mathematics (see MathStackExchange and Leinster). In principle (using an extension of ETCS) you can prove the same results with both approaches. So the above mentioned philosophies of both concepts are not claiming a fundamental distinction in what you can express or what results you can prove.

The expressions set and membership in ZFC are abstract concepts just like the concepts of categories or any other axiomatic system and can mean anything. So from this formal viewpoint, to claim, that ZFC is concerned with the inner structure of sets whereas categories deal with the outer relations of objects to each other seems inappropriate. On the other hand this seems to be the philosophy or intuition of the regarding theories.

However in practice you will prefer a certain Approach for e.g. the sake of clarity or simplicity or because some concept or a connection to another area evolves more naturally than elsewhere.

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References

Spivak.Category theory for scientists

Leinster.Rethinking set theory

Lawvere.An elementary theory of the category of sets

MathStackExchange.Category theory without sets