The original definition of Category

[martinbn]

Maclane, in his book Categories for the working mathematician, seems to restrict himself to categories where Ob is a set, and each object is thus a "small" set, then defines the (large) category of all "small" groups. No doubt you are right in anticipating the desire to have categories of categories.

I think these are called small categories, but there are also large categories.

 

[fresh_42]

I tend to follow nLab when it comes to categories. Their definition of GRP has no restrictions:
https://ncatlab.org/nlab/show/Grp
and they cite
https://www.cambridge.org/core/jour...-des-groupes/BDD4B7EA2FD122F7E0F1C81236162FE4
as a reference for an axiomatic approach.

 

[lavinia]

In my experience, meaning mostly studying Algebraic Topology, foundational questions never enter. One speaks of homology functors from the category of topological spaces and continuous maps to the category of abelian groups and group homomorphisms. There seems less to be concern about set theoretic issues and instead, allowing the idea of a topological space and continuous map as a mathematical object. This idea cannot define a set. So are we to say that it isn't really an idea?

This reminds me for some reason of defining the first infinite ordinal as representing the idea of adding one to generate the next finite ordinal.

 

[mathwonk]

here is a (legitimate link to a) reference that purports to treat such questions of relative size of gadgets used in category theory:
http://katmat.math.uni-bremen.de/acc/acc.pdf

".............
Hence sets are special classes. Classes that are not sets are called proper classes. They cannot be members of any class. Because of this, Russell’s paradox now translates into the harmless statement that the class of all sets that are not members of themselves is a proper class. Also the universe U, the class of all vector spaces, the class of all topological spaces, and the class of all automata are proper classes.
Notice that in this setting condition 2.1(4)(a) above gives us the Axiom of Replacement : (5) there is no surjection from a set to a proper class.
This means that each set must have “fewer” elements than any proper class.
Therefore sets are also called small classes, and proper classes are called large classes. This distinction between “large” and “small” turns out to be crucial for many categorical considerations.5
The framework of sets and classes described so far suffices for defining and investigating such entities as the category of sets, the category of vector spaces, the category of topological spaces, the category of automata, functors between these categories, and natural transformations between such functors. Thus for most of this book we need not go beyond this stage. Therefore we advise the beginner to skip from here, go directly to §3, and return to this section only when the need arises.
The limitations of the framework described above become apparent when we try to per- form certain constructions with categories; e.g., when forming “extensions” of categories or when forming categories that have categories or functors as objects. Since members of classes must be sets and U is not a set, we can’t even form a class {U} whose only member is U, much less a class whose members are all the subclasses of U or all functions from U to U. In order to deal effectively with such “collections” we need a further level of generality:
2.3 CONGLOMERATES
The concept of “conglomerate” has been created to deal with “collections of classes”. In particular, we require that:

1. (1) every class is a conglomerate,
2. (2) for every “property” P , one can form the conglomerate of all classes with property
   P,
3. (3) conglomerates are closed under analogues of the usual set-theoretic constructions outlined above (2.1); i.e., they are closed under the formation of pairs, unions, products (of conglomerate-indexed families), etc."

 

[mathwonk]

@martinbn: I thought so too, but I was quoting directly from Maclane, page 12, so he seems to use these terms differently. maybe for him a small set is just what we call a set, and a set is what we call a class.

well, on pages 21-24, Maclane gives a more detailed discussion of the foundational issues, but I find it highly confusing.

 

[mathwonk]

I still am far from understanding, but I gather we mainly want to avoid self contradictory statements. so we have to be careful about what is an admissible statement, and have to rule out statements that would be both true and false. e.g. in this setting, if x and y are classes such that the statement "x is an element of y" is not an admissible statement, then it need not be assigned a value of true or false.

i.e. the collection U above, of all sets that do not belong to themselves as elements, is apparently a proper class, hence the statement "U belongs to U" is inadmissible. We can only form the relation "A belongs to B", when A is small enough, i.e. a "set", or "small set" in Maclane's language.

Although called Russell's paradox, this all seems to go back to more basic logic paradoxes such as "this statement is false". we have to eliminate such statements, which are true if and only if they are false.

This also reminds me of the philosopher wanting to know the way to town when he encountered a man known only to be either a strict truth teller or a total liar. How to ask directions successfully? You probably recall the solution: "Sir, did you know there is free beer being served in town today?" one then simply follows the man to town.

 

[billtodd]

I still am far from understanding, but I gather we mainly want to avoid self contradictory statements. so we have to be careful about what is an admissible statement, and have to rule out statements that would be both true and false. e.g. in this setting, if x and y are classes such that the statement "x is an element of y" is not an admissible statement, then it need not be assigned a value of true or false.

i.e. the collection U above, of all sets that do not belong to themselves as elements, is apparently a proper class, hence the statement "U belongs to U" is inadmissible. We can only form the relation "A belongs to B", when A is small enough, i.e. a "set", or "small set" in Maclane's language.

Although called Russell's paradox, this all seems to go back to more basic logic paradoxes such as "this statement is false". we have to eliminate such statements, which are true if and only if they are false.

This also reminds me of the philosopher wanting to know the way to town when he encountered a man known only to be either a strict truth teller or a total liar. How to ask directions successfully? You probably recall the solution: "Sir, did you know there is free beer being served in town today?" one then simply follows the man to town.

Well some philosophers still don't abandon Naive Set Theory.
https://press.uchicago.edu/ucp/books/book/distributed/U/bo3535532.html

I'll find the time to read it carefully after I'll finish my coursework next year.

 

[weirdoguy]

Fun fact about me studying physics at Warsaw University (as if anyone cares): our lecturer who was teaching us Analysis 1* (first semester, first week of our studies, we're freash out of high school) decided to start with category theory So it has a veeery special place in my heart. Somehow it didn't contribute to my depression.

*In Poland what in most countries is called "calculus" is called "analysis". But our analysis had also elements of topology, and other advanced stuff so I guess the name was more justified.

Sorry for the off-topic.

 

[martinbn]

Fun fact about me studying physics at Warsaw University (as if anyone cares): our lecturer who was teaching us Analysis 1* (first semester, first week of our studies, we're freash out of high school) decided to start with category theory So it has a veeery special place in my heart. Somehow it didn't contribute to my depression.

*In Poland what in most countries is called "calculus" is called "analysis". But our analysis had also elements of topology, and other advanced stuff so I guess the name was more justified.

Sorry for the off-topic.

Did he use it in that course?

 

[weirdoguy]

He did actually, but at one point some "higher power" suggested that he should tone down a little bit.

 

[billtodd]

Fun fact about me studying physics at Warsaw University (as if anyone cares): our lecturer who was teaching us Analysis 1* (first semester, first week of our studies, we're freash out of high school) decided to start with category theory So it has a veeery special place in my heart. Somehow it didn't contribute to my depression.

*In Poland what in most countries is called "calculus" is called "analysis". But our analysis had also elements of topology, and other advanced stuff so I guess the name was more justified.

Sorry for the off-topic.

Also in my university there was a teacher that taught calculus 1 (or was it 2) straight ahead with Topology. I on the other hand didn't take this course with him.

My first acquaintance with category theory in university was in a UG course on Topology which covered basically Munkres book. Or at least Munkres was more rigorous than the teacher, who prefered drawing pictures over giving lengthy arguments.