The original definition of Category

[billtodd]

I was a little bit rereading the entry on Category Theory here: https://plato.stanford.edu/entries/category-theory/

And it's written there:

Definition (Eilenberg & Mac Lane 1945): A category C𝐶 is an aggregate Ob𝑂𝑏 of abstract elements, called the objects of C𝐶, and abstract elements Map, called mappings of the category.

What on earth is 'aggregate', and what are those abstract elements?
Every possible mathematical object or philosophical object?

Well mathematical object constitutes as part of philosophical objects anyways.
What did they have in mind?

 

[fresh_42]

A category consists of objects and morphisms between those. Objects in this sense are sets of some kind. They can be for instance groups, vector spaces, topological spaces, algebras, rings, etc., or simply sets without an additional structure.

Morphisms are mappings between those objects that respect the given structure: linear mappings in the case of vector spaces, continuous mappings in the case of topological spaces, group homomorphisms in the case of groups, etc., or just maps in the case of sets.

So instead of listing examples and in order to emphasize the fact that not only objects but also morphisms belong to a category, the authors spoke of an aggregate of both: (Obj , Mor). It is not really a pair since both components consist of many instances and objects and morphisms are of a different quality (sets versus mappings), so an aggregate is a better name for it than a pair.

Category theory deals with properties that are common to all categories alike: isomorphism theorems, nine lemma, tensor product, projective limits, direct products, etc.

 

[billtodd]

A category consists of objects and morphisms between those. Objects in this sense are sets of some kind. They can be for instance groups, vector spaces, topological spaces, algebras, rings, etc., or simply sets without an additional structure.

Morphisms are mappings between those objects that respect the given structure: linear mappings in the case of vector spaces, continuous mappings in the case of topological spaces, group homomorphisms in the case of groups, etc., or just maps in the case of sets.

So instead of listing examples and in order to emphasize the fact that not only objects but also morphisms belong to a category, the authors spoke of an aggregate of both: (Obj , Mor). It is not really a pair since both components consist of many instances and objects and morphisms are of a different quality (sets versus mappings), so an aggregate is a better name for it than a pair.

Category theory deals with properties that are common to all categories alike: isomorphism theorems, nine lemma, tensor product, projective limits, direct products, etc.

Well, but does this notion of categories help for example solving some difficult nonlinear integral equation and/or PDE? I know that Category Theory pops up in AT and AG, and those can be used in the theory of PDE, but is this notion of Category can help solve such difficult equation analytically?

I once in my search found a book called:"Categories in Continuum Mechanics", so I guess there is a use of this abstract nonsense also in applied analysis.

 

[billtodd]

Instead of Mechanics it should be Physics. I get confused between the two sometimes every physics constitutes to me as some sort of mechanics...

 

[fresh_42]

Well, but does this notion of categories help for example solving some difficult nonlinear integral equation and/or PDE?

Not that I knew of. But this isn't the intention of category theory. It was originally meant to define and prove things once like the tensor product instead of doing the same in every single category. The tensor product of vector spaces and the tensor product of algebras have essentially the same properties. And depending on the degree of abstraction, many things are true for all categories.

I know that Category Theory pops up in AT and AG, and those can be used in the theory of PDE, but is this notion of Category can help solve such difficult equation analytically?

No, not in this generality. Here are some examples of relationships
https://www.mdpi.com/2227-7390/9/16/1946
but I doubt that you could actually solve something.

I once in my search found a book called:"Categories in Continuum Mechanics", so I guess there is a use of this abstract nonsense also in applied analysis.

... assuming that category means the same thing which I am not yet convinced of!

Maybe you want to read
https://www.physicsforums.com/insights/higher-category-theory-physics/

As far as I could see ...
https://ncatlab.org/nlab/show/continuum+mechanics
... it was primarily F. WILLIAM LAWVERE who tried to establish such a connection,
http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf .

 

[WWGD]

Somehow it seems Category Theory pops up mostly in Algebra, Algebraic Topology. I was surprised to see that Metric Completion, i.e., the completion of a Metric Space, is a functor.

 

[martinbn]

What on earth is 'aggregate', and what are those abstract elements?

The word is used for the collection of objects because they need not form a set, so the don't want to say a set of objects. For example the category of sets has all sets as objects, and there is no the set of all sets.

On the other hand the morphisms between two objects $Hom(X, Y)$ is a set.

 

[martinbn]

Somehow it seems Category Theory pops up mostly in Algebra, Algebraic Topology. I was surprised to see that Metric Completion, i.e., the completion of a Metric Space, is a functor.

Why were you surprised! The a space you associate another space, its completion, and this respects maps between the spaces. That's what a functor is whether you call it a functor or not. You know functor by any other name..

 

[WWGD]

Why were you surprised! The a space you associate another space, its completion, and this respects maps between the spaces. That's what a functor is whether you call it a functor or not. You know functor by any other name..

I meant most functors seem to arise within Algebra, Algebraic Topology.

 

[lavinia]

I was a little bit rereading the entry on Category Theory here: https://plato.stanford.edu/entries/category-theory/

And it's written there:

What on earth is 'aggregate', and what are those abstract elements?
Every possible mathematical object or philosophical object?

Well mathematical object constitutes as part of philosophical objects anyways.
What did they have in mind?

The abstract idea of category does not depend on the idea of a set though there are categories whose objects do form a set. That I would think is the reason for the word "aggregate".

Categories include things that are too large to be sets. For instance there is no set of all vector spaces but there is a category of vector spaces and linear maps. That is another reason why the word "aggregate" is used.

 

[WWGD]

The abstract idea of category does not depend on the idea of a set though there are categories whose objects do form a set. That I would think is the reason for the word "aggregate".

Categories include things that are too large to be sets. For instance there is no set of all vector space but there is a category of vector spaces. That is another reason why the word "aggregate" is used.

Is there a category that includes the set of all sets?

 

[lavinia]

Is there a category that includes the set of all sets?

The category of sets. I would think that the natural morphisms are inclusion maps. There is no set of all sets.

 

[lavinia]

Categories have an idea of a mapping between them although "mapping" is not a good word since it implies a function between two sets. Instead these are called "functors". A functor assigns to each object in one category an object in a second and to each morphism in the first a morphism in the second.

For instance the derivative is a functor from the category smooth manifolds and differentiable maps to the category of smooth vector bundles and smooth vector bundle morphisms.

For each positive integer there is a homology functor from the category of topological spaces and continuous maps to the category of abelian groups and group homomorphisms.

These functors play a key role in much of mathematics and may be one of the inspirations for considering categories.

 

[martinbn]

Is there a category that includes the set of all sets?

There is no set of all sets, buy the there is a catogory whose objects are all sets.

 

[WWGD]

There is no set of all sets, buy the there is a catogory whose objects are all sets.

Thanks, are the morphisms in the Category of sets, bijections?

 

[WWGD]

The abstract idea of category does not depend on the idea of a set though there are categories whose objects do form a set. That I would think is the reason for the word "aggregate".

Categories include things that are too large to be sets. For instance there is no set of all vector spaces but there is a category of vector spaces and linear maps. That is another reason why the word "aggregate" is used.

Are the collections of objects, morphisms sets, or are they ( potentially at least), too large?

 

[martinbn]

Thanks, are the morphisms in the Category of sets, bijections?

No, all maps. Of course you can consider the category whose objects are all sets and the set of morphisims between two objects is the set of all binections between them or the empty set if they are not bijective.

 

[mathwonk]

On the set theoretic foundational aspect, the authors themselves say, in their General Theory of Natural Equivalences (1945):

"We remarked in §3 that such examples as the "category
of all sets," the "category of all groups" are illegitimate. The difficulties and antinomies here involved are exactly those of ordinary intuitive Mengenlehre; no essentially new paradoxes are apparently involved. Any rigorous foundation capable of supporting the ordinary theory of classes would equally well support our theory. Hence we have chosen to adopt the intuitive standpoint, leaving the reader free to insert whatever type of logical foundation (or absence thereof) he may prefer. …

Perhaps the simplest precise device would be to speak not of the category of groups, but of a category of groups (meaning, any legitimate such category). …

One can also choose a set of axioms for classes as in the Fraenkel-von Neumann-Bernays system. A category is then any (legitimate) class in the sense of this axiomatics. Another device would be that of restricting the cardinal number, considering the category of all denumerable groups, of all groups of cardinal at most the cardinal of the continuum, and so on. "...

 

[martinbn]

On the set theoretic foundational aspect, the authors themselves say, in their General Theory of Natural Equivalences (1945):

"We remarked in §3 that such examples as the "category
of all sets," the "category of all groups" are illegitimate. The difficulties and antinomies here involved are exactly those of ordinary intuitive Mengenlehre; no essentially new paradoxes are apparently involved. Any rigorous foundation capable of supporting the ordinary theory of classes would equally well support our theory. Hence we have chosen to adopt the intuitive standpoint, leaving the reader free to insert whatever type of logical foundation (or absence thereof) he may prefer. …

Perhaps the simplest precise device would be to speak not of the category of groups, but of a category of groups (meaning, any legitimate such category). …

One can also choose a set of axioms for classes as in the Fraenkel-von Neumann-Bernays system. A category is then any (legitimate) class in the sense of this axiomatics. Another device would be that of restricting the cardinal number, considering the category of all denumerable groups, of all groups of cardinal at most the cardinal of the continuum, and so on. "...

So, what exactly is the problem with the "category of all groups"?

 

[fresh_42]

So, what exactly is the problem with the "category of all groups"?

I guess you can define a group structure on it. $G^{-1}=G$ and $(G,H) \mapsto G\times H.$ Something like this.

 

[mathwonk]

I am not expert on these foundations and tend to ignore them. But there apparently is a problem with naive set theory, in which every "property" is assumed to define a set of objects having that property. The classic example of a contradiction involved in this theory is to consider the "set of all sets". Given any set S, one can ask whether S is actually an element of itself or not, although this may sound unlikely.
But then, within the set of all sets, there should be a subset U of those sets which are not elements of themselves. If this makes sense, then one should be able to decide whether the set U belongs to itself or not. However, by the definition of U, we see that if U does not belong to itself, then it does belong to itself, and vice versa! Thus this property, which is formed using the most elementary terms in set theory, does not define a set.

I have heard that one approach to this problem is to decide what properties are allowable, and to call any well described aggregate a "class", and then to reserve the term "set", for a class that does belong to some other class. But I don't really know. I have always heard people refer to the category of (presumably all) groups, but maybe those people understand the subtleties involved and suppress them.

 

[martinbn]

I am not expert on these foundations and tend to ignore them. But there apparently is a problem with naive set theory, in which every "property" is assumed to define a set of objects having that property. The classic example of a contradiction involved in this theory is to consider the "set of all sets". Given any set S, one can ask whether S is actually an element of itself or not, although this may sound unlikely.
But then, within the set of all sets, there should be a subset U of those sets which are not elements of themselves. If this makes sense, then one should be able to decide whether the set U belongs to itself or not. However, by the definition of U, we see that if U does not belong to itself, then it does belong to itself, and vice versa! Thus this property, which is formed using the most elementary terms in set theory, does not define a set.

I have heard that one approach to this problem is to decide what properties are allowable, and to call any well described aggregate a "class", and then to reserve the term "set", for a class that does belong to some other class. But I don't really know. I have always heard people refer to the category of (presumably all) groups, but maybe those people understand the subtleties involved and suppress them.

But how does the category of groups have this problem?

 

[mathwonk]

I suppose one can ask whether a group occurs as an element of itself.

 

[WWGD]

I am not expert on these foundations and tend to ignore them. But there apparently is a problem with naive set theory, in which every "property" is assumed to define a set of objects having that property. The classic example of a contradiction involved in this theory is to consider the "set of all sets". Given any set S, one can ask whether S is actually an element of itself or not, although this may sound unlikely.
But then, within the set of all sets, there should be a subset U of those sets which are not elements of themselves. If this makes sense, then one should be able to decide whether the set U belongs to itself or not. However, by the definition of U, we see that if U does not belong to itself, then it does belong to itself, and vice versa! Thus this property, which is formed using the most elementary terms in set theory, does not define a set.

I have heard that one approach to this problem is to decide what properties are allowable, and to call any well described aggregate a "class", and then to reserve the term "set", for a class that does belong to some other class. But I don't really know. I have always heard people refer to the category of (presumably all) groups, but maybe those people understand the subtleties involved and suppress them.

IIRC, that's called Russell's Paradox, with the parallel metaphor of the town barber who only shaves those who don't shave themselves. Then the barber should only shave himself if he doesn't shave himself.

 

[martinbn]

I suppose one can ask whether a group occurs as an element of itself.

How does this relate to categories?

 

[mathwonk]

Almost the only place I have ever seen any mathematician consider such niceties is in the treatment of Cech cohomology by Hirzebruch, in his Topological methods in algebraic geometry, where he says:

"An open covering U of X is an indexed system U = {Ui} of open sets of X whose union is equal to X. The index i runs through the given index set I and so it is possible for the same open set to occur several times in the covering. Since the index set is arbitrary there are logical difficulties in discussing the set of all open coverings of X. These difficulties can be avoided by considering the set of all proper coverings of X. An open covering U = {Ui} is proper if distinct indices i, j in I determine distinct open sets Ui , Uj and if the index set is chosen, in the natural way, as the set of all open sets of the covering. Each proper covering is then a subset of the set of all subsets of X."

 

[mathwonk]

the connection with categories is whether the collection Ob of all objects in the category "makes sense", and how one goes about describing sub collections forming subcategories. But I really do not appreciate these subtleties enough to explain them well.

 

[martinbn]

the connection with categories is whether the collection Ob of all objects in the category "makes sense", and how one goes about describing sub collections forming subcategories.

And going back to my quetsion, what is the issue with the category of groups?

 

[mathwonk]

the problem seems to be what does one mean by "the set of all groups", which is Ob in the category of (all) groups?

I.e. presumably, if there is a category of all groups, then there should be a subcategory of those groups that are not elements of themselves, which there is not. But again, I am not expert.

I think the fact that you are able to ask your question, several times, is evidence that this issue should not matter to you, as it also does not to me.

or rather, more accurately, it is evidence that I do not understand the problem well enough to explain why it is a problem. so I'll stop trying.

 

[martinbn]

the problem seems to be what does one mean by "the set of all groups", which is Ob in the category of (all) groups?

I.e. presumably, if there is a category of all groups, then there should be a subcategory of those groups that are not elements of themselves, which there is not. But again, I am not expert.

I think the fact that you are able to ask your question, several times, is evidence that this issue should not matter to you, as it also does not to me.

or rather, more accurately, it is evidence that I do not understand the problem well enough to explain why it is a problem. so I'll stop trying.

But the definition does not say that Ob needs to be a set.