Explore the Endless Possibilities of Categories

[meteor]

There's an infinite number of categories?

 

[matt grime]

Yes. There is a proper class of categories.

 

[marcus]

More basic stuff about categories

Hello Meteor, and Matt Grime
would it be all right for this to be a general purpose thread for
basic category theory?

the concepts of "fully faithful" functor and
"opposite category" and
"contravariant functor" came up in some recent posts.

these are not the same as Baez idea of a *-category but they are interesting to talk about and know the definitions of.

Here is a bit that I believe is cribbed from Saunders MacLane "Categories for the Working Mathematician"

--------full and faithful functors---
A functor T : C -> B is 'full' when to every pair c, c' of objects of C
and to every arrow g : Tc -> Tc' of B, there is an arrow f : c -> c' of C
with g = Tf. Clearly the composite of two full functors is a full functor.

A functor T : C -> B is 'faithful' (or an embedding) when to every pair
c, c' of objects of C and to every pair f_1, f_2 : c -> c' of parallel
arrows of C the equality Tf_1 = Tf_2 : Tc -> Tc' implies f_1 = f_2.
Again, composites of faithful functors are faithful. For example,
the forgetful functor Grp -> Set is faithful but not full and
not a bijection on objects.

These two properties may be visualized in terms of hom-sets (see (2.5)).
Given a pair of objects c, c' in C, the arrow function of T : C -> B
assigns to each f : c -> c' an arrow Tf : Tc -> Tc' and so defines
a function:

T_c,c' : hom(c, c') -> hom(Tc, Tc'), f ~> Tf.

Then T is full when every such function is surjective, and faithful
when every such function is injective. For a functor which is both
full and faithful (i.e., "fully faithful"), every such function is
a bijection, but this need not mean that the functor itself is an
isomorphism of categories, for there may be objects of B not in
the image of T.

Saunders Mac Lane,
'Categories for the Working Mathematician', pp. 14-15.
2nd edition, Springer, New York, NY, 1997.
http://stderr.org/pipermail/inquiry/2003-July/000634.html

---------contravariant functor-------

A functor F is called contravariant if it reverses the directions of arrows, i.e., every arrow f:X->Y is mapped to an arrow T(f): T(Y)->T(X).

--------definition of functor---
is everyone clear on this one? probably since it is so basic.

 

[matt grime]

And the point of that is?

 

[marcus]

And the point of that is?

point of what?

I am hoping BTW that one of us will supply a way of defining Baez concept of a *-category in these terms. would you like to do the honors? that is, say what a *-category is

 

[matt grime]

The point of your post. It looks like you've just hijacked a thread.

And the paper you refer to defines star category: it is one equipped with a contrqavariant functor (equivlance I believe as it's invertible) that is the identity on objects and whose square is the identity.

 

[marcus]

And the paper you refer to defines star category: it is one equipped with a contrqavariant functor (equivlance I believe as it's invertible) that is the identity on objects and whose square is the identity.

this is in line with what you said back there
------quote from Matt back in LQG etc forum----
"the star is a contravariant idempotent functor from Hilb to Hilb that is the identity on objects."
------------------------------------------------

The "paper" I refer to was MacLane "Categories for the Working Mathematician" but it does not AFAIK define "star-category"
however using some concepts in MacLane you (or I) can construct
a possible alternative definition of the concept

It will take me a little time to mull over what you say. I don't think what you say actually captures Baez definition.

For him the * is not a functor from Hilb to Hilb. the idea is not limited to Hilb. You get a different star
for each category. Your definition "contravariant idempotent functor that is the identity on objects" doesn't capture one of the essential features of a star category.

but you could add something to it and say
"contravariant idempotent functor that is the identity on objects and so-and-so-and-so" and then it might be equivalent to Baez concept!

your rephrase doesn't quite do it either:
"... star category: it is one equipped with a contravariant functor that is the identity on objects and whose square is the identity."

If you keep working on this you will get something equivalent to B's definition I think.

 

[matt grime]

I am not particularly interested in this, and if you look in Baez's paper you will see that what I've written above (in another thread, please pick one) is exactly what a star category is. What do you think I am omitting from this definition? I might well be missing something but I can't think of it right now. Perhaps you want takes Id to Id, but that might follow from the other observations. I get the impression you're trying to encourage me to think about this, please don't. It's fairly obvious what's going on.

 

[marcus]

I get the impression you're trying to encourage me to think about this, please don't. .

OK, you're the boss

 

[marcus]

Perhaps you want takes Id to Id, but that might follow from the other observations...

might or might not, wd require proof
maybe someone else here will check that one out, prove or
offer counterexample

 

[marcus]

Hey meteor!
look how your "Category Theory" thread has grown.
we actually have a fair number of people conversant
maybe some of the others will bring questions or definitions and
grow this thread some more

 

[matt grime]

Are you at least now accepting the fact that * is a contravariant functor from some category to itself satisfying certain properties?

 

[matt grime]

perhaps, as we're doing double posts, you might care to think of star category of hilb spaces as to category of vect spaces as C-star alg is to banach space...

 

[quantumdude]

The off-topic discussion about gravity can be found here:

Gravity.

 

[lethe]

.. I don't think what you say actually captures Baez definition...

...Your definition "contravariant idempotent functor that is the identity on objects" doesn't capture one of the essential features of a star category.

which feature does he not capture?

 

[matt grime]

I should have said unipotent (with some indication of order) not idempotent, as it happens (trying to use too few wrods), or an 'involution' but that almost certainly wouldn't be standard. I haven't checked that Id*=Id ought to be a definition or not, but T*=(IdT)* = T*Id* [and is id*T*] for all T. It would then depend what kind of categories you were thinking about, so perhaps it ought to be in the definition (if the category had monos and epis it would follow that Id*=Id).

 

[meteor]

Hi, I don't see any harm that you discuss about category theory in the thread
It's funny that each category has a different name, for example Hilb (the category of Hilbert spaces with linear operators as morphisms), or nCob (the category of (n-1)-dimensional manifolds with n-dimensional manifolds as morphisms). Since there are an infinite number of categories, there will not be possible never to make a book with a complete list of the names of categories!
I really would appreciate a definition of what's a sheaf (even a roughly definition). The entry in wikipedia is too technical for me

 

[matt grime]

When I said that there are an infinite number of categories I was strictly speaking of it in this sense:

Let w be any cardinal, then there is a group of cardinality w. To each group we can assign a category with one object and whose hom set is the group. There are a proper class of cardinals.

Moreover, given any category, we can replace any element with w isomorphic copies of itself, the resulting category is equivalent to the original category.

If you want to ask are there infinitely many "kinds" of category you need to tell us what you mean by "kind".

 

[meteor]

Hi matt
by different kind i mean that two categories don't have the same objects AND the same morphisms.
For example, consider the category Ring that has rings like objects and ring homomorphisms like morphisms. (this category really exist, i saw it somewhere)
Now imagine that there exist a category that have rings like objects and differentiable maps like morphisms. (I'm not sure if this category actually exist)
Then i would say that those categories are not of the same kind

but, say, you have the category Met of metric spaces like objects and short maps like morphisms (this category really exist). And you have another category with metric spaces like objects and short maps like morphisms. Then i would say that those categories are of the same kind

That was my question, if there's an infinite number of categories all of different kind

 

[matt grime]

That isn't a particularly consistent definition of kind. My group as a categories are both the same and different from each each other by it.

 

[Organic]

Matt,

How to you know that they are both the same AND also different?

 

[matt grime]

Because the first criterion is satisfied, the morphisms are different, thus the categories are "different". But from the rest of the post it seems that they are not "essentially" different enough to count as "different" categories because I used the same words to describe them, and count as the same by the last criterion in the example of Met. The inexactitudes of language, something I'm sure you are becoming expert on.

 

[WWW]

To what category these connected disconnected categories belong?

 

[matt grime]

Modulo some set theory every category is in CAT the category of categories.

If you read the definition of a category, you'll realize that meteor's request is ill-defined. A category is essentially a collection of objects and arrows between them, so on some level all categories are the same in his wording.

Yet the derived categories of q-c sheaves on P^n is equivalent to the category of modules for some quiver, so those categories are different in his wording, yet they are essentially the same object in CAT.

Just saying we call them different if we use different words to describe the objects and morphisms doesn't provide a sensible way of distinguishing things.

We can sort of see what is being got at, but it still needs work to make it a well defined question.

 

[WWW]

Modulo some set theory every category is in CAT the category of categories.

What about CAT and russell's paradox?

 

[matt grime]

not an issue, we don't claim it is a set of objects.

 

[WWW]

Modulo some set theory every category is in CAT the category of categories.

What is the difference between an abstract object like Category and a Category?

 

[matt grime]

I don't understand the question. Which is phrased in exactly the same way as one Organic would ask.

 

[WWW]

You used the words 'set theory' to say something on 'Category' and also you say that you don't claim it is a set of objects, so I am a little confused.

 

[matt grime]

there are hom sets too. in some people's systems hom sets must be genuine sets. as the hom sets in CAT are natural functors it is a debatable point whether or not we say it is a category without further qualification. there are usually a proper class of objects in a category, we can often replace them with ones that are skeletally small for instance.

 

[WWW]

If objects can be replaced between categories, do we get mathematical structures that can be ordered by their abilities to replace objects, and if so, how this CAT structure will look like?

 

[matt grime]

Your last post looks like something that organic would post and makes no sense at all so I correspondenigly refuse to answer it.

 

[meteor]

I'm wondering if there exist a category in which the objects are tensors; if this category exist, please what's its name and what morphisms it has.

 

[matt grime]

Whilst there are tensor categories you ought to bear in mind these tensors are not the tensors I'm guessing you know about (invariant under rotation), so don't be mislead.