category theory

meteor

There's an infinite number of categories?

matt grime

Yes. There is a proper class of categories.

marcus

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/...ly/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

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

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 dont 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" doesnt 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 doesnt 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

OK, you're the boss

marcus

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....

Tom Mattson

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

Gravity.

lethe

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