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## category theory

There's an infinite number of categories?
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 Recognitions: Homework Help Science Advisor Yes. There is a proper class of categories.
 Recognitions: Gold Member Science Advisor 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.

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## category theory

And the point of that is?

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 Quote by matt grime 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
 Recognitions: Homework Help Science Advisor 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.

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 Quote by matt grime 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."
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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.
 Recognitions: Homework Help Science Advisor 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.

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OK, you're the boss

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 Quote by matt grime 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
 Recognitions: Gold Member Science Advisor 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
 Recognitions: Homework Help Science Advisor Are you at least now accepting the fact that * is a contravariant functor from some category to itself satisfying certain properties?
 Recognitions: Homework Help Science Advisor 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....
 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus The off-topic discussion about gravity can be found here: Gravity.

 Quote by marcus .. I dont think what you say actually captures Baez definition... ...Your definition "contravariant idempotent functor that is the identity on objects" doesnt capture one of the essential features of a star category.
which feature does he not capture?
 Recognitions: Homework Help Science Advisor 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).
 Blog Entries: 4 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