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Robert Pare | 1 Jan 2010 15:48
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Small is beautiful


I would like to add a few thoughts to the "evil" discussion.

My 30+ years involvement with indexed categories have led me
to the following understanding. There are two kinds of categories,
small and large (surprise!). But the difference is not mainly one
of size. Rather it's how well we can pin down the objects. The
distinction between sets and classes is often thought of in terms
of size but Russell's problem with the set of all sets was not one of
size but rather of the nature of sets. Once you think you have the set
of all sets, you can construct another set which you had missed.
I.e. the notion is changing, slippery. There are set theories where
you can have a subclass of a set which is not a set (c.f. Vopenka, e.g.)
Smallness is more a question of representability: a functor may fail to
be representable because it's too big (no solution set) or, more often,
because it's badly behaved (doesn't preserve products, say). Subfunctors
of representables are not usually representable.

In our work on indexed categories, Schumacher and I had tried to treat
this question by considering categories equipped with a groupoid of
isomorphisms, which we called *canonical*, and then consider functors
defined up to canonical isomorphism. In small categories only identities
were canonical whereas in large categories, all isomorphisms were canonical.
Our ideas were a bit naive and not well developed and earned us some ridicule,
so we quietly stopped talking about it. Recently, Makkai developed
an extensive theory of functors defined up to isomorphisms, FOLDS, but
did not consider the possibility of specifying which isomorphisms ahead
of time, so small categories were not included.

When I used to teach category theory, before Dalhousie made me chuck my
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Fred E.J. Linton | 1 Jan 2010 05:44
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Re: Quantum computation and categories

Peter Selinger offered the thought that, considering

> ... the category of finite dimensional complex
> vector spaces vs. the category of finite dimensional Hilbert spaces.
> They are equivalent ...

Hmmm ... you mean just *any* linear transformation is 
allowed between two Hilbert spaces? Isn't the category 
of f.d. Hilbert spaces a subcategory of the category of 
Banach spaces (with linear maps of bound ≤ 1)?

If so, I'm not so sure my Hilbert spaces are the same as yours :-) .

Cheers, and Happy New Year, -- Fred

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John Baez | 1 Jan 2010 20:06
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Re: Quantum computation and categories

Happy New Year!

Peter wrote:

Consider the following two categories:
>
> (a) the category of finite dimensional complex vector spaces and linear
> maps, and (b) the category of finite dimensional Hilbert spaces and linear
> maps.
>

> Specifically, take Toby's proposal, and consider two different objects A,B
> of (b) such that both A and B are two-dimensional Hilbert spaces. Let u:A->B
> be some non-unitary isomorphism.

Using "u" to stand for a non-unitary morphism!  Reminds me of the joke:

Teacher: Suppose p is a prime number...

Student: But what if it's not?

Teacher: Well then it wouldn't be called "p", now, would it!

> Then you can easily find an equivalence of categories which identifies both
> A and B with the two-dimensional vector space C^2, and which identifies u
> with the identity morphism on C^2. At this point, you have not equipped the
> category (a) with anything useful, because it does not induce a notion
> of unitary map on C^2.
>
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Dusko Pavlovic | 2 Jan 2010 23:22
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Re: in defense of evil

hi peter,

happy new year! may the daggers over hilbert spaces be the worst evil cast
upon us :)

but i think that this "defense" view of evil suggests that you may be
missing the point.

1) THE PROBLEM OF EVIL

if someone gives you two big complicated hilbert spaces, or two big
complicated groups --- how do you decide whether they are the same? what
does it mean that to be the same group?

well, one thing you could try is to map the elements of one to the
elements of the other one. this may give you an isomorphism.

OR you may look at the actual presentations, say of the group structures
that they gave you, and see whether these are completely equal. if you are
extremely lucky, the two groups may be given by the same equational
presentation. otherwise, if you live in a set theoretic universe, each of
the groups would be given to you as a set, and the two sets can be equal,
or not. in a computer system, each of the groups would be given as a
software module, and they may, or may not be identical as executable
binaries.

the main problem with this is that the various implementation conventions
need to be consistent and interoperable. i think it was peter freyd who
once asked something like: "How can you tell that the set corresponding to
the Monster Group is not the same as the set corresponding to an initial
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Vaughan Pratt | 3 Jan 2010 08:57
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Re: Small is beautiful

Robert Pare wrote:
> Then
> there are the small categories which are used to study the large ones.
> These are syntactic in nature.

Don't get me started.  Oops, too late.

> For these, one can't expect the kinds of
> universal constructions that large categories have,

Not following.  FinSet is an essentially small category, what do you
mean that it doesn't enjoy universal constructions?  It's even a topos.

Then there are the categories enriched in small categories, again
subject to cardinality restrictions, which too are perfectly capable of
enjoying universal constructions.

> but now it's okay,
> even necessary, to consider equality between objects.

For small as opposed to essentially small categories, yes in some cases.
    But consider the category of ordinals truncated at say beth_2,
certainly a small category when the morphisms are the inequalities.  Are
you comfortable defining equality on the objects of this category?  (PTJ
would correctly accuse me of being inconsistent on this point.)

> Well, after these ramblings, perhaps my message is lost. So here it is:
> Small categories -> equality of objects okay
> Large categories -> equality of objects not okay
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Peter Selinger | 3 Jan 2010 08:23
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the definition of "evil"

Dear all,

sorry for sending yet another message on the topic of "evil"
structures on categories. After some interesting private replies, as
well as Dusko's latest message (which should have appeared on the list
by the time you read this), I noticed that not everyone is agreeing on
the technical meaning of the term "evil". I will therefore attempt to
state a more precise technical definition of the term as I have used
it. Perhaps 2-category theorists already have another name for this.

The information definition I had used is that a structure is "evil" if
it does not "transport along equivalences of categories". I thought it
was reasonably obvious what was meant by "transport along", but there
is actually a lot of variation in what people understand this phrase
to mean.

John Baez gave a pointer to a website containing a technical
definition of "evil": http://ncatlab.org/nlab/show/evil.
Unfortunately, this site only speaks of properties, not structures. It
is easy to state what it means for a property of categories to be
transported along equivalences: namely, if C has the property, and C
and C' are equivalent, then C' has the property. Structures are more
tricky.

Certainly, it should not just mean that if C admits such a structure,
and C' is a category equivalent to C, then C' admits such a structure.
(Then "admitting a structure" would merely be a property).  This seems
to be the definition Dusko has used. If we used this definition, there
would be almost no evil structures; in particular, the original
(strict) notion of dagger category is not evil in this sense.  Dagger
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David Roberts | 3 Jan 2010 11:01
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Re: evil (fwd) Re: Quantum computation and categories

> but in general, evil exists. every
> functor can be factored as an identity-on-the-objects-functor (ioof),
> followed by an embedding. the embedding is good, but ioofs are evil, and i
> think that they deserve their name. lord knows how much we use them.

An ioof is 'evil' in a subtly different way to what was discussed, in
my opinion, in that the property of being such a functor is not
invariant under natural isomorphism. This is then really a
2-categorical notion of evil. Are there many examples/other commonly
used properties of functors that are evil in this way?

Happy new year

David

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Ross Street | 3 Jan 2010 22:42
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Re: Small is beautiful

On 03/01/2010, at 6:57 PM, Vaughan Pratt wrote:

>> For these, one can't expect the kinds of
>> universal constructions that large categories have,
>
> Not following.  FinSet is an essentially small category, what do you
> mean that it doesn't enjoy universal constructions?  It's even a  
> topos.

Dear Vaughan

Part of what Bob Paré was arguing, I believe, was that we should be  
flexible
(pun intended) about what "small" means. If "small" means "finite"  
then FinSet
is not "essentially small". Also, "small" could mean "no more than one  
element".

Ross

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John Baez | 3 Jan 2010 18:53
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re: the definition of "evil"

Dear Categorists -

I'm glad Peter is trying to formulate a definition of structures that can be
transported along equivalences, and I like the spirit of his definition,
namely in terms of a "lifting property" where one has a 2-functor

U: XCat -> Cat

and one is trying to lift equivalences from Cat to XCat.

But it makes me nervous when he says "isomorphic [not equivalent!]".  Just
as evil in category theory typically arises from definitions that impose
equations between objects instead of specifying isomorphisms, evil in
2-category theory typically arises when we specify isomorphisms between
objects instead of specifying equivalences.

It would be sad, or at least intriguing, if the definition of "evil" was
itself evil.

Best,
jb

  DEFINITION. Let X be some structure on categories. By this, I mean
>  that there is a given 2-category called X-Cat, whose objects are
>  called X-categories, whose morphisms are called X-functors, and whose
>  2-cells are called X-transformations, and for which there is a given
>  2-functor U to Cat, called the forgetful functor.
>
>  We say that X is "transported along equivalences of categories" if the
>  following holds. Given an X-category D', with underlying category D =
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David Roberts | 4 Jan 2010 01:39
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bigroupoid valued functors

Hi all,

In the following let Bigpd denote the _category_ of bigroupoids and
2-functors. I have a category C with a full subcategory i:D --> C and
a pair of functors

P_C: C --> Bigpd
P_D: D --> Bigpd

and a natural transformation t : P_D => P_C\circ i. The components of
t are biequivalences, and in fact (non-strict) inclusions of
sub-bigroupoids.

I would like to know if it is possible to define a functor P' : C -->
Bigpd that agrees (on objects) with P_D on D and with P_C on the rest
of C.

Cheers,

David

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Gmane