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* Re: equality is beautiful
@ 2010-03-14  8:51 David Leduc
  2010-03-15 11:25 ` Toby Bartels
                   ` (2 more replies)
  0 siblings, 3 replies; 17+ messages in thread
From: David Leduc @ 2010-03-14  8:51 UTC (permalink / raw)
  To: Richard Garner; +Cc: categories

Dear Richard,

On Mon, Jan 11, 2010 at 11:26 AM, Richard Garner <rhgg2@hermes.cam.ac.uk> wrote:
> Instead one takes a category internal to the type theory to be given by a
> type of objects O, together with an OxO indexed family of hom-setoids
> O(x,y), composition and identities which are maps of setoids, and
> associativity and unitality witnessed by elements of the hom-setoid
> equality. In this setting, we may not talk of equality of objects (since O
> is not a setoid but only a type) but may talk of the equality of any pair of
> parallel arrows.

What about the characterization of limits in terms of products and
equalizers? It states that the limit of a functor F:J->C is
constructed by products indexed by the set(oid) of objects and the
set(oid) of arrows of J. But if you don't allow equality on objects in
J, you only have a preset of object, not a set(oid).

David


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^ permalink raw reply	[flat|nested] 17+ messages in thread
* Re: equality is beautiful
@ 2010-03-21 21:32 Bas Spitters
  0 siblings, 0 replies; 17+ messages in thread
From: Bas Spitters @ 2010-03-21 21:32 UTC (permalink / raw)
  To: David Leduc; +Cc: Richard Garner, categories

Thorsten Altenkirch suggested the arrow category as an example where
we would want equality on objects:
http://www.mail-archive.com/epigram@durham.ac.uk/msg00285.html

Bas

On Sat, Mar 20, 2010 at 8:18 AM, David Leduc
<david.leduc6@googlemail.com> wrote:
> Dear all,
>
> On 3/16/10, Richard Garner <rhgg2@hermes.cam.ac.uk> wrote:
>> Is this really the case?
>
> I did some research on internet and found a document by Mathieu Dupont
> where, at the beginning of Section 4, he claims that it is the case.
> But he did not write where equality on objects in necessary. I am
> confused...
>
>  http://breckes.org/dokumenty/warning.pdf
>
> David
>


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^ permalink raw reply	[flat|nested] 17+ messages in thread
* the definition of "evil"
@ 2010-01-03  7:23 Peter Selinger
  2010-01-05 20:04 ` dagger not evil Joyal, André
  0 siblings, 1 reply; 17+ messages in thread
From: Peter Selinger @ 2010-01-03  7:23 UTC (permalink / raw)
  To: Categories List

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
structure is reflected by full and faithful functors, and therefore by
one half of an equivalence. The point is that the other half won't
respect it.

At least to me, "transported" suggests that the given equivalence
respects the structure in some sense. So here is my attempt at a
definition.

 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 =
 U(D'), and a category C, and an equivalence (F,G,e,h) of categories D
 and C, where F: D -> C, G: C -> D, and e: FG -> id_C, h: GF -> id_D,
 it is then possible to find:

 (1) an X-category C' whose underlying category U(C') is isomorphic
     [not equivalent!] to C. Let c : U(C') -> C be the isomorphism
     (i.e., an invertible functor in Cat) with inverse c': C -> U(C');

 (2) an X-equivalence of X-categories (F',G',e',h'), where
     F': D' -> C', G': C' -> D', e': F'G' -> id_C', and h': G'F' -> id_D'
     [the concept of equivalence makes sense in any 2-category];

 such that

 (3) (U(F'), U(G'), U(e'), U(h')) = (cF, Gc', cec', h).

 Here, cF and Gc' denotes composition of functors, and cec' denotes
 whiskering.

 The structure X is called "evil" iff it is not transported along
 equivalences of categories.

 This finishes the definition.

More informally, "transported along equivalences" therefore means that
if D and C are equivalent, and D has an X-structure, then there is a
way to equip C with an X-structure and to lift the original equivalence
to an X-equivalence.

There was a need for the isomorphism c in the definition, because the
forgetful functor U : X-Cat -> Cat may not be strictly speaking
surjective onto 0-cells in some real-life examples (and in any case,
this forgetful functor may sometimes only be well-defined up to
isomorphism). It is important that c is an isomorphism, rather than an
equivalence, because else the definition becomes vacuous (and we are
precisely interested in notions that are not well-defined up to
equivalence).

Also note that I didn't require the data (C',F',G',e',h') to be
unique, not even up to equivalence in X-Cat. Although in practice, it
will often be unique in this sense. So my definition allows for a
given structure to be transported "in essentially more than one way"
along a given equivalence. I am open to strengthening the definition
to forbid this.

It is clear that the definition generalizes to any 2-category instead
of Cat, so one might for example speak of structures on monoidal
categories, or on categories-with-a-distinguished-subcategory, or even
on dagger categories, as being evil or not.

Here are some examples of structures:

* monoidal structure on categories is non-evil (for concreteness,
  taken with strong monoidal functors and monoidal natural
  transformations).

* strict monoidal structure is evil, when taken with strict monoidal
  functors. With strong monoidal functors, I think it is still evil,
  but I am not sure at this late hour.

* dagger structure is evil. More generally, any structure X with which
  one can equip FHilb (the category of finite dimensional Hilbert
  spaces), and which allows a definition of unitary map that includes
  all identities and that coincides with the usual one on FHilb, and
  for which the full and faithful X-functors preserve and reflect
  unitary maps, is evil. Here is the technical argument again, as it
  seems to have been misunderstood. The forgetful functor
  F : FHilb -> FVect induces an equivalence, whose other half
  G : FVect -> FHilb requires a choice of inner product on each finite
  dimensional vector space. Define such a G in some way. Fix some
  X-structure on FVect. Let V be some non-trivial vector space, and
  let i and j be two different inner products on V. Then (V,i) and
  (V,j) are Hilbert spaces, so different objects of FHilb.  Consider
  the morphism (in FHilb) f:(V,i) -> (V,j) given by f(v) = v. It is
  evidently not unitary. However, we have F(f) = id_V: V -> V, which
  is unitary, no matter the X structure that was chosen on FVect.
  So F does not reflect unitary maps. QED.

  Note that it is F, not G, that is causing problems. As remarked
  above, since G is full and faithful, it is possible to successfully
  reflect the dagger structure along G to FVect. This amounts to
  arbitrarily choosing some inner product on each vector space. But it
  won't be compatible with F.

  Also note that this argument is independent of the definition of the
  2-cells of X-Cat. So it is even valid for some weaker definitions of
  "evil", for example, if one only requires F and G to lift to
  X-functors, rather than to an X-equivalence.

  I will argue that any structure X that claims to be a "weak" version
  of dagger structure should at least satisfy the conditions I listed
  as preconditions for the argument above. This is the basis for my
  claim that no construction such as Toby's or Dusko's can succeed in
  producing a non-evil equivalent of dagger structure.

* the structure of "being equipped with a chosen Frobenius structure
  on each object" is evil, relative to monoidal categories.

* the structure of "being equipped with an identity-on-objects
  covariant functor" is evil.

* more generally, the structure of "being equipped with a chosen
  subcategory" is evil, unless the subcategory is required, as part of
  the structure, to contain all isomorphisms (in which case it is not
  evil).

* poset-enrichment (with composition f o g monotone in f and g) is
  non-evil.

* The following structure is evil: equip a category with a partial
  order on each hom-set, so that composition f o g is monotone in g,
  but not necessarily in f. Proof: Given such a structure on any
  category, define g:A->B to be "monotone" if (X,g) : (X,A) -> (X,B)
  is monotone for all X.  Consider the category whose objects are
  partially-ordered sets, and whose morphisms are *all* functions
  (thanks to Fred Linton for this example). It can be equipped with
  the aforementioned structure, by giving the pointwise ordering to
  the functions in each hom-set.  As a category, it is equivalent to
  Set. The rest of the argument proceeds as above for Hilbert spaces,
  with "monotone" instead of "unitary": take some non-trivial set with
  two different partial orders, then the identity is non-monotone,
  etc.

The last example is almost an enrichment in Poset, but instead of the
usual cartesian product on Poset, we have used another bifunctor on
Poset, given by cartesian product P x Q of the underlying sets, with
the non-standard order defined by (p,q) <= (p',q') iff p=p' and
q<=q'. This operation is bifunctorial and associative, but not quite
monoidal, because it lacks a right unit. More generally, taking
"enrichments" in such almost-monoidal categories often yields evil
structures. An analogous example works for the category of finite
abelian groups and all set-theoretic functions.

Happy new year to all, -- Peter


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^ permalink raw reply	[flat|nested] 17+ messages in thread

end of thread, other threads:[~2010-03-22 16:15 UTC | newest]

Thread overview: 17+ messages (download: mbox.gz / follow: Atom feed)
-- links below jump to the message on this page --
2010-03-14  8:51 equality is beautiful David Leduc
2010-03-15 11:25 ` Toby Bartels
2010-03-16  1:59   ` Michael Shulman
     [not found]   ` <4B9EE601.5070801@uchicago.edu>
2010-03-16  8:03     ` Richard Garner
2010-03-20  7:18       ` David Leduc
2010-03-21  2:17       ` Michael Shulman
     [not found]   ` <c3f821001003201917w4476a777i53fda02cb9bece66@mail.gmail.com>
2010-03-21 17:54     ` Richard Garner
2010-03-21 19:36       ` Toby Bartels
2010-03-22  9:17 ` Thomas Streicher
2010-03-22 16:15 ` Michael Shulman
  -- strict thread matches above, loose matches on Subject: below --
2010-03-21 21:32 Bas Spitters
2010-01-03  7:23 the definition of "evil" Peter Selinger
2010-01-05 20:04 ` dagger not evil Joyal, André
     [not found]   ` <B3C24EA955FF0C4EA14658997CD3E25E370F5672@CAHIER.gst.uqam.ca>
     [not found]     ` <B3C24EA955FF0C4EA14658997CD3E25E370F5673@CAHIER.gst.uqam.ca>
2010-01-09  3:29       ` equality is beautiful Joyal, André
2010-01-10 17:17         ` Steve Vickers
2010-01-10 19:54         ` Vaughan Pratt
2010-01-11  2:26         ` Richard Garner
2010-01-13 11:53         ` lamarche
2010-01-13 21:29           ` Michael Shulman

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