re: the definition of "evil"

re: the definition of "evil"

[Fred E.J. Linton]

After Mike Shulman's remarks beginning with

> It looks to me like there are (at least) two different ideas of "evil"
> floating around.

I'm tempted to ask: which idea(s) of "evil" does 
the notion of a category's being *skeletal* embody?

And is that *really* "evil" in the everyman's sense of the word?

Cheers, -- Fred




[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

re: the definition of "evil"

[Michael Shulman]

It looks to me like there are (at least) two different ideas of "evil"
floating around.


1. A property or structure (on objects of a 2-category) is "non-evil" if
it can be transported along equivalences.

This is clearly a property of a forgetful 2-functor, which I agree that
it makes the most sense to formulated as a lifting property for entire
(adjoint) equivalences, including both functors and the unit and counit.
 (I'm surprised that Peter didn't require (F,G,e,h) and (F',G',e',h') to
be *adjoint* equivalences in his definition; that seems to me likely to
be the more correct notion.)  Thus, whether a given structure is "evil"
in this sense depends on what you are forgetting down to.  Dagger
structure is evil as structure on a category, but it is not evil as
structure on a "category equipped with a distinguished subgroupoid."
(Of course, a distinguished subgroupoid is evil as structure on a category.)

I also think that this notion must necessarily be "2-evil" in the way
that makes John sad, for anything at all is always "transportable along
an equivalence up to equivalence"!  In other words, if we are serious
about avoiding evil, even at a higher-categorical level, then we
shouldn't even be talking about evil in the first place.  (-:

(Of course, that also suggests that we probably can't construct, by
purely non-2-evil (e.g. bicategorical) means, the 2-category of dagger
categories from the 2-category of categories.  But we can construct
something pretty close, e.g. if we weaken "distinguished subgroupoid" to
"faithful functor with groupoidal domain.")


2. A categorical structure is "evil" if it involves talking about
equality of objects.

For this sense, one has to be careful, because lots of notions in
category theory involve equality of objects.  In order to compose two
morphisms f:A-->B and g:B-->C one has to know that the target of f is
*equal* to the source of g.  Likewise, to say that f:A-->B is an
isomorphism, one has to say that there is an inverse g:B-->A whose
source and target are *equal* to the target and source of f,
respectively.  However, as Toby says, there is a precise way to say that
something is "not evil" in this sense while still admitting all of these
"natural" constructions.  Namely, we work in a dependent type theory
with a type Ob of objects, and for each pair of objects x,y a dependent
type Hom(x,y) of arrows, and stipulate that our theory contains an
equality predicate only for the types Hom(x,y) and not for Ob.
(Makkai's FOLDS, which Toby mentioned, is a generalization of this
appropriate for higher-categorical structures.)

The point is that specifying the source and target of an arrow should
not be thought of as "talking about equality of objects," but rather as
a *typing assertion*.  What is forbidden is rather asking whether two
already *given* objects are equal, not introducing an arrow whose source
and target are ("equal to") some pair of already given objects.  The
notion of "dagger category" can be formulated in this dependently-typed
language, as Toby has said, so it is *not* evil in this sense.  This is
related to the observation that dagger-categories are still
"implementation-independent" relative to membership-based set theory,
e.g. for the dagger-structure on Hilb it doesn't matter whether you
define the real numbers as Cauchy sequences or Dedekind cuts.


The relationship between these two notions is not immediately obvious to
me.  Clearly evil (1) does not imply evil (2), because of the example of
dagger-categories.  Does evil (2) imply evil (1)?

Best,
Mike

John Baez wrote:
> 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
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: the definition of "evil"

[Urs Schreiber]

On Sun, Jan 3, 2010 at 8:23 AM, Peter Selinger <selinger@mathstat.dal.ca> wrote:

> 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.

Fortunately, though, everybody can and is invited add to this site! So
eventually it may speak also of structures --  and much more.

This site was created for exactly the kind of situation we have here:
we have an intersting technical discussion on a mailing list or
similar forum. After a while it will end and a bunch of scattered
messages will remain in the archives of the mailing list. The
important insight gained or exhibited in the discussion will be
non-trivial to find and deduce from the archived discussion threads.
It'll be a shame if all the valuable insight of various participants,
all the energy they invested into composing these messages, find no
more focused and polished incarnation than that.

The above site is meant to provide a place where results of such
discussion is collected in a more useful form. I am hoping that
eventually the upshot of the discussion on "evil" had here on the list
will eventually find a nice incarnation on that site. Everyone can
help to make that come true. Just hit the "edit" button at the bottom
of the page:

http://ncatlab.org/nlab/edit/evil

Best,
Urs


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

re: the definition of "evil"

[John Baez]

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 =
>  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.
>
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

re: the definition of "evil"

[Claudio Hermida]

Peter Selinger wrote:

 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.



Dear all,

A quick pointer to a variation on this matter, which has already been around
(I presented it at CT07).
The nature of 2-categorical structures that transport along equivalences is
nicely captured by the notion of Equ-iso-fibration:

DEFINITION
A 2-functor F:E → B is an Equ-iso-fibration if:

 - it admits cartesian liftings of (adjoint) equivalences and

 - every hom F(X,Y):E(X,Y) → B(FX,FY) admits cartesian lifitings of
isomorphism 2-cells, and such liftings are preserved by precomposition
(pointwise).

Notice that a cartesian lifting of an adjoint equivalence is another such.
The notions of cartesian liftings used here are those for 2-fibrations (as
in the references below); they involve the expected 2-dimensional property
for 1-cells.
The liftings are strict.

The most interesting example of such a structure is the forgetful functor of
a 2-cateogory of pseudo-algebras
V: Ps-T-Alg → K, for a 2-monad T on a 2-category K (or a pseudo-monad on a
bicategory). They include all the known interesting examples of
pseudo-structures, the classical such being monoidal categories. The
morphisms in Ps-T-Alg are strong ones, preserving structure up to coherent
isomorphism. The observation that pseudo-algebras do transport along
equivalences is credited to Max Kelly in Power´s article below.

I used this notion to capture the following universality of coherence:
coherence for such pseudo-algebras (meaning that we can strictify every
pseudo-algebra into a equivalent strict T-algebra) is equivalent to the
statement

THM: The inclusion 2-functor J: T-alg → Ps-T-alg makes V: Ps-T-alg → K the
FREE Equ-iso-fibration over
U:T-alg → K

In other words, the passage from strict algebras (with their strict
morphisms) to pseudo-algebras (and strong morphisms) is the universal way of
achieving transportability along equivalences.

Claudio

References

- C.Hermida, Some properties of Fib as a fibred 2-category, JPAA, *134 (1),
83-109, 1999*

- C.Hermida, Descent on 2-fibrations and 2-regular 2-categories, Applied Cat
Str, *12(5-6), 427--459, 2004*

- A.J.Power, A general coherence result,


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

the definition of "evil"

[Peter Selinger]

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


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]