Peter wrote:

In order to focus on the math and not on the terminology, let me today use the word "XXXX" instead of "evil".

Good. I think you're secretly on my side, though, because you're using four X's. :-)

I don't think the notion used in your examples is general enough. For example, fix some groupoid C, and consider the property of an object x: "x is isomorphic to exactly 3 objects of C".

What a fiendishly clever example!

To me, this is clearly XXXX, because it is not invariant under equivalences of

C. Yet, according to the definition you used in this email, it extends

to a functor C -> {F,T}, and therefore is non-XXXX.

True. By the way, in case anyone out there forgets: "Extending to a functor C -> {F,T}" was merely a pedantic way of saying "being a property of objects of C that is invariant under isomorphisms" - a pedantic way that lets us easily generalize this notion to "being a structure on objects of C that is covariant under isomorphisms". To generalize, we just replace {F,T} by Set. It may not be clear to everybody why I like this pedantic approach, so I should probably explain why. A (-1)-category is a truth value, a 0-category is a set, and a 1-category is a category. So, when we replace {F,T} by Set, we are replacing the 0-category of (-1)-categories by the 1-category of 0-categories. Since we are just increasing a certain parameter by 1, it becomes easy to see how to continue this game indefinitely. For more details, try this: http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=12 For a property P of objects x of a category C, "being invariant under

isomorphisms of objects in C" is strictly weaker than "being invariant

under equivalences of C".

Yes. Here's my rejoinder: I had been fixing a groupoid C and asking whether a property of objects of that category was invariant under isomorphisms. When you say "x is isomorphic to exactly 3 objects of C", you are actually treating C not as fixed but as variable. The more things we let vary, the more invariance properties we can demand! In particular, there's a 2-groupoid Cat_* where an object is a "pointed category" (C,x), that is, a category C with chosen object x. I can treat "being an object x that is isomorphic to exactly 3 objects in C" as a property of pointed categories. And, I would call this property evil... whoops, I mean XXXX... because it determines a function Ob(Cat_*) -> {F,T} that does not extend to a 2-functor Cat_* -> {F,T} Again, this is just a pedantic way of saying what you're saying. I'm just trying to point out that I can fit it into my philosophy.

I tried to give a more general and precise 2-categorical definition on the categories list on January 3, 2010, but I am not sure I got it quite right.

I remember enjoying that post, but I'll need to reread it to remember what you said.

I think it was Mark Weber who also pointed out, around the same time, that one person's XXXX concept is another person's non-XXXX concept - in a different 2-category.

Very much so! And you've also noticed here that sometimes a property of objects in a fixed category arises from a property of pointed categories.... so that we can take either a 1-categorical or a 2-categorical approach to the XXXXness of this property.

So I don't think it is correct to identify the concept of XXXX with "having to talk about equality".

I agree. I take it as a *rule of thumb* that when somebody writes down a property of categories that involves equations between objects, they're running the risk that this property is not invariant under equivalence of categories. But I don't know the general theorems that make this rule of thumb precise. Can every property of categories that is invariant under equivalence be expressed in some language that doesn't include equations between objects? Or conversely? Or what precise conditions are needed to get theorems along these lines? Best, jb

Andre wrote:

I guess you have introduced the "evil" terminology because you wanted people to pay attention to the fact that certain constructions in category theory and higher category theory are not invariant under equivalences. If this is so, you have succeeded in your goal.

Good! But I never really thought of it as a piece of "terminology" that I "introduced". I've never used in any published paper, for example. I've always thought of it as a JOKE. In the future I'll try to avoid telling this joke in the presence of people who find it upsetting. This should be easy, because I'm not working on pure math anymore, except for a few projects that I'm trying to finish up. I'm working on environmental issues. The planet is headed for an ecological disaster within this century. We cannot prevent it, we can only minimize the damage. I hope some people on the category theory mailing list can help out here: http://www.azimuthproject.org/azimuth/show/Azimuth+Project If you aren't sure how to help, send me an email. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

[Note from moderator: This is a very long post and off topic. Further discussion of the subject must happen elsewhere. ] To the moderator - while this post is a bit off topic, I hope you can indulge me and post it, if only because it explains why I am no longer working on n-categories. The math is near the end. Dear Andre - You write:

You wrote that the situation about climate change is hopeless. I can understand your pessimism, but how do you really know?

First of all, climate change is already upon us. So far this year has been the hottest in recorded history, and we're seeing precisely the erratic precipitation patterns that we'd expect from global warming. Droughts and heat waves in Russia have caused that country - the 3rd largest grain exporting country - to ban exports of wheat. Floods displaced 2 million people in Pakistan, 2 million in Nigeria, and hundreds of thousands Uganda, Kenya and Sudan. We're also seeing unusual things like tornadoes in New York City, and coral reefs dying from overheated water in Indonesia. Etcetera. Second of all, once carbon dioxide is in the atmosphere, a large portion stays there for hundreds or thousands of years. So, to prevent the CO2 concentration from rising above 450 ppm, truly astounding actions would be required STARTING NOW. Let me quote Stewart Brand's summary of a talk by the engineer Saul Griffith:

What would it take to level off the carbon dioxide in the atmosphere at 450 parts per million (ppm)? That level supposedly would keep global warming just barely manageable at an increase of 2 degrees Celsius. There still would be massive loss of species, 100 million climate refugees, and other major stresses. The carbon dioxide level right now is 385 ppm, rising fast. Before industrialization it was 296 ppm. America's leading >climatologist, James Hansen, says we must lower the carbon dioxide level to 350 ppm if we want to keep the world we evolved in.

The world currently runs on about 16 terawatts (trillion watts) of energy, most of it burning fossil fuels. To level off at 450 ppm of carbon dioxide, we will have to reduce the fossil fuel burning to 3 >terawatts and produce all the rest with renewable energy, and we have to do it in 25 years or it's too late. Currently about half a terawatt comes from clean hydropower and one terawatt from clean >nuclear. That leaves 11.5 terawatts to generate from new clean sources.

That would mean the following. (Here I'm drawing on notes and extrapolations I've written up previously from discussion with Griffith):

"Two terawatts of photovoltaic would require installing 100 square meters of 15-percent-efficient solar cells every second, second after second, for the next 25 years. (That's about 1,200 square miles of solar cells a year, times 25 equals 30,000 square miles of photovoltaic cells.) Two terawatts of solar thermal? If it's 30 percent efficient all told, we'll need 50 square meters of highly >reflective mirrors every second. (Some 600 square miles a year, times 25.) Half a terawatt of biofuels? Something like one Olympic swimming pool of genetically engineered algae, installed every second. (About 15,250 square miles a year, times 25.) Two terawatts of wind? That's a 300-foot-diameter wind turbine every 5 minutes. (Install 105,000 turbines a year in good wind locations, times 25.) Two terawatts of geothermal? Build three 100-megawatt steam turbines every day — 1,095 a year, times 25. Three terawatts of new nuclear? That's a 3-reactor, 3-gigawatt plant every week — 52 a year, times 25".

[...]

Meanwhile for individuals, to stay at the world's energy budget at 16 terawatts, while many of the poorest in the world might raise their standard of living to 2,200 watts, everyone now above that level would have to drop down to it.

I believe actions of this scale will not happen in time, and thus it's hopeless to prevent a disaster. However, that does not mean we should give up! Even if a disaster of some sort is certain, there are different degrees of disaster, and it’s our responsibility to minimize the disaster.

You also wrote that not all human beings will die. Is this a source of optimism?

It is for me! I like people.

Who should be saved? Who can be abandoned?

Luckily for me, there are lots of useful things I can do without knowing the answer to this question.

You also wrote that we should consider using geo-engineering.

More precisely, I think we should study geo-engineering. People are already considering it, regardless of what I say. The pressure to use it will become intense as things get worse. Some forms, such as biochar, might be quite safe if managed properly. Others, which do not reduce the CO2 level, could be very dangerous. (People disagree on whether biochar counts as geo-engineering. It simply means turning agricultural waste into charcoal and burying it: http://www.azimuthproject.org/azimuth/show/Biochar)

How different is your position from Bjorn Lomborg's?

I haven't read Lomborg's new book yet, so I'm not sure. However, I get the feeling that he advocates geo-engineering as a cost-effective way to prevent global warming. My position would then be different. I believe we won't do anything significant about global warming until it's too late and there are massive social disruptions. Then there will be extreme pressure to try anything, including geo-engineering. So, I think it's important to study geo-engineering along with all other solutions. If people who don't like it refuse to study it, only people who like it will study it - so they'll be the ones that governments will consult. Now, about mathematics:

I hope you will not leave math. Everybody likes your web pages on maths and physics.

Thanks very much! I'm trying to figure out how to combine my interest in math with my desire to help save the planet. There are lots of options; the problem is finding one that achieves a significant effect. To do pure math, I just needed to follow the beauty. But this is different. There are easy things, and harder things. Everyone who teaches math can incorporate real-world examples in their teaching, and use them to educate people about the world we live in. For example: overfishing can cause fish populations to crash. This can be seen in a very simplified way using the equation dP/dt = kP - c or in more realistic ways using more complicated equations. A mathematician friend of mine was shocked when two colleagues of his, experts on differential equations, didn't know this. When I heard his story, I realized we should be talking about overfishing every time we teach kids how to solve separable differential equations. Another slightly more sophisticated example involves the role of feedback in global warming. I'm sure there are many more. I'll collect them and make them easy to find. More generally, everyone who teaches math or science can help students think clearly about real-world problems. This is urgent! Logic and statistics are vital. Mathematicians and scientists should also spend more of their research time on environmental issues. For students this should be easy: these issues will become ever more important, so working on them is a good road to a successful career - much easier than, say, category theory. Other people may feel they're too old to change directions. But in fact, I've found that the best way to become young is to try something new. Being tenured makes this very easy to do. The hardest part is figuring out which actions will have the optimal effect. For me, the first step is to quit pure math, work on environmental issues, and trying to convince large numbers of scientists to turn their attention in that direction. But the last part will only work if I can find a path forward that looks attractive. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

On Tue, Sep 28, 2010 at 3:18 AM, Thomas Streicher <streicher@mathematik.tu-darmstadt.de> wrote:

The notion of Grothendieck fibration is a property of functors and not an additional structure. However, the notion of fibration in a(n abstract) 2-category can be formulated only postulating a certain kind of structure which, however, is unique up to canonical isomorphism. But this amounts to defining Grothendieck fibrations in terms of cleavages (which certainly are all canonically isomorphic). But choosing cleavages amounts to accepting very strong choice principles which is maybe no real problem but at least aesthetically moderately pleasing.

I'm not sure what you mean here. The notion of fibration in a 2-category can be defined as a property if you like: a morphism E --> B in a 2-category K is a fibration if all the induced functors K(X,E) --> K(X,B) are fibrations and all commutative squares induced by morphisms X --> X' are morphisms of fibrations (preserve cartesian arrows). This is equivalent to giving some structure on E --> B, but that structure is unique up to unique isomorphism when it exists. (One can argue that both ordinary fibrations and fibrations in a 2-category are actually "property-like structures," or "properties that are not necessarily preserved by morphisms," since their forgetful functors are pseudomonic but not full. But that applies equally to both.) It is true that if one has a Grothendieck fibration between categories in the ordinary sense, then it only becomes an internal fibration in the naively defined 2-category Cat if we have the axiom of choice, since the latter amounts to saying that we can simultaneously choose cartesian liftings for any families of objects of E and morphisms of B we might want to pull them back along. (I don't think any "global choice" is necessary, since the definition doesn't require us to make such a choice simultaneously for every possible family--only that for any particular family, we -could- make such a choice.) However, in the absence of the axiom of choice, the naive definition of "functor" is not very well-behaved; it's better to use "anafunctors," or equivalently to invert the weak equivalences (fully-faithful and essentially surjective functors) in the naively defined 2-category Cat. In the resulting bicategory, I think any ordinary Grothendieck fibration will indeed be an internal fibration, without any need for choice. (Of course, "internal fibration" should probably now be interpreted in the looser sense of Street, as appropriate when working in a non-strict 2-category.) Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear Mike,

I'm not sure what you mean here. The notion of fibration in a 2-category can be defined as a property if you like: a morphism E --> B in a 2-category K is a fibration if all the induced functors K(X,E) --> K(X,B) are fibrations and all commutative squares induced by morphisms X --> X' are morphisms of fibrations (preserve cartesian arrows). This is equivalent to giving some structure on E --> B, but that structure is unique up to unique isomorphism when it exists.

The definition you give entails that a "generalised" fibration is actually a Grothendieck fibration (since Cat(1,E) is isomorphic to E). This way you don't get closure under precomposition by equivalences. I also don't see why cartesiannness of the functors induced by X --> X' should amount to a choice of structure (cartesiannness of a functor is a property and not additional structure). Moreover, this requirement is a property of Grothendieck fibrations which can be established when having strong choice available. I was rather alluding to the notion of fibration in 2-cats as can be found in part B of the Elephant where a fibration is defined as a 1-arrow together with additional structure. The definition you gave above (which is not more general) is the obvious thing to do in case K is not wellpointed enough (as Cat is). Moreover, your definition of fibration in a 2-category is based on Grothendieck fibrations and thus employs equality of 1-cells. Since 1-cells are objects of a category it should be "evil" to speak about their equality. Not that it were a problem to me... Thomas PS Your definition of fibration in a 2-cat looks much simpler than what I could find in the papers by Street and Johnstone. That's nothing to complain about but where is it from? It seems to me the appropriate one when generalising from Cat to more general 2-cats. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear Michael,

Does this clarify what I meant?

Yes, indeed that has been very helpful.

Perhaps "not well-behaved" was a poor choice of words since it implies a value judgement, but I think it is correct to say that in the absence of choice, category theory becomes very unfamiliar unless we replace functors with anafunctors. For instance, if we insist on using only functors, then a category with finite products does not necessarily become a monoidal category, as there is no "product" functor from A×A to A. Also, without choice the 2-category Cat using functors is not even a regular 2-category, let alone a 2-topos. Since the defining characteristics of Set include that it is a well-pointed 1-topos, it seems unlikely to me that one will be able to get much of anywhere with a version of Cat that is not a 2-topos.

Well, but then we can work with categories with a chosen structure, e.g. chosen products; that's what is recommended in the Elephant and it looks to me as close to practice; it's very unlikely to have a nonconstructive proof of existence of products. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

On Sun, Sep 26, 2010 at 10:36 PM, John Baez <baez@math.ucr.edu> wrote:

Can every property of categories that is invariant under equivalence be expressed in some language that doesn't include equations between objects? Or conversely? Or what precise conditions are needed to get theorems along these lines?

The converse is very easy, and it's something that I and others have frequently mentioned in these discussions: if we write category theory in dependent type theory with arrows dependent on their source and target and no equality predicate on objects, then all formulas and constructions in this language are easily proven to be invariant under equivalence and isomorphism. The forward direction is trickier, but essentially the answer is yes: I believe theorems along these lines can be found in: 1) Peter Freyd, "Properties invariant within equivalence types of categories", 1976 2) Georges Blanc, "Équivalence naturelle et formules logiques en théorie des catégories", 1979 3) Michael Makkai, "First-order logic with dependent sorts, with applications to category theory," http://www.math.mcgill.ca/makkai/folds/foldsinpdf/FOLDS.pdf (and perhaps others that I'm unaware of). Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]