Robert Dawson wrote: Surely we should start with the set of (-1)-categories? Actually we should start at least a little bit before that, with (-2)-categories. Let me explain.... Once upon a time I showed what I called the "periodic table" to Chris Isham, a physicist who works on quantum gravity. It starts like this: k-tuply monoidal n-categories n = 0 n = 1 n = 2 k = 0 sets categories 2-categories k = 1 monoids monoidal monoidal categories 2-categories k = 2 commutative braided braided monoids monoidal monoidal categories 2-categories k = 3 " " symmetric sylleptic monoidal monoidal categories 2-categories k = 4 " " " " symmetric monoidal 2-categories k = 5 " " " " " " and it extends infinitely in both directions. The basic idea is that a "k-tuply monoidal n-category" is a weak (n+k)-category with only one j-morphism for j < k. There's a lot of evidence from homotopy theory and elsewhere that each column of this table must "stabilize" when k reaches n + 2. Of course, this observation needs to be made more precise before it can become a theorem, or even a conjecture, so James Dolan and I called it the "stabilization hypothesis". Carlos Simpson found one way to make it precise and prove it: On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani's weak n-categories, math.CT/9810058. but everyone who has a definition should take a whack at it! Anyway, when I showed this pattern to Chris Isham, I was very proud of it, so I was annoyed when he instantly found fault with it. He said: what about the (-2)-categories, (-1)-categories, and monoidal (-1)-categories? You've drawn this big triangle, but it's missing the upper left-hand corner! I told him I'd have to think about that. After that, I kept trying to guess what (-2)-categories, (-1)-categories and monoidal (-1)-categories should be. Clearly a monoidal (-1)-category should be a set with just one element. But what about the other two? Later, when explaining the concepts of "property", "structure" and "stuff" to Toby Bartels, James Dolan figured out what (-1)-categories are. Toby then helped him figure out what (-2)-categories were, too. Actually, I should be a bit careful here: they really figured out what (-1)-groupoids and (-2)-groupoids are. However, I believe that these coincide with (-1)-categories and (-2)-categories. I have a lot to do today, and this article is already getting too long, so I'll stop here and leave these as a puzzle for all of you. It's sort of fun! I should however mention this: after James and I came to understand this stuff, someone pointed out an error in our definition of n-categories, and we were very perturbed until we realized it could be fixed by changing just one number in our existing definition - which would have been obvious from the start if we'd understood about (-1)-categories. John Baez
Frank Atanassow emailed me saying that he's "dying to know" the answer to my puzzle about (-1)-categories and (-2)-categories. That's good! So far the silence has been deafening, and I can't tell if it indicates bewilderment, lack of interest, or disgust with an imprecisely posed question. I guess I'll give away the answer. As I said, the trick is to figure out what (-1)-groupoids and (-2)-groupoids are, and then cross our fingers and hope that the answer is the same for (-1)-categories and (-2)-categories. We start with the basic principle that people use when trying to cook up definitions of "weak n-groupoid": weak n-groupoids should be the same as nice topological spaces of homotopy dimension n. Here "nice" means something like a CW complex, and "the same" must be taken in a suitably n-categorical/homotopical spirit. But what does "homotopy dimension n" mean? Well, the usual definition is that a space has homotopy dimension n if all its homotopy groups above the nth are trivial. But if we carefully unpack this, we'll see it's an interesting condition even when n is -1 or -2. Here goes: A topological space X has homotopy dimension n if given k > n, any continuous map from the k-sphere to X extends to a continuous map from the (k+1)-disc to X. So: X has homotopy dimension 0 if its arc-components have vanishing homotopy groups. If X is nice, this means it's a disjoint union of contractible spaces, so it's the same as a *set*. That's good: 0-groupoids should be sets. (See? I'm using "the same" in a suitably homotopical spirit! A disjoint union of nice contractible spaces is homotopy equivalent to a set with the discrete topology.) Next: X has homotopy dimension -1 if it has homotopy dimension 0 and also any continuous map from the 0-sphere to X extends to a continuous map from the 1-disc to X. In other words, X is either empty, or it consists of a single arc-component with vanishing homotopy groups. If X is nice, this means it's the same as a *set with cardinality 0 or 1*. So we declare that a (-1)-groupoid is a set with cardinality 0 or 1. Next: X has homotopy dimension -2 if it has homotopy dimension -1 and also any continuous map from the (-1)-sphere to X extends to a continuous map from the 0-disc to X. The 0-disc is a single point, and its boundary the (-1)-sphere is the empty set. So X must be a *nonempty* space with homotopy dimension -1. So X consists of a single arc-component with vanishing homotopy groups. If X is nice, this means it's the same as a *set with cardinality 1*. So we declare that a (-2)-groupoid is a set with cardinality 1. Crossing our fingers, we therefore guess: A (-1)-category is a set with cardinality 0 or 1. A (-2)-category is a set with cardinality 1. This may seem silly, but it's not! There is a nice relation between all this business and the notion of "n-stuff". But I'm getting worn out, so instead of explaining that, I'll just quote some articles that James Dolan and Toby Bartels wrote on sci.physics.research when they were first figuring out this "n-stuff" stuff. Frank also asked if the error in our definition of n-categories appears in HDA3. Yes! We will correct it in HDA5, which will be about Feynman diagrams and "n-stuff". As I said, it's easy to fix: just one number is wrong. John Baez ......................................................................... From: "james dolan" <jdolan@math.ucr.edu> Subject: Re: Just Categories now Date: 16 Nov 1998 00:00:00 GMT Message-ID: <72pusp$3o5$1@pravda.ucr.edu> Organization: Department of Mathematics, University of California, Riverside Newsgroups: sci.physics.research X-Newsposter: Pnews 4.0-test50 (13 Dec 96) toby bartels writes: -john baez wrote: ->I will leave it to James Dolan to explain the technical ->distinction between "extra properties", "extra structure", ->and "extra stuff" - there is a nice category-theoretic way ->of making this precise. - -Ooh, let me guess! - -Given a functor U: C -> D, interpret U as a forgetful functor. -Then C is D with extra *structure* if U is surjective on the -objects and, given a pair of objects, injective on the -morphisms between them; and C is D with extra *properties* if -U is injective on the morphisms (meaning injective on the -objects and on the morphisms between a given pair); Otherwise, -I guess C is just D with extra *stuff* if, given a pair of -objects, U is injective on the morphisms between them. here's my classification: given groupoids c,d and a functor u:c->d, the objects of c can be thought of via the forgetful functor u as objects of d with an extra _property_ iff u is full and faithful, as objects of d with extra _structure_ iff u is faithful, and as objects of d with extra _stuff_ regardless. (some category-theoretic jargon: 1. a "groupoid" is a category where all the morphisms are invertible. it may very well be interesting to generalize the subject matter of this discussion to the case where c and d are not necessarily groupoids, but to keep things simple for now i won't do that in this post. 2. a functor u:c->d is "full" iff for any pair c1,c2 of objects in c, the map from the hom-set hom(c1,c2) to the hom-set hom(u(c1),u(c2)) given by u is surjective. 3. a functor u:c->d is "faithful" iff for any pair c1,c2 of objects in c, the map from the hom-set hom(c1,c2) to the hom-set hom(u(c1),u(c2)) given by u is injective.) one reason i don't (as i think toby was suggesting) require the forgetful functor u to be surjective on (isomorphism classes of) objects in order for the objects of c to qualify as objects of d with extra "structure" is as follows: consider for example the case where c is the category of rings, d is the category of groups, and u is the functor assigning to each ring its underlying additive group. clearly the objects of c are objects of d with extra "structure" in the intuitive sense that i'm trying to capture; we can say that "a ring is defined to be a group (henceforward referred to as "the underlying additive group of the ring") equipped with an extra multiplication operation on it satisfying certain equational laws...", and although this may sound like the equational laws only constrain the ring structure on the additive group, they in fact also implicitly constrain the additive group itself: it's easy to show that even if you don't explicitly require the additive group of a ring to be commutative, it will automatically be forced to be commutative by the other clauses in the usual definition of "ring" (left and right distributivity plus multiplicative unit laws, in combination with the group axioms for addition, should do it, i think). thus this example is supposed to demonstrate the fact that as soon as you generally allow yourself to invent a new "type of structure that an object of d can be equipped with" by starting with an arbitrary existing such type of structure and constraining the structures to satisfy some property, it's awkward to prevent an arbitrary "property that can be predicated of an object of d" from being considered as a "type of structure that an object of d can be equipped with" by being looked at as a constraint on [the degenerate "type of structure that an object of d can be equipped with" given by "no extra structure at all"]. thus you should probably broaden your concept of "type of structure that an object of d can be equipped with" to include "property that can be predicated of an object of d" as a special case. for similar reasons you should probably broaden your concept of "type of stuff that an object of d can be equipped with" to include "type of structure that an object of d can be equipped with" as a special case, if it isn't that broad already. -For example, the forgetful functor Groups -> Sets shows tha -groups are sets with extra structure, while the forgetful -functor Abelian Groups -> Groups shows that Abelian groups are -groups with extra properties. i agree with those examples (at least if i interpret them in accordance with my self-imposed restriction to consider only the case where all of the morphisms in c and d are invertible). -Or you can turn around and use -the free functor Sets -> Groups and say that sets are groups -with extra properties (to wit, the property of being free). i disagree with that example, for reasons that hopefully are clear from my explanations above. thus i would _not_ say that a set is a group with the extra property of being free; rather i'd say that a set is a group with the extra _structure_ of being equipped with a favored "basis" of mutually free mutual generators. -OTOH, the Abelianization functor Groups -> Abelian groups is -surjective on the objects (and on the morphisms for that -matter), but groups are not Abelian groups with extra -structure, because the functor isn't injective on the -morphisms between a given pair. i think i agree with this, but it sounds like you're using my rules here rather than the rules i thought you tried to spell out in your post. another example of an object equipped with extra "stuff" would be a set equipped with another _set_; that is, take c to be the category of ordered pairs of sets, d to be the category of sets, and u to be the "projection" functor assigning to an ordered pair (x,y) of sets its first coordinate x. i hope this example helps to show why i consider the terminology "stuff" reasonably descriptive of the intuition involved. another example (maybe or maybe not causing some additional (?) number of people to see this post as having some relevance to physics) of an object equipped with extra "stuff" rather than merely with extra "structure" is a manifold equipped with an unfortunately so-called "spin structure". the point is that if we define the concept of "morphism between spin manifolds" in what seems to me to be the most advantageous way, then taking c to be the category of spin manifolds, d the category of manifolds, and u the hopefully obvious forgetful functor assigning to a spin manifold its underlying ordinary manifold, u is not faithful. thus a "spin structure" is not merely "structure"; it's "stuff". so what is this extra "stuff"?? you can think of it as "spin frames" if you want to. (a "spin frame" is what a spin manifold has two of where an ordinary manifold has only one.) or you can think of it as "spinors"; morphisms between spin manifolds have an extra discrete degree of freedom to flip the sign of spinors even after their action on ordinary manifold points has been completely nailed down. a deeper understanding of how the classification offered here arises involves the relationship between groupoid theory and homotopy theory, as follows: for any integer n greater than or equal to -1, a space x is defined to be of "homotopy dimension n" iff for any integer j strictly greater than n, every continuous map from the j-dimensional sphere s^j to x is homotopic to a constant map. using this terminology, every space of homotopy dimension n is also of homotopy dimension m for any integer m greater than n. a crucial fact is that the world of spaces of homotopy dimension 1 is secretly isomorphic in a very strong way to the world of groupoids; there's an amazingly perfect "dictionary" linking concepts from the world of spaces of homotopy dimension 1 to their secret equivalents in the world of groupoids. the groupoid corresponding to a space x of homotopy dimension 1 is called the "fundamental groupoid" of x, and the space of homotopy dimension 1 corresponding to a groupoid g is called the "classifying space" of g. inside the world of spaces of homotopy dimension 1 are of course the sub-world of spaces of homotopy dimension 0, and the sub-sub-world of spaces of homotopy dimension -1. the secret equivalent inside the world of groupoids of the sub-world of spaces of homotopy dimension 0 is the sub-world of so-called "discrete groupoids", and the secret equivalent of the sub-sub-world of spaces of homotopy dimension -1 is the sub-sub-world of just those special discrete groupoids which have either just one object and one morphism, or no objects and morphisms at all. the discrete groupoids are also known as "sets", or (exploiting the [homotopy dimension 1]/groupoids dictionary) "groupoids of homotopy dimension 0". the special discrete groupoids corresponding to the spaces of homotopy dimension -1 are called "truth values", or "groupoids of homotopy dimension -1". the groupoid with just one object and one morphism is called "true" (aka "the terminal groupoid" aka "yes" aka "in") while the empty groupoid is called "false" (aka "the initial groupoid" aka "no" aka "out"). given a pair c,d of groupoids and a functor u:c->d and an object d1 in d, we can construct a new groupoid called "the homotopy fiber of u over d1". roughly, the homotopy fiber of u over d1 is the groupoid of "objects of c equipped with designated isomorphisms from their images under u to d1"; the morphisms in the homotopy fiber are required to preserve the designated isomorphisms. as you might guess from the name "homotopy fiber", the groupoid-theoretic concept of "homotopy fiber" has a very direct equivalent in homotopy theory. we can now re-state the definitions of "property", "structure", and "stuff" in terms of homotopy dimension of homotopy fibers, as follows: given groupoids c,d and a functor u:c->d, the objects of c can be thought of via the forgetful functor u as objects of d with an extra _property_ iff the homotopy fibers of u are all of homotopy dimension -1, as objects of d with extra _structure_ iff the homotopy fibers of u are all of homotopy dimension 0, and, and as objects of d with extra _stuff_ iff the homotopy fibers of u are all of homotopy dimension 1. hopefully this makes the intuition behind the concepts a bit clearer. a "property" is something which, if you possess it at all, then you have no choice in _how_ to possess it, you just do. a "structure" is something which if you possess it then possessing it involves picking a particular structure in a way analogous to picking an element of a set. "stuff" is something which if you possess it then possessing it amounts to picking some particular stuff in a way analogous to picking an object of a groupoid. of course as with most concepts of groupoid theory, the concepts discussed here should be generalized to the case of "higher-dimensional groupoid theory" which corresponds to the homotopy theory of spaces with arbitrary homotopy dimension in the same way that ordinary groupoid theory corresponds to the homotopy theory of spaces of homotopy dimension 1. thus the stunted progression property, structure, stuff becomes a genuine open-ended progression: property, structure, stuff, eka-stuff, eka-eka-stuff, ... . thus given arbitrary spaces c,d and a continuous map u:c->d, we should say that "the objects of the fundamental infinity-groupoid of c can be thought of via the forgetful infinity-functor induced by u as objects of the fundamental infinity-groupoid of d equipped with extra eka^n-stuff" iff all of the homotopy fibers of u are of homotopy-dimension n+1. From: james dolan <jdolan@math.ucr.edu> Subject: Re: Just Categories now Date: 05 Jan 1999 00:00:00 GMT Message-ID: <76rulr$h7d$1@pravda.ucr.edu> References: <726r22$qr$1@news-1.news.gte.net> <3647A029.6FB4@easyon.com> <72agsd$ook$1@pravda.ucr.edu> <72b3pd$709@gap.cco.caltech.edu> Organization: Department of Mathematics, University of California, Riverside Newsgroups: sci.physics.research X-Newsposter: Pnews 4.0-test50 (13 Dec 96) toby bartels writes: -james dolan <jdolan@math.ucr.edu> wrote: - ->given groupoids c,d and a functor u:c->d, the objects of c can ->be thought of via the forgetful functor u as objects of d with ->an extra _property_ iff u is full and faithful, as objects of d ->with extra _structure_ iff u is faithful, and as objects of d ->with extra _stuff_ regardless. - ->A "groupoid" is a category where all the morphisms are ->invertible. it may very well be interesting to generalize the ->subject matter of this discussion to the case where c and d are ->not necessarily groupoids, but to keep things simple for now i ->won't do that in this post. - -You seem to agree with John Baez's classification, -but he doesn't feel the need to limit to groupoids; -perhaps a word on how you think that complicates things? it complicates things in the obvious way: a single concept in groupoid theory (for example the concept of "faithful functor between groupoids") may bifurcate into non-equivalent concepts in category theory (for example the concepts of "faithful functor between categories" and "functor between categories which is faithful on isomorphisms"); the necessity of worrying about the distinctions between such non-equivalent concepts is eliminated by discussing only the groupoid case. but presumably you're also asking why it is that in this tradeoff between simplicity and generality i chose simplicity, so i'll try to say something about that too. -Or is it just that groupoids are needed for the deep homotopy connection? that's part of my motivation by now, but i think my original motivation had less to do with the "dictionary" that relates groupoid theory to a special part of homotopy theory than with a different but in its own way equally powerful "dictionary" relating groupoid theory to a special kind of predicate logic. in the world of predicate logic there's an obvious sense in which adding extra "properties" to the models of a theory means adding new axioms to the theory, adding extra "structure" to the models means adding new predicate symbols (possibly supplemented by new axioms) to the theory, and adding extra "stuff" to the models means adding new "types" (possibly supplemented by new predicate symbols and axioms) to the theory. this property/structure/stuff distinction in predicate logic matches perfectly the property/structure/stuff distinction in groupoid theory if groupoids are interpreted as a certain sort of logical theories in a certain way. the more i think about this the more it seems that there should be some nice big picture that links together the predicate logic aspects of the situation with the homotopy theory aspects of the situation, but if so it's a bit too big for me to fully grasp yet so i won't try to say any more about it at the moment. i will say though that if someone would show how to generalize the correspondence between groupoids and logical theories of a certain sort to a correspondence between categories and logical theories of some more general sort, then i might be willing to agree that there is some obvious way of extending the property/structure/stuff classification of groupoid theory to apply to category theory as well. i have a vague suspicion that in fact this has already been done and that the logical theories corresponding to categories differ from the logical theories corresponding to groupoids more or less precisely in being "intuitionistic" rather than "classical", but i'm not at all clear on the details of how this works if it's even correct. ->a deeper understanding of how the classification offered here ->arises involves the relationship between groupoid theory and ->homotopy theory, as follows: - ->for any integer n greater than or equal to -1, a space x is ->defined to be of "homotopy dimension n" iff for any integer ->j strictly greater than n, every continuous map from the ->j-dimensional sphere s^j to x is homotopic to a constant map. - -You can even generalize this to n = -2, noting that s^{-1} is the empty set. yes, very much so, though i don't think i thought about this until afterwards. -Of course, no map from s^{-1} to any space can ever be homotopic to a -constant, yet there is always some map from s^{-1} to any space (the -empty map), so no space has homotopy dimension -2, which must be why -nobody talks about it. hmm. first of all, i think i should revise my definition of homotopy dimension to eliminate the idea of "homotopic to a constant map", because people seem to disagree on the meaning of "constant map" when the domain is empty. (some people think that constantness of maps is the property of factoring through the one-point set, others think it's the _structure_ of being equipped with a specific factorization through the one-point set, and toby apparently thinks it's the property of having the one-point set as image.) here's the revised version: for any integer n greater than or equal to -2, a space x is defined to be of "homotopy dimension n" iff for every continuous map m from the [n+1]-dimensional sphere s^[n+1] to x, the space of extensions of m to the [n+2]-dimensional disk d^[n+2] is contractible. (here the sphere s^[-1] is defined to be empty, the disk d^[j+1] is defined to be the mapping cylinder of the map s^j->1, and the sphere s^[j+1] is defined to be the pushout d^[j+1] +_[s^j] d^[j+1]. "contractible" means equivalent to the 1-point space.) hopefully with this revised definition it's still true that being of homotopy dimension n implies being of homotopy dimension n+1. the spaces of homotopy dimension -2 are the contractible spaces, and the spaces of homotopy dimension n for higher n are hopefully just as before. the spaces of homotopy dimension n taken from a sufficiently "convenient" category s of spaces form a cartesian closed category s_n, and the spaces of homotopy dimension n+1 in s are the spaces equivalent to classifying spaces of groupoids enriched over s_n. the class of continuous maps with all homotopy fibers of homotopy dimension -2 is the class of all homotopy equivalences. in the world of groupoids this corresponds to the class of all functors that are "invertible up to natural isomorphism". thus eka^[-3]-stuff is _vacuous_ properties; that is, given groupoids c,d and a functor f:c->d with f invertible up to natural isomorphism, objects of c can be thought of as objects of d equipped with a _vacuous_ property. (as throughout this discussion, we are interested in everything only "up to natural isomorphism" or "up to homotopy" in groupoid theory or in homotopy theory theory respectively.) notice that the class of all maps with all homotopy fibers of homotopy dimension n is closed under composition because the homotopy fibers of a composite map fg are themselves the total spaces of fibrations with base spaces which are homotopy fibers of g and fibers which are homotopy fibers of f, and because the class of spaces of homotopy dimension n is closed under the process of forming a new space as the total space of a fibration with its base and all its fibers in the class. finally, if there's anything such as "spaces of homotopy dimension -3", i don't want to hear about it. From: jdolan@galaxy.ucr.edu (james dolan) Subject: Re: Just Categories now Date: 15 Jan 1999 00:00:00 GMT Message-ID: <77k9fr$va6$1@pravda.ucr.edu> References: <3647A029.6FB4@easyon.com> <72b3pd$709@gap.cco.caltech.edu> <76rulr$h7d$1@pravda.ucr.edu> <77h2qd$ohj@gap.cco.caltech.edu> Organization: none Newsgroups: sci.physics.research toby bartels wrote: -james dolan <jdolan@math.ucr.edu> wrote: - ->Toby Bartels <toby@ugcs.caltech.edu> wrote: - ->>Or is it just that groupoids are needed for the deep homotopy ->>connection? - ->that's part of my motivation by now, but i think my original ->motivation had less to do with the "dictionary" that relates groupoid ->theory to a special part of homotopy theory than with a different but ->in its own way equally powerful "dictionary" relating groupoid theory ->to a special kind of predicate logic. in the world of predicate logic ->there's an obvious sense in which adding extra "properties" to the ->models of a theory means adding new axioms to the theory, adding extra ->"structure" to the models means adding new predicate symbols (possibly ->supplemented by new axioms) to the theory, and adding extra "stuff" to ->the models means adding new "types" (possibly supplemented by new ->predicate symbols and axioms) to the theory. this ->property/structure/stuff distinction in predicate logic matches ->perfectly the property/structure/stuff distinction in groupoid theory ->if groupoids are interpreted as a certain sort of logical theories in ->a certain way. - -OK, I tried to think about this, but I don't really know where to -start. Give me a clue: what famous groupoid corresponds to what I've -been taught to regard as the basic predicate calculus: ordinary logic -with forall, forsome, and equality? the correspondence is between individual groupoids and individual _theories_ of a particular form of predicate logic. the particular form of predicate logic involved is pretty much just "the basic" form, with the allowed syntactic constructions including: 1. the usual finitary boolean connectives obeying the usual finitary boolean equational laws 2. the universal quantifier "for all" (and therefore also the existential quantifier "for some" via the equivalence between "for some x, p(x)" and "not (for all x, (not p(x)))") 3. the built-in binary predicate "equality" with it's usual built-in reflexivity, symmetry, transitivity, and substitutability properties plus one more construction going beyond what's ordinarily considered "the basic": 4. the restriction in #1 above against the _infinitary_ boolean connectives (such as n-fold conjunction for an arbitrary infinite cardinality n) is lifted. given a theory t expressed in this kind of logic, we obtain the groupoid of models of t. when all the i's are dotted and the t's crossed in the right way, this process of passing from the theory t to the groupoid of models of t becomes a "bi-equivalence from the bi-category of theories to the bi-category of groupoids". for example, let t be the theory presented by giving no predicate symbols, plus the one axiom "there are exactly seven things". (of course this axiom can be expressed using the allowed syntatic constructions.) the groupoid of models of t is the groupoid of seven-element sets. this groupoid has just one isomorphism class because the theory t is "categorical" (in a sense of the word "categorical" having not much relationship to category theory!). that's not a complete exposition of the situation, rather just a clue of the sort i hope you wanted. i will mention further though that to develop the full correspondence between theories and groupoids, the theories should be allowed to be "multi-typed". if only "single-typed" theories are considered then the most straightforward correspondence is not with "abstract" groupoids but rather with "concrete" groupoids, a "concrete groupoid" being a groupoid equipped with a faithful functor to the groupoid of sets. it might be a good idea to develop the correspondence between single-typed theories and concrete groupoids before developing the full correspondence between multi-typed theories and abstract groupoids. one of the basic lemmas you should try to understand is as follows: let x be a set. let c be the collection of all pairs (s,p) with s a (possibly infinite) set and p an s-ary relation on x. let d be the hyper-collection of all sub-collections of c that are closed under all of the operations on relations alluded to in #1-#4 above. then d is in canonical bijection with the set of subgroups of the group of permutations of x (taking "permutation" to mean "auto-bijection"). (in the above lemma, among the operations that should count as "alluded to" is the operation of replacing an s-ary relation by the obvious corresponding t-ary relation given a bijection from s to t, even though this operation was perhaps _not_ very explicitly alluded to.)
Steve Vickers asked by email whether in constructive mathematics a (-1)-category should be allowed to be an arbitrary "subsingleton", i.e. a set with at most one element. I think that yes, these should be allowed - but maybe other things too! Indeed, a space has homotopy dimension -1 iff 1) given 2 points in the space there exists a path joining them, and 2) given 2 paths joining them, there exists a path of paths joining *them* and 3) given 2 paths of paths joining *them*, there exists a path of paths of paths joining THEM, and so on ad infinitum. As a nonconstructivist, I would say that either the space is empty in which case clause 1) is vacuous, or it's nonempty in which case we go on and read the resulting infinite list of clauses. But a constructive mathematician would have to proceed differently here, not being allowed to use the excluded middle. Do the remaining clauses provide any extra challenges for the constructivist? Then there's the bit where, having gotten a space that's either empty or contains one arc-component with vanishing homotopy groups, I conclude that if it's *nice* (e.g. a CW complex) it's homotopy equivalent to a space that's either empty or one point. I don't know how this reasoning (which uses the Whitehead theorem) gets affected by constructivism. Steve's idea sounds interesting, for this reason. If you plow through the detailed exchange between James Dolan and Toby Bartels, you'll see that (-1)-categories secretly represent TRUTH VALUES. In any approach to math where "truth values" are more interesting than merely 0 or 1, (-1)-categories will be correspondingly more interesting than merely sets with 0 or 1 elements. I guess this is familiar from topos theory. But I don't know if there's an extra twist due to all the "higher-dimensional" stuff going on in my reasoning above. Are truth values for an omega-categorical constructivist still more interesting than for an ordinary constructivist? The ordinary constructivist may not know whether two things are equal. The omega-categorical constructivist may not know whether all morphisms between two things are related by a 2-morphism, or whether all such 2-morphisms are related a 3-morphism, and so on....
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baez＠newmath.UCR.EDU