Subject: categories: (-1)-categories and (-2)-categories

From: [email-masked]
Date: Thu, 14 Dec 2000 13:07:32 -0800 (PST)

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" <[email-masked]>
Subject: Re: Just Categories now
Date: 16 Nov 1998 00:00:00 GMT
Message-ID: <[email-masked]>
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 <[email-masked]>
Subject: Re: Just Categories now
Date: 05 Jan 1999 00:00:00 GMT
Message-ID: <[email-masked]>
References: <[email-masked]> <[email-masked]> <[email-masked]> <[email-masked]>
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 <[email-masked]> 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: [email-masked] (james dolan)
Subject: Re: Just Categories now
Date: 15 Jan 1999 00:00:00 GMT
Message-ID: <[email-masked]>
References: <[email-masked]> <[email-masked]> <[email-masked]> <[email-masked]>
Organization: none
Newsgroups: sci.physics.research

toby bartels wrote:

-james dolan <[email-masked]> wrote:
-
->Toby Bartels <[email-masked]> 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.)