-
->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.