james dolan <jdo...@math.ucr.edu> wrote:
>Toby Bartels <t...@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.

>>james dolan <jdo...@math.ucr.edu> wrote:
>>>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.
>>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.)

>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.

>finally, if there's anything such as "spaces of homotopy dimension
>-3", i don't want to hear about it.

>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].