Timeline for History of classifying spaces
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 12, 2011 at 5:48 | comment | added | Andy Putman | @Harry : I have no idea about when Grothendieck started thinking about these things. It's all speculation. In any case, Thomason came onto the scene in the late '70's, long after Quillen's work. | |
| Jan 12, 2011 at 4:50 | comment | added | Harry Gindi | @Andy: The classifying space of a category is extremely important in Grothendieck's topos-theoretic approach to homotopy theory. I remember reading that he was already thinking about this kind of stuff in the mid-1960s (presumably since topoi completely subsume classical notions of "space" vis-a-vis geometry, so a homotopy theory of topoi would generalize homotopy theory to these new kinds of spaces). I believe that this program was the one that inspired Thomason's work (both indirectly and directly (as Thomason proved a number of relevant conjectures of Grothendieck). | |
| Jan 12, 2011 at 3:45 | comment | added | Andy Putman | I don't think Milnor ever considered anything more general. Really, I don't know many serious applications of classifying spaces of categories before Quillen (and the ones that come to mind just involve classifying spaces of monoids). | |
| Jan 12, 2011 at 1:25 | comment | added | Dr Shello | Milnor's "infinite join" construction for topological groups was published in 2 papers from 1956, but I don't think he ever considered anything more general. | |
| Jan 12, 2011 at 0:30 | comment | added | Harry Gindi | @Andy: Good point! If I had to guess, I would say that the notion must be have been known to Kan, who is responsible for the observations that a.) CSS complexes are presheaves of sets on $\Delta$ and b.) the definition of a Kan extension. I don't know him, but if somebody could e-mail him and ask him, he would know the answer. | |
| Jan 11, 2011 at 23:15 | comment | added | Andy Putman | @Harry : Classifying spaces for topological groups were first constructed by Milnor using other methods. | |
| Jan 11, 2011 at 22:36 | comment | added | Harry Gindi | @Dr. Shello: The Grothendieck reference is most probably to the use of the Cech nerve in Grothendieck's definition of Cech cohomology. | |
| Jan 11, 2011 at 22:33 | comment | added | Harry Gindi | the "simplicial nerve" functor that originates in the work of Kan and Dwyer in the late 1970s and early 1980s. Segal's construction bypasses, in particular, the nasty definition of the "simplicial nerve", which arises naturally as the result of applying the bar construction with respect to the free category comonad. | |
| Jan 11, 2011 at 22:30 | comment | added | Harry Gindi | I can see why Segal would attribute the original ideas to other people, since I'm reasonably certain that by that time, the notion was already folklore. I don't think that Segal's major contribution in that paper is the definition of the nerve/classifying space, but his constructions of classifying spaces of, for instance, topological groups. The "right definitions" of such objects are not at all clear with the mathematics of 1968. Indeed, the modern way of doing it is to take the singular complex, which gives a simplicial group, considering it as a simplicial category, then applying... | |
| Jan 11, 2011 at 22:22 | comment | added | Harry Gindi | Dan Kan's defintion of the subdivision functor in his paper "On CSS complexes" from 1957 does not take the natural step of extending the subdivision functor from simplices to simplicial sets by composing with the nerve, so I would say that is a pretty conclusive lower bound. However, the key idea behind the nerve is definitely there lurking in the shadows in that paper, as well as Kan's other early work on simplicial homotopy theory (all of the work is done when we realize that "CSS complexes" defined by the simplicial identities are exactly presheaves on $\Delta$. | |
| Jan 10, 2011 at 22:05 | comment | added | Dr Shello | What were the early constructions/theorems of Grothendieck's that used his definition of nerve? | |
| Jan 10, 2011 at 21:15 | comment | added | David Roberts♦ | Certainly Grothendieck seems to have been the one who defined the nerve of a category, and if people at that time were thinking of simplicial sets as spaces... | |
| Jan 10, 2011 at 20:46 | history | answered | Andy Putman | CC BY-SA 2.5 |