Holmstrom Space

There are of course many different notions of “space” in mathematics. Here is a survey of Connes treating many of them.

http://www.ncatlab.org/nlab/show/space

http://golem.ph.utexas.edu/category/2010/10/cohesive_toposes.html

http://golem.ph.utexas.edu/category/2010/11/structures_in_a_cohesive_topos.html

nLab on space and quantity

http://ncatlab.org/nlab/show/infinity-space

http://ncatlab.org/nlab/show/%28infinity%2C1%29-quantity

http://ncatlab.org/nlab/show/structured+%28infinity%2C1%29-topos

For some approaches to generalized spaces in differential geometry, see the preface to Vassiliou: Geometry of Principal Sheaves, in Homol alg folder.

See n-category cafe for links to stuff about “generalized smooth spaces”.

Compactly generated spaces: See Borceaux, vol 2 chapter 7.

Something by Grothendieck on categories as spaces: http://mathoverflow.net/questions/76505/in-which-situations-can-one-see-that-topological-spaces-are-ill-behaved-from-the

nLab page on Space