Reference for higher categorical analogue of algebraic cycle? Are there higher categorical analogues of algebraic cycles?
What are some references?
This question arise in an attempt to generalize algebraic cycles towards higher dimensional algebra. Has there been some progress here?
 A: I think I understand what you want. A first good environment for your "generalize algebraic cycles" should be to consider Stacks coming from algebraic geometry (usually Deligne-Mumford Stacks, or Artin Stacks). A stack can be though like a "family of algebraic varieties" and generalize schemes, etc. So DM-stacks for example, are a kind of 1-categorification of scheme. If in any schemes you have a notion of "algebraic cycles", you should be able to find such notion for DM-stacks, that we can call a "1-algebraic cycle". If you are able to understand carefully what are DM-stacks, and find how to define its "1-algebraic cycle", then you should be on the good way to generalize it in high dimension. In higher dimension many candidates of \infty-stacks are proposed, usually they are fibrant objects in some appropriate Quillen model structure (See the beautiful work of Joyal-Tierney (title is almost : folk Quillen model structure on Cat), where they build a canonical Quillen structure on the category of internal categories in a Grothendieck topos, and have recover a notion a stacks, similar to DM-stacks, as fibrants objects in their Quillen model structure. Many generalizations have been proposed after their work. Probably these notions of \infty stacks where used in algebraic geometry by Jacob Lurie ? Anyway such \infty-stacks should give the good environment for a notion of "higher algebraic cycle", and in fact, should be the good environment for all "higher notions" of classical notions we can found in AG.
A: Daniel, sorry, its just to give some intuitions (I guess coming from orbifolds, which depending on the geometry involved, could be differential stacks, or algebraic stacks, etc.). However someone ask the question : "...Let X be a stack of n-groupoids on the site of affine schemes over a fixed base, with the etale topology. If n=1 then for X to be Deligne-Mumford, aside from having an etale atlas from an algebraic space, one must also impose certain separation conditions on its diagonal. In DGA-V, Lurie remarks that what he defines as higher Deligne-Mumford stacks lacks separation axioms, but that they may be added in by hand later. My question is, what separation axioms should be added? Here, I do not mean "separated" or "quasicompact". What I mean to ask is on which morphisms do I put the appropriate separation conditions? It seems to be that simply imposing them on the diagonal may be too naive, but perhaps I am wrong, hence this question.."
Which means that some people already works on higher DM-stacks !! I believe you can build these higher DM-stacks by using my approach of higher groupoids (Published in TAC recently). I do believe Higher dimensional categories could build higher dimensional Algebraic Geometry ! So generalization of such higher algebraic cycles should be possible. Perhaps see the work of Bertrand Toen here : http://arxiv.org/abs/math/0604504 He is a great specialist in AG and in higher things as well, but also the work of Jacob Lurie on higher topos and higher algebras.
