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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?

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  • $\begingroup$ I'm trying to imagine what you mean. Could you provide a bit more detail as to what sort of analogy you are aiming for? Do you mean algebraic cycles in (gluings of) spectra of categorified rings (in some sense)? $\endgroup$
    – David Roberts
    Aug 10, 2015 at 1:29
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    $\begingroup$ Algebraic cycles are not a category-theoretical concept but an algebraic-geometric one. I don't understand the question. $\endgroup$ Aug 10, 2015 at 5:32
  • $\begingroup$ You might consider the approach to cycles pursued by Grothendieck, Borel-Serre, Manin, etc., via an appropriate filtration on the Grothendieck group of the category of coherent sheaves. $\endgroup$ Aug 10, 2015 at 12:44
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    $\begingroup$ -1: You really need to be more specific. $\endgroup$
    – S. Carnahan
    Aug 11, 2015 at 8:04
  • $\begingroup$ Morphisms from a point in Voevodsky's derived category of mixed motives are related to algebraic cycles, right? So maybe look at the higher morphisms... $\endgroup$
    – Will Sawin
    Aug 12, 2015 at 13:24

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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.

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    $\begingroup$ 'A stack can be though like a "family of algebraic varieties"': I don't see why you say this. $\endgroup$ Aug 11, 2015 at 0:25
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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.

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  • $\begingroup$ Yes, there is by now a well-studied theory of higher stacks. One could ask whether there is a good theory of algebraic cycles on them, but it is not clear if this is what the OP is asking. I don't think your answers are useful, would vote -1 if I could. $\endgroup$ Aug 12, 2015 at 22:19
  • $\begingroup$ Indeed, there's some reasonable cycle theory for algebraic stacks (work of Edidin, Toen, etc. give various points of view). That said, I think the "intuition" you're giving about stacks is at best misleading. $\endgroup$ Aug 13, 2015 at 7:13

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