Spahn
the higher derived cahiers topos (Rev #2)

The Cahiers topos is the sheaf topos on the site ThCartSp of infinitessimally thickened cartesian spaces. More generally the higher cahiers topos is the (,1)(\infty,1)-sheaf (,1)(\infty,1)-topos on the (,1)(\infty,1)-site ThCartSp.

However the (,1)(\infty,1)-topos arising in this way is (still) a 1-localic (i.e. localic) (,1)(\infty,1)-topos; in other words this notion of higher cahiers topos is no more intelligible than just the classical Cahiers topos.

Requisites

Let SpSp denote the ∞-category of spectra, E Ring:=CAlg(Sp)E_\infty Ring:=CAlg(Sp) the ∞-category of commutative algebra objects in SpSp, for RE RingR\in E_\infty Ring let Mod R(Sp)Mod_R(Sp) denote the category of RR-module objects in SpSp. A derived moduli problem is defined to be a functor X:E RingGrpdX:E_\infty Ring\to \infty Grpd. For a field kk let CAlg kCAlg_k denote the coslice of E RingE_\infty Ring over kk and call it the ∞-category of E E_\infty-algebras; such a kk-algebra AA is called to be discrete if its homotopy groups vanish for i0i\neq 0. An object of the symmetric monoidal (by the usual tensor product) category Chain kChain_k of chain complexes over kk is called a commutative differential graded algebra over kk. There are functors Chain kMod kChain_k\to Mod_k and CAlg(Chain k)CAlg(Mod k)CAlg kCAlg(Chain_k)\to CAlg(Mod_k)\simeq CAlg_k. Aquasi-isomorphism* in CAlg dgCAlg_{dg} is defined to be morphism inducing an isomorphism between the underlying chain complexes. There is a notion of smallness for kk-module spectra and E E_\infty-algebras over kk; the corresponding full sub ∞-categories are denoted by Mod k sm{Mod_k}_sm resp. CAlg k sm{CAlg_k}_sm. A formal moduli problem over kk is defined to be a functor X:CAlg k smGrpdX:{CAlg_k}_{sm}\to \infty Grpd such that X(k)X(k) is contractible and XX preserves pullbacks of maps inducing epimorphisms between the 00-th homotopy groups.

References

  • Jacob Lurie, Formal moduli problems, containing: DAGX: Formal Moduli Problems, 2011, (166 p.). And another more condensed (30 p.) version of this text titled “Moduli Problems and DG-Lie Algebras”. In particular Theorem 5.3 in the second version

  • Vladimir Hinich, DG coalgebras as formal stacks, (arXiv:math/9812034

Revision on February 13, 2013 at 05:14:31 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.