the higher derived cahiers topos


The Cahiers topos being the sheaf topos on the site SDCartSp of infinitessimally thickened cartesian spaces is an important example of a differentially cohesive toppos. More generally the higher cahiers topos being the (,1)(\infty,1)-sheaf (,1)(\infty,1)-topos on the (,1)(\infty,1)-site SDCartSp is an important example of a differentially cohesive (,1)(\infty,1)-topos.

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. In fact there hasn’t been described any example of a differentially cohesive non-localic (,1)(\infty,1)-topos.

Apology and Redemption

What is SDCartSpS D Cart Sp? The construction is a follows: Let T:=CartSp smoothT:=Cart Sp_{smooth} denote the opposite of the category of cartesian spaces of finite dimension (as \mathbb{R}-vector spaces, so these are essentially of the form n\mathbb{R}^n). TT is the syntactic category of the Lawvere theory of smooth algebras. Define InfPointTAlg opInf Point\hookrightarrow T Alg^{op} to be the subcategory of Weil algebras; i.e. the subcategory on those objects having as vector space at least dimension 11 and which are nilpotent as algebras. Then SDCartSpS D Cart Sp is defined to be the category of objects being of the form a product n×D\mathbb{R}^n\times D with DInfPointD\in Inf Point and nn\in \mathbb{N}.

By substituting into this receipt InfPoint:=CAlg k sm op\infty Inf Point:={{C Alg_k}_{sm}}^{op} (see below) for InfPointInf Point we obtain the notion of higher derived Cahiers topos which is not nn-localic for any n<n\lt\infty. In the following shall be argued that it is differentially cohesive.


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 (There is also a notion of classical moduli problem where an instance is called to be enhanced by an associated derived moduli problem). For a field kk let CAlg k:=k/E RingCAlg_k:=k/E_\infty Ring 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. A quasi-isomorphism in CAlg dgCAlg_{dg} is defined to be a 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 (see below); 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.

The (Grothendieck) tangent space of a formal moduli problem X:CAlg k smGrpdX:{CAlg_k}_{sm}\to \infty Grpd is defined to be a map T X(0):=X(k[ϵ]/ϵ 2)X(k)T_X(0):=X(k[\epsilon]/\epsilon^2)\to X(k). T X(0)GrpdT_X(0)\in \infty Grpd is a topological space. Define T X(n):=X(kk[n])T_X(n):=X(k\otimes k[n]) where k[n]k[n] denotes the nn-fold shift of kk (as a kk-module spectrum). One can elaborate that T X(n1)T_X(n-1) is the loop space of T X(n)T_X(n); define the tangent complex of the formal moduli problem XX to be the sequence T X:=(T X(n)) n0T_X:=(T_X(n))_{n\ge 0}; T XT_X is a kk-module spectrum. The operation T ()T_{(-)} reflects equivalences.

Let kk be a field of characteristic zero. A differential graded Lie algebra over kk is defined to be a Lie algebra object in Chain kChain_k: a chain complex gg equipped with a binary operation [;]:ggg[-;-]:g\otimes g\to g such that [x,y]+(1) d(x)d(y)[y,x]=0[x,y]+(-1)^{d(x)d(y)}[y,x]=0 and (1) d(z)d(x)[x,[y,z]]+(1) d(x)d(y)[y,[z,x]]+(1) d(y)d(x)[z,[x,y]]=0(-1)^{d(z)d(x)}[x,[y,z]]+(-1)^{d(x)d(y)}[y,[z,x]] + (-1)^{d(y)d(x)}[z,[x,y]]=0 for homogenous elements xg d(x),yg d(y),zg d(z)x\in g_{d(x)},y\in g_{d(y)},z\in g_{d(z)}. The category of differential graded Lie algebras over kk localized at quasi-isomorphisms is denoted by Lie k dgLie_k^{dg} and just also called the category of differential graded Lie algebras over kk.

(Theorem 5.3): Let kk be a field of characteristic zero, let ModuliFun(CAlg k sm,Grpd)Moduli\hookrightarrow Fun({CAlg_k}_{sm},\infty Grpd) the full subcategory spanned by formal moduli problems over kk, let Lie k dgLie_k^{dg} denotes the ∞-category of differential graded Lie algebras over kk. Then there is an equivalence ModuliLie k dgModuli\stackrel{\sim}{\to}Lie_k^{dg}.


We first summarize the proof for the 11-localic case from [Schr11] Proposition 4.5.8: A covering family in SDCartSpS D Cart Sp is define to be of the form {U i×D(f,id)U×D}\{U_i\times D\stackrel{(f,id)}{\to}U\times D\} where {U iU}\{U_i\to U\} is a covering family in CartSp smoothCart Sp_{smooth}. Hence such a covering family by definition does not depend on the thickening components DD. ?: Since all DD are contractible a morphism VUV\to U is an epimorphisms iff D×VD×UD\times V\to D\times U is an epimorphism. Thus it suffices to show that CartSp topCart Sp_{top} is an (,1)(\infty,1)-cohesive site: CartSp topCart Sp_{top} has finite products given by m× n m+n\mathbb{R}^m\times \mathbb{R}^n\simeq \mathbb{R}^{m+n}. Every object has a point *= 0 n*=\mathbb{R}^0\to \mathbb{R}^n. Let {U iU} i\{U_i\to U\}_i be a good open covering family. This implies that the Cech nerve ζ( iU iU)[CartSp op,sSet]\zeta(\coprod_i U_i\to U)\in [Cart Sp^{op}, s Set] is degree-wise a coproduct of representables. Hence the nerve theorem implies colimζ( iU i)colimU=*colim \zeta(\coprod_i U_i)\stackrel{\sim}{\to} colim U=* is an equivalence (the statement of the nerve theorem is that colimζ(U iU)SingUcolim \zeta(U_i\to U)\simeq Sing U is an equivalence, our statement is then implied by the fact that UU as a cartesian space is contractible). Finally limζ( iU i)limU=CartSp loc(*,U)lim \zeta(\coprod_i U_i)\stackrel{\sim}{\to} lim U= Cart Sp_{loc}(*,U) is an equivalence: The morphism \sim has the right lifting property wrt. all boundary inclusions and hence it is an equivalence.

Now we define the (,1)(\infty,1)-site SDCartSp\infty S D Cart Sp: Let kk be a field. Define E Ring:=CAlg(Sp)E_\infty Ring:=C Alg (Sp), CAlg k:=k/E RingC Alg_k:=k/E_\infty Ring. For an associative ring RR let Mod R:=Chain R/qiMod_R:=Chain_R/q-i; i.e. the category of chain complexes of RR-modules modulo quasi-isomorphisms. (Definition 4.4): An object VMod kV\in Mod_k is called to be small if (1) For every integer nn, the homotopy group π n(V)\pi_n(V) is a finite dimensional kk-vector space. (2) π n(V)\pi_n(V) vanishes for n<0n\lt 0 and n>>0n\gt\gt 0. An object ACAlg kA\in CAlg_k is called to be small if it is small as a kk-module spectrum and satisfies (3): The commutative ring π 0A\pi_0 A has a unique maximal ideal pp and the composite map kπ 0Aπ 0A/pk\to \pi_0 A\to \pi_0 A/p is an isomorphism. The full subcategory of Mod kMod_k spanned by the small kk-module spectra is denoted by Mod k sm{Mod_k}_{sm}. The full subcategory of CAlg kC Alg_k spanned by the small E E_\infty-algebras over kk is denoted by CAlg k sm{CAlg_k}_{sm}. And we take

InfPoint:=CAlg k sm\infty Inf Point:={CAlg_k}_{sm}

and we could write InfPointSmoothAlg:=CAlg k=k/E Ring=k/CAlg(Sp)=CAlg(Mod k)\infty Inf Point\hookrightarrow\infty Smooth Alg:={CAlg_k}=k/E_\infty Ring=k/CAlg(Sp)=CAlg(Mod_k) (where CAlg k=CAlg(Mod k)CAlg_k=CAlg(Mod_k) by Remark 4.3). We define

SDCartSp:=Mod kInfPoint\infty S D Cart Sp:=Mod_k\ltimes \infty Inf Point

meaning the semi-direct product of categories: there is a functor Mod kInfPointMod kMod_k\ltimes \infty Inf Point\to Mod_k which we identify with the


Semidirect Product of Categories

Semidirect product of categories are described in (Kock p.12). In more modern terminology and generalized to quasicategories this reads: Let DD be a category which is (left) tensored over CC. Then the category DCD\ltimes C is defined to have as objects pair (d,c)D×C(d,c)\in D\times C and a morphism (d 1,c 1)(d 2,c 2)(d_1,c_1)\to (d_2,c_2) is defined to be a pair (f:d 1d 2c 1,ϕ:c 2c 1)(f:d_1\to d_2\coprod c_1,\phi:c_2\to c_1). Composition of this morphism with (g:d 2d 3c 2,γ:c 3c 2)(g:d_2\to d_3\coprod c_2,\gamma:c_3\to c_2) is defined to be the pair

(d 1fd 2c 1gc 1d 3c 2c 1d 3?d 3c 1,c 3ϕγc 1)(d_1\stackrel{f}{\to}d_2\coprod c_1\stackrel{g\coprod c_1}{\to} d_3\coprod c_2\coprod c_1\stackrel{d_3\coprod ?}{\to}d_3\coprod c_1, c_3\stackrel{\phi \gamma}{\to}c_1)

The identity morphism in DCD\ltimes C is defined to be (dIdIdc,id c)(d\simeq I\stackrel{d\coprod I}{\to} d\coprod c, id_c) where II denotes the initial object of CC. There is a (full) embedding DDCD\hookrightarrow D\ltimes C given by d(d,I)d\mapsto (d,I) and f(f,id I)f\mapsto (f,id_I). This embedding preserves all limits which are preserved by all c{-}\coprod c. If DD has exponential object which are preserved by all ()c(-)\coproduct c (in that y xc(yc) xy^x\coprod c\simeq (y\coprod c)^x) and if ()c(-)\coprod c preserves finite products, then the embedding preserves exponential objects.

Lemma: The semidirect product of an ∞-cohesive ∞-site with an ∞-site equipped with trivial topology is an ∞-cohesive ∞-site in that the ∞-sheaf ∞-topos on is is a cohesive ∞-topos.

Proof: adapt the proof of Schr11, Proposition 3.4.9, p.198-199: