+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### $(\infty,1)$-Topos Theory +--{: .hide} [[nlab:!include (infinity,1)-topos - contents]] =-- #### Higher geometry +--{: .hide} [[nlab:!include higher geometry - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea For $\mathcal{G}$ a [[nlab:geometry (for structured (∞,1)-toposes)]] a $\mathcal{G}$-[[nlab:structured (∞,1)-topos]] $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is _locally representable_ if it is locally equivalent to $Spec U$ for $U \in Pro(\mathcal{G})$ (or $U \in \mathcal{G}$ if it is _locally finite presented_ ). This generalizes * the notion of [[nlab:smooth manifold]] from [[nlab:differential geometry]]; * the notion of [[nlab:scheme]] from [[nlab:algebraic geometry]]. * etc. ## Definition Let $\mathcal{G}$ be a [[nlab:geometry (for structured (∞,1)-toposes)]]. Write $\mathcal{G}_0$ for the underlying discrete geometry. The identity functor $$ p : \mathcal{G} \to \mathcal{G}_0 $$ is then a morphism of geometries. Recall the notation $LTop(\mathcal{G})$ for the [[nlab:(∞,1)-category]] of $\mathcal{G}$-[[nlab:structured (∞,1)-topos]]es and [[nlab:geometric morphism]]s between them. ### Affine $\mathcal{G}$-schemes +-- {: .un_theorem} ###### Theorem ( [[nlab:Structured Spaces|StSp]] 2.1.1 ) There is a pair of [[nlab:adjoint (∞,1)-functor]]s $$ p^* : LTop(\mathcal{G}) \stackrel{\leftarrow}{\to} LTop(\mathcal{G}_0) : \mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}} $$ with $\mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}}$ left adjoint to the canonical functor $p^*$ given by precomposition with $p$. =-- +-- {: .un_remark} ###### Remark ( [[nlab:Structured Spaces|StSp]] p. 38 ) There is a canonical morphism $$ can : Pro(\mathcal{G})^{op} \to LTop(\mathcal{G}_0) $$ =-- +-- {: .un_defn} ###### Definition ( affine $\mathcal{G}$-scheme, [[nlab:Structured Spaces|StSp]] 2.3.9) Write $\mathbf{Spec}^{\mathcal{G}}$ for the [[nlab:(∞,1)-functor]] $$ \mathbf{Spec}^{\mathcal{G}} : Pro(\mathcal{G})^{op} \stackrel{can}{\to} LTop(\mathcal{G}_0) \stackrel{ \mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}} }{\to} LTop(\mathcal{G}) \,. $$ A $\mathcal{G}$-[[nlab:structured (∞,1)-topos]] in the image of this functor is an **affine $\mathcal{G}$-scheme**. =-- ### $\mathcal{G}$-Schemes +-- {: .un_defn} ###### Definition (geometric scheme, [[nlab:Structured Spaces|StSp]] 2.3.9) Let $\mathcal{G}$ be a [[nlab:geometry (for structured (∞,1)-toposes)]]. A $\mathcal{G}$-[[nlab:structured (∞,1)-topos]] $(\mathcal{X},\mathcal{O}_{\mathcal{X}})$ is a **$\mathcal{G}$-scheme** if * there exists a collection $\{U_i \in \mathcal{X}\}$ such that * the $\{U_i\}$ cover $\mathcal{X}$ in that the canonical morphism $\coprod_i U_i \to {*}$ (with ${*}$ the [[nlab:terminal object]] of $\mathcal{X}$) is an [[nlab:effective epimorphism]]; * for every $U_i$ there exists an equivalence $$ (\mathcal{X}/{U_i}, \mathcal{O}_{\mathcal{X}}|_{U_i}) \simeq \mathbf{Spec}^{\mathcal{G}} A_i $$ of structured $(\infty,1)$-toposes for some $A_i \in Pro(\mathcal{G})$ (in the [[nlab:(∞,1)-category]] of [[nlab:pro-object]]s of $\mathcal{G}$). =-- +-- {: .un_defn} ###### Definition (pregeometric scheme, [[nlab:Structured Spaces|StSp]], 3.4.6) For $\mathcal{T}$ a pregeometry, a $\mathcal{T}$-[[nlab:structured (infinity,1)-topos]] $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is a **$\mathcal{T}$-scheme** if it is a $\mathcal{G}$-scheme for [[nlab:generalized the|the]] geometric envelope $\mathcal{G}$ of $\mathcal{T}$. This means that for $f : \mathcal{T} \to \mathcal{G}$ [[nlab:generalized the|the]] geometric envelope and for $\mathcal{O}'_{\mathcal{X}}$ [[nlab:generalized the|the]] $\mathcal{G}$-structure on $\mathcal{X}$ such that $\mathcal{O}_{\mathcal{X}} \simeq \mathcal{O}'_{\mathcal{X}} \circ f$, we have that $(\mathcal{X}, \mathcal{O}'_{\mathcal{X}})$ is a $\mathcal{G}$-scheme. =-- ### Smooth $\mathcal{G}$-schemes Let $\Tau$ be a [[nlab:pregeometry (for structured (∞,1)-toposes)]] and let $\Tau \hookrightarrow \mathcal{G}$ be an inclusion into an enveloping [[nlab:geometry (for structured (∞,1)-toposes)]]. We think of the objects of $\Tau$ as the _smooth_ test spaces -- for instance the [[nlab:cartesian product]]s of some affine line $R$ with itsef -- and of the objects of $\mathcal{G}$ as affine test spaces that may have singular points where they are not smooth. The idea is that a _smooth_ $\mathcal{G}$-scheme is a $\mathcal{G}$-structured space that is locally not only equivalent to objects in $\mathcal{G}$, but even to the very nice -- "smooth" -- objects in $\mathcal{Tau}$. +-- {: .un_defn} ###### Definition ( smooth $\mathcal{G}$-scheme, [[nlab:Structured Spaces|StSp]] 3.5.6) With an envelope $\Tau \hookrightarrow \mathcal{G}$ fixed, a $\mathcal{G}$-scheme is called **smooth** if there the affine schemes $\mathbf{Spec}^{\mathcal{G}} A_i$ appearing in its definition may be chosen with $A_i$ in the image of the includion $\tau \hookrightarrow \mathcal{G}$. =-- ### Examples #### Ordinary schemes See the discussion at [[nlab:derived scheme]] for how ordinary [[nlab:scheme]]s are special cases of [[nlab:generalized scheme]]s. #### Ordinary Deligne-Mumford stacks See the discussion at [[nlab:derived Deligne-Mumford stack]] for how ordinary [[nlab:Deligne-Mumford stack]]s are special cases of [[nlab:derived Deligne-Mumford stack]]s. #### Derived schemes +-- {: .un_defn} ###### Definition (derived scheme, [[nlab:Structured Spaces]], 4.2.8) Let $k$ be a commutative ring. Recall the pregoemtry $\mathcal{T}_{Zar}(k)$. A **[[nlab:derived scheme]]** over $k$ is a $\mathcal{T}_{Zar}(k)$-scheme. =-- #### Derived Deligne-Mumford stacks +-- {: .un_defn} ###### Definition (derived Deligne-Mumford stack, [[nlab:Structured Spaces]], 4.3.19) Let $k$ be a commutative ring. Recall the pregeometry $\mathcal{T}_{et}(k)$ A **[[nlab:derived Deligne-Mumford stack]]** over $k$ is a $\mathcal{T}_{et}(k)$-scheme. =-- #### Derived schemes with $E_\infty$-ring valued structure sheaves The above [[nlab:derived scheme]]s have structure sheaves with values in [[nlab:simplicial object|simplicial]] commutative rings. There is also a notion of derived scheme whose structure sheaf takes values in [[nlab:E-infinity ring]]s. The theory of these is to be described in full detail in * [[nlab:Jacob Lurie]], _[[nlab:Spectral Schemes]]_ . An indication of some details is in * [[nlab:Paul Goerss]], _[[nlab:Topological Algebraic Geometry - A Workshop]]_ ##### Derived smooth manifolds * [[nlab:derived smooth manifold]]. ## Related concepts * [[nlab:geometry (for structured (∞,1)-toposes)]] * [[nlab:structured (∞,1)-topos]] * **locally representable structured (∞,1)-topos** ## References Generalized schemes are definition 2.3.9 of * [[nlab:Jacob Lurie]], _[[nlab:Structured Spaces]]_ The definition of affine $\mathcal{G}$-schemes (absolute spectra) is in section 2.2. [[nlab:!redirects locally representable structured (∞,1)-topos]] [[nlab:!redirects locally representable structured (∞,1)-toposes]] [[nlab:!redirects locally representable structured (infinity,1)-toposes]] [[nlab:!redirects spectral Deligne-Mumford stack]] [[nlab:!redirects spectral Deligne-Mumford stacks]]