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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Q-category} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory} [[!include topos theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{motivation}{Motivation}\dotfill \pageref*{motivation} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{PresheafQCategory}{(Co)Presheaves on a $Q$-category}\dotfill \pageref*{PresheafQCategory} \linebreak \noindent\hyperlink{DomainAndCodomainFibration}{Domain and codomain fibration}\dotfill \pageref*{DomainAndCodomainFibration} \linebreak \noindent\hyperlink{qcategory_of_cones}{Q-category of cones}\dotfill \pageref*{qcategory_of_cones} \linebreak \noindent\hyperlink{sieves}{Sieves}\dotfill \pageref*{sieves} \linebreak \noindent\hyperlink{the_qcategory_factoring_a_fully_faithful_factor}{The Q-category factoring a fully faithful factor}\dotfill \pageref*{the_qcategory_factoring_a_fully_faithful_factor} \linebreak \noindent\hyperlink{quasicosites}{Quasi-(co)-sites}\dotfill \pageref*{quasicosites} \linebreak \noindent\hyperlink{InfinitesimalThickening}{Infinitesimal thickenings}\dotfill \pageref*{InfinitesimalThickening} \linebreak \noindent\hyperlink{ASheaves}{$\mathbb{A}$-Sheaves}\dotfill \pageref*{ASheaves} \linebreak \noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak \noindent\hyperlink{FormalSmoothness}{Formal smoothness and $Alg_k^{inf}$-sheaves}\dotfill \pageref*{FormalSmoothness} \linebreak \noindent\hyperlink{relation_to_other_concepts}{Relation to other concepts}\dotfill \pageref*{relation_to_other_concepts} \linebreak \noindent\hyperlink{RelationToCohesiveToposes}{Relation to cohesive toposes}\dotfill \pageref*{RelationToCohesiveToposes} \linebreak \noindent\hyperlink{sheafification_versus_the_gabriel_localization_}{Sheafification versus the Gabriel localization $G_{\mathcal{F}} = H^2_{\mathcal{F}}$}\dotfill \pageref*{sheafification_versus_the_gabriel_localization_} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{$Q$-category} is nothing but a [[coreflective subcategory]] and a \emph{$Q^\circ$-category} is nothing but a [[reflective subcategory]]. Since both of these encode reflective [[localization]]s, following \hyperlink{Rosenberg}{Rosenberg} the ``Q'' is for \emph{quotient} and is to indicate that in this context one is interested in notions similar to, but different from, the standard notion of [[sheaves]]: for $\mathbb{A} := (A \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}} \bar A)$ a $Q^\circ$-category there is canonically induced a quadruple of [[adjoint functor]]s between the corresponding [[presheaf categories]] \begin{displaymath} PSh(\bar A) \stackrel{\overset{u_!}{\to}}{\stackrel{}{\stackrel{\overset{u^*}{\leftarrow}}{\stackrel{\overset{u_*}{\to}}{\underset{u^!}{\leftarrow}}}}} PSh(A) \end{displaymath} and a presheaf $F \in PSh(A)$ is called an \emph{$\mathbb{A}$-sheaf} if the canonical morphism \begin{displaymath} u^* F \to u^! F \end{displaymath} is an [[isomorphism]] in $PSh(\bar A)$. More generally, there are generalizations of this condition where presheaves of sets can be replaced with presheaves with values in other categories, notably in [[abelian categories]]. In a central motivating class of examples $\mathbb{A}$ is a category of [[sieve]]s on objects in a [[small category]] $C$ that are regarded as being [[covering]] but which do not necessarily satisfy the axioms of a [[Grothendieck topology]] and not even of a [[coverage]]. \hypertarget{motivation}{}\subsection*{{Motivation}}\label{motivation} The [[Yoneda embedding]] is continuous but not cocontinuous functor. Hence the Grothendieck topologies are used to define smaller codomain of Yoneda, a sheaf subcategory of the category of presheaves such that for that corestricted embedding a desired class of covering cones will stay covering cones. More general families of diagrams than the sieves of a Grothendieck topology may be involved. The important properties of the categories of diagrams for doing the sheaf theory can be expressed in terms of an adjoint pair of functors; this adjoint pair gives an example of a $Q$-category. This generalization of sheaf theory can rephrase categorically also properties like formal smoothness and formal etaleness of functors. The sheafification and the construction of a Gabriel localization of an Abelian category can in this formalism be seen as special cases of the same construction. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{QCategory}\hypertarget{QCategory}{} An \textbf{almost quotient category}, or a \textbf{$Q$-category} is \begin{itemize}% \item a pair of [[adjoint functor]]s \begin{displaymath} \mathbb{A} : (u^* \dashv u_*) : \bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}} A \,, \end{displaymath} with $u^*$ the [[left adjoint]]; \item such that $u^*$ is [[full and faithful functor|full and faithful]] \end{itemize} In other words, $A$ is equipped with an equivalence with a [[coreflective subcategory]] of $\bar{A}$. \end{defn} \begin{note} \label{QTerminology}\hypertarget{QTerminology}{} \begin{itemize}% \item In motivating classes of examples $\bar A$ and $A$ are [[topos]]es and $\mathbb{A}$ is a [[geometric morphism]] between them. Therefore one generally speaks of $u_*$ as the \textbf{[[direct image]]} and of $u^*$ as the \textbf{[[inverse image]]} of the Q-category. \item The definition \ref{QCategory} is nothing but the definition of a [[coreflective subcategory]]. However the term \emph{$Q$-category} is used when the pair is used in a specific meaning useful to constructions in (generalized) sheaf theory (similarly like presheaf of objects in $D$ and contravariant functor to $D$ are synonyms, but a different word refers to a different context and intuition). \item One sometimes write the above data as $\bar A \stackrel{\overset{u}{\leftarrow}}{\to} A$. \end{itemize} \end{note} \begin{defn} \label{QCategoryMorphism}\hypertarget{QCategoryMorphism}{} A \textbf{[[morphism]] of Q-categories} from $\mathbb{A} : (u^* \dashv u_*) : \bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}} A$ to $\mathbb{B} : (v^* \dashv v_*) : \bar B \stackrel{\overset{v^*}{\leftarrow}}{\underset{v_*}{\to}} B$ is a triple $(\Phi,\bar{\Phi},\phi)$ where $\Phi : A\to B$, $\bar{\Phi}:\bar{A}\to\bar{B}$ are functors and $\phi:\Phi u_*\Rightarrow v_*\bar{\Phi}$ is a [[natural isomorphism]] of functors. The composition is given by \begin{displaymath} (\Phi,\bar{\Phi},\phi)\circ(\Phi',\bar{\Phi}',\phi') = (\Phi\Phi',\bar{\Phi}\bar{\Phi}',\phi\bar{\Phi}' \circ \Phi\phi') \end{displaymath} A \textbf{transformation of morphisms of Q-categories} is a pair $(\alpha,\bar{\alpha}):(\Phi,\bar{\Phi},\phi)\to (\Psi,\bar{\Psi},\psi)$ of [[natural transformation]]s $\alpha:\Phi\to\Psi$ and $\bar{\alpha}:\bar{\Phi}\to\bar{\Psi}$ such that the diagram \begin{displaymath} \itexarray{ \Phi u_* & \stackrel{\phi}\longrightarrow & v_* \bar{\Psi}\\ \alpha u_*\downarrow && \downarrow v_*\bar{\alpha}\\ \Psi u_* &\stackrel{\psi}\longrightarrow& v_* \bar{\Psi} } \end{displaymath} [[commuting diagram|commutes]]. \end{defn} Small Q-categories, morphisms of Q-categories and natural transformations of morphisms form a [[2-category]] of small Q-categories. \begin{defn} \label{QopCategory}\hypertarget{QopCategory}{} A $Q^\circ$-category is a pair of functors $Q: A\leftrightarrow\bar{A}: I$, where $Q$ is fully faithful and right adjoint to $I$. In other words, $A$ is equipped with an equivalence with a [[reflective subcategory]] of $\bar{A}$. \end{defn} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{PresheafQCategory}{}\subsubsection*{{(Co)Presheaves on a $Q$-category}}\label{PresheafQCategory} The [[category of presheaves]] over any $Q$-category canonically inherits itself the structure of a Q-category. \begin{prop} \label{PresheafQCategories}\hypertarget{PresheafQCategories}{} Let $\mathbb{A} = \bar A \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} A$ be a $Q$-category. Let $C$ be any category. Then forming [[copresheaves]] with values in $C$ yields a Q-category of the form \begin{displaymath} [\mathbb{A},C] : [\bar A, C] \stackrel{\overset{C^R = (-)\circ R}{\leftarrow}}{\underset{C^L = (-)\circ L}{\to}} [A,C] \,. \end{displaymath} \end{prop} \begin{proof} The $(C^R \dashv C^L)$-[[unit of an adjunction|unit]] is $C^\eta$ induced by the original unit $\eta: 1_A\to R L$ \begin{displaymath} C^{\eta} : Id_{C^A} \to C^L \circ C^R = C^{R L} \end{displaymath} and the counit $C^\epsilon$ is induced by the original counit $\epsilon: L R\to 1_{\bar{A}}$ \begin{displaymath} C^\epsilon : C^R\circ C^L = C^{L R}\to Id_{C^{\bar{A}}} \,. \end{displaymath} The only thing is who is adjoint -- now $C^R$ is the left adjoint. The triangle identities for $C^R\dashv C^L$ can be obtained by expanding. For $R: \bar{A}\to A$, one has $C^R : C^A\to C^{\bar{A}}$ is given by $C^R : G\mapsto GR$, and for $L:A\to\bar{A}$ one has $C^L:F\mapsto F L$. Then $C^\eta : Id_{C^A}\to C^R C^L = C^{LR}$ has the components $(C^\eta)_G : (Id_{C^A})(G) \Rightarrow C^R C^L (G)$ given by $G \eta : G\to G L R$. Thus for each functor $G\in C^{\bar{A}}$, the composition \begin{displaymath} G R\stackrel{G\eta R}\longrightarrow G R L R \stackrel{G R \eta}\longrightarrow G R \end{displaymath} is the identity by the triangle identity for $L\dashv R$, but this is precisely the $G$-component of the transformation \begin{displaymath} C^R \stackrel{C^R C^\eta}\longrightarrow C^R C^L C^R \stackrel{C^\epsilon C^R}\longrightarrow C^R. \end{displaymath} Similarly the $F$-component of \begin{displaymath} C^{L} \stackrel{C^\eta C^L}\longrightarrow C^L C^R C^L \stackrel{C^L C^\epsilon}\longrightarrow C^L, \end{displaymath} for a functor $F\in C^A$ reads \begin{displaymath} F L \stackrel{F L \eta}\longrightarrow F L R L \stackrel{F \epsilon L}\longrightarrow F L \end{displaymath} and this composition equals $Id_{F L}$ by another triangle identity for $L\dashv R$. It is clear that if $\eta$ is iso then the composition with $\eta$ is also iso. Thus we obtain a $Q$-categories. In other words, since the [[left adjoint]] being a [[full and faithful functor]] is equivalent to the unit of the [[adjunction]] being an [[isomorphism]], it follows from $L$ being full and faithful that $C^R$ is full and faithful. \end{proof} This appears as (\hyperlink{KontsevichRosenbergSpaces}{Kontsevich-Rosenberg, 2.7}). \begin{note} \label{ExtraAdjointsForPresheafQCategories}\hypertarget{ExtraAdjointsForPresheafQCategories}{} Assume in the context of prop. \ref{PresheafQCategories} that \begin{itemize}% \item $A$ and $\bar A$ are [[small categories]]; \item $C$ has all small [[limit]]s and [[colimit]]s (for instance $C$ = [[Set]]). \end{itemize} Then there is an [[adjoint quadruple]] \begin{displaymath} (u_! \dashv u^* \dashv u_* \dashv u^!) : [\bar A, C] \stackrel{\overset{u_! = Lan R}{\to}}{\stackrel{\overset{u^* = C^R}{\leftarrow}}{\stackrel{\overset{u_* = C^L}{\to}}{\underset{u^! = Ran L}{\leftarrow}}}} [A,C] \,, \end{displaymath} where $Lan R$ denotes the left [[Kan extension]] along $R$ and $Ran L$ the right Kan extension along $L$. \end{note} \begin{proof} By general properties of [[Kan extension]]s. \end{proof} \begin{defn} \label{SubcategoryOfPresheafQCategories}\hypertarget{SubcategoryOfPresheafQCategories}{} Any subcategory $B\subset C^{\bar{A}}$ containing $Im(C^R)$ determines a Q-subcategory $C^A\leftrightarrow B$. \end{defn} \hypertarget{DomainAndCodomainFibration}{}\subsubsection*{{Domain and codomain fibration}}\label{DomainAndCodomainFibration} \begin{defn} \label{DomainAndCodomainFibration}\hypertarget{DomainAndCodomainFibration}{} For any [[category]] $A$, write $\bar A := A^I$ for its [[arrow category]]. This comes equipped with the [[codomain fibration]] $cod : \bar A \to A$ and the [[domain cofibration]] $dom : \bar A \to A$. Both have a joint [[section]] $\epsilon : A \to \bar A$ by a [[full and faithful functor]] that assigns identity morphisms. These form a triple of [[adjoint functor]]s \begin{displaymath} (codom \dashv \epsilon \dashv dom) : A^I \stackrel{\overset{codom}{\to}}{\stackrel{\overset{\epsilon}{\leftarrow}}{\underset{dom}{\to}}} A \,. \end{displaymath} Taking this apart, we have that \begin{displaymath} A^{dom} : A^I \stackrel{\overset{\epsilon}{\leftarrow}}{\underset{dom}{\to}} A \end{displaymath} is a $Q$-category and \begin{displaymath} A^{dom} : A^I \stackrel{\overset{cod}{\to}}{\underset{\epsilon}{\leftarrow}} A \end{displaymath} is a $Q^o$-category. \end{defn} This appears as (\hyperlink{KontsevichRosenbergSpaces}{Kontsevich-Rosenberg, 2.5}). Several classes of examples of Q-categories of interest arise as sub-Q-categories of those of this form. For instance the Q-categories of infinitesimal thickenings \hyperlink{InfinitesimalThickening}{below}. \begin{note} \label{PresheavesOnCodomainFibration}\hypertarget{PresheavesOnCodomainFibration}{} Passing to copresheaf Q-categories as in prop. \ref{PresheafQCategories} we have the Q-category \begin{displaymath} [A^I,Set] \stackrel{\overset{u^* = (-)\circ dom}{\leftarrow}}{\stackrel{\overset{u_* = (-)\circ \epsilon}{\to}}{\overset{u^! = (-)\circ codom}{\leftarrow}}} [A,Set] \end{displaymath} where the extra [[right adjoint]] is given by precomposition with the domain evaluation. \end{note} \hypertarget{qcategory_of_cones}{}\subsubsection*{{Q-category of cones}}\label{qcategory_of_cones} Let $C$ be a category and $LC$ be the category whose objects are [[cones]] $\alpha: x\to d$ over (small) diagrams $d: D\to C$ where $D$ are variable small categories; and the morphisms from $x\stackrel{\alpha}\to d$ to $x'\stackrel{\alpha'}\to d'$ are triples of the form $(f,\rho,\bar{f})$ where $f:x\to x'$ is a morphism in $C$, $\rho : D' \to D$ is a diagram (= functor), and $\bar{f}:d \circ\rho \to d'$ is a morphism of diagrams (= natural transformation) such that \begin{displaymath} \itexarray{ x & \stackrel{f}\to & x'\\ \alpha \star\rho \downarrow && \downarrow\alpha'\\ d \circ \rho&\stackrel{\bar{f}}\to & d' } \end{displaymath} commutes and $\alpha \star \rho$ denotes the horizontal composition (= [[Godement product]]) of natural transformations. Then one defines composition of morphisms by the formula \begin{displaymath} (f_1, \rho_1,\bar{f_1})\circ(f_2, \rho_2,\bar{f_2}) \stackrel{def}{=}(f_1\circ f_2, \rho_2\circ\rho_1, \bar{f_1} \circ (\bar{f_2} \star \rho_1)). \end{displaymath} There is a fully faithful functor $Q_C:C\to LC$ that to any $x\in C$ assigns the trivial cone $id_x :x\to x$ and to any morphism the corresponding morphism of trivial cones. Its right adjoint is the morphism $I_C:LC\to C$ defined by sending the cone $\alpha: x\to d$ over a diagram $d:D\to C$ its vertex $x$ and to a cone morphism $(f,\rho,\bar{f})$, the morphism of vertices $f$. Then $I_C\circ Q_C = Id_C$. The identity transformation can be thus taken as the unit of the adjunction. The counit of the adjunction $\epsilon: Q_C\circ I_C \to Id_{LC}$ is constructed as follows: to a cone $\alpha:x\to d$ assign the morphism $(1_x, const, \alpha)$ where $const: D\to C$ is the constant diagram which is the unique diagram from $D = dom(d)$ to the final category $1=\{\star\}$. One can check that these data indeed define an adjoint pair $Q_C\dashv I_C$ of functors. $Q_C: C\leftrightarrow I_C: LC$ is therefore a Q-category, and it is called the \textbf{Q-category of cones}. If $\mathcal{L}\subset Cat$ is a family of small categories, then one considers the full subcategory $L_{\mathcal{L}}C$ of cones whose domains are in $\mathcal{L}$; the rest of the construction restricts to obtain a Q-category $Q^{\mathcal{L}}_C : C\leftrightarrow L_{\mathcal{L}}C : I_C^{\mathcal{L}}$. The most classical case is when $\mathcal{L}$ is the (say skeletal) category $Dis$ of small discrete categories (=just identity morphisms), one obtains then the Q-category $Q^{Dis}_C: C\leftrightarrow L_{Dis}C$. A \textbf{semicosite} (or semicositus pl. semicositi) is a Q-category of the form $C\leftrightarrow \bar{C}$ where $\bar{C}$ is a full subcategory of $L_{Dis}C$ and the adjoint pair is obtained by the restriction. A semicosite is a \textbf{precosite} (=Grothendieck precotopology) if (i) $id_x\in Ob(\bar{C})$ whenever $x\in Ob(C)$. (ii) $\{x\stackrel{\phi_i}\to x_i\}_{i\in I}\in Ob(\bar{C})$ and $\{x_i\stackrel{\phi_{ij}}\to x_{ij}\}_{j\in J_i}\in Ob(\bar{C})$ then $\{x\stackrel{\phi_{ij}\circ\phi_i}\to x_{ij}\}\in Ob(\bar{C})$ (iii) $\{x\to x_i\}_{i\in I}\in Ob(\bar{C})$ and $g\in C(x,y)$, then the family of pushouts $\{y\mapsto x_i\coprod_x y\}_{i\in I}$ exists and belongs to $Ob(\bar{C})$. An example of a cosite is a cosite of closed sets of a topological space. \hypertarget{sieves}{}\subsubsection*{{Sieves}}\label{sieves} The $Q$-category of [[sieves]]. The $Q$-subcategory of the $Q$-category of (all) sieves corresponding to the subcategory of sieves corresponding to the Grothendieck topology\ldots{} (needs explanation) \hypertarget{the_qcategory_factoring_a_fully_faithful_factor}{}\subsubsection*{{The Q-category factoring a fully faithful factor}}\label{the_qcategory_factoring_a_fully_faithful_factor} Any fully faithful functor among small categories $F: A\to B$ factors canonically into the composition $A\stackrel{u^*}\to \bar{A}\hookrightarrow B$ where $\bar{A}\subset B$ is the full subcategory of $B$ whose objects are all $b$ in $Ob B$ such that $a\mapsto B(F(a),b)$ is a representable functor $A^{op}\to Set$, and $u^*$ is the corestriction of $F$ to $\bar{A}$. This corestriction makes sense: $F$ is fully faithful, hence $B(F(a),F(a)) = B(a,a)$, i.e. $F(a)\in \bar{A}$ for all $a$ in $Ob A$. For each $b\in \bar{A}$, define now $u_*(b)$ as the functor representing $B(F(-),b)$, i.e. by $\bar{A}(u^*(a),b) = B(F(a),b) \cong B(a,u_*(b))$ (KR NcSpaces A1.1.1). This relation on objects extends to an adjunction $u^*\dashv u_*$ with $u^*$ fully faithful. \hypertarget{quasicosites}{}\subsubsection*{{Quasi-(co)-sites}}\label{quasicosites} (\ldots{}) \begin{defn} \label{QuasiCoSite}\hypertarget{QuasiCoSite}{} Let $A$ be a [[category]] and $\mathcal{T}$ a map that assigns to every [[object]] a collection of co[[sieve]]s on that object ([[subfunctor]]s of $A(x,-)$), which includes maximal cosieve (functor $A(x,-)$). Write $\bar A_{\mathcal{T}}$ for the category whose objects are co[[sieve]]s on $A$, and whose morphisms are morphisms in $A$ that respect the corresponding cosieves. This yields a Q-category \begin{displaymath} \bar A_{\mathcal{T}} \stackrel{\leftarrow}{\to} A \,. \end{displaymath} Call this a \textbf{quasi-cosite} if \begin{enumerate}% \item for any two cosieves in $\mathcal{T}$ their intersection is also in $\mathcal{T}$; \item for any cosive in $\mathcal{T}$ any cosieve containing it is also in $\mathcal{T}$. \end{enumerate} \end{defn} This is (\hyperlink{KontsevichRosenberg}{KontsevichRosenberg, 2.2}). \begin{defn} \label{QuasiCoSiteAssociatedToQCategory}\hypertarget{QuasiCoSiteAssociatedToQCategory}{} For any Q-category $\mathbb{A}$, the \textbf{quasi-cosite associated} with $\mathbb{A}$ is the Q-category $\mathcal{T}\mathbb{A}$ defined by\ldots{} \end{defn} The following is supposed to be the standard quasi-cosite for non-commutative geometry. \begin{defn} \label{NCQuasiCosite}\hypertarget{NCQuasiCosite}{} Let $k$ be a [[ring]] and $A := Alg_k$ the category of [[associative algebra]]s over $k$. Let $\bar A \subset A^I$ be the [[full subcategory]] of the \href{DomainAndCodomainFibration}{domain fibration} whose objects are the \textbf{faithfully flat} morphisms, i.e. those morphisms $\phi : R \to T$ in $Alg_k$ such that the induced \begin{displaymath} \phi^* : R Mod \to T Mod \end{displaymath} is an [[exact functor|exact]] and [[full and faithful functor]]. This forms a Q-category. Write $\mathcal{T}Alg_k$ for the quasi-cosite associated with this Q-category by def. \ref{QuasiCoSiteAssociatedToQCategory}. This is the \textbf{standard quasi-cosite for [[noncommutative geometry]]}. \end{defn} This is (\hyperlink{KontsevichRosenberg}{KontsevichRosenberg, A.1.9.2}). \hypertarget{InfinitesimalThickening}{}\subsubsection*{{Infinitesimal thickenings}}\label{InfinitesimalThickening} Generally, [[infinitesimal object|infinitesimal thickenings]] are characterized by [[coreflective subcategory|coreflective embeddings]]: A characteristic property of an [[infinitesimally thickened point]] $D$ is that for any object $X$ without infinitesimal thickening, there are in general nontrivial morphisms $D \to X$, but there is only a unique morphism $X \to D$, reflecting the fact that $D$ has only a single global point. Thus if by $i : C \to \bar C$ we denote the inclusion of objects $X$ without infinitesimal thickening into the collection of possibly infinitesimally thickened objects, and by $p : \bar C \to C$ the projection that contracts away the infinitesimal extension, we have indeed an [[adjunction]] \begin{displaymath} \bar C(i(X), D) \simeq C(X, p(D)) \simeq C(X,*) \simeq * \,. \end{displaymath} This general concept is described at . See also the discussion below at \hyperlink{RelationToCohesiveToposes}{Relation to cohesive toposes}. The following is one realization of this general concept. \begin{prop} \label{InfinitesimalThickening}\hypertarget{InfinitesimalThickening}{} Let $k$ be a [[field]] and $A := CAlg_k$ be the category of commutative [[associative algebra]]s over $k$. Let $\bar A \subset A^I$ be the [[full subcategory]] of the [[codomain fibration]] Q-category from prop. \ref{DomainAndCodomainFibration} on those morphisms of commutative algebras which are [[epimorphism]]s and whose [[kernel]] is nilpotent. Then \begin{displaymath} CAlg_k^{inf} = (\epsilon \dashv dom) : \bar A \stackrel{\overset{\epsilon}{\leftarrow}}{\underset{dom}{\to}} A \end{displaymath} is a Q-category. The analogous statement is true with $A = Alg_k$ the category of all [[associative algebra]]s, not necessarily commutative. \end{prop} This appears as (\hyperlink{KontsevichRosenbergSpaces}{Kontsevich-Rosenberg, 2.6}). \begin{remark} \label{DiscussionOfTheInfinitesimalThickeningFormalization}\hypertarget{DiscussionOfTheInfinitesimalThickeningFormalization}{} Here we think of an algebra epimorphism $\phi : \mathbf{B} \to B$ with nilpotent kernel -- an [[infinitesimal ring extension]] -- as the infinitesimal thickening of $Spec B$ by $Spec ker \phi$ to $Spec \mathbf{B}$. The functor $\epsilon$ builds the trivial (empty) infinitesimal thickenings \begin{displaymath} \epsilon B : B \stackrel{id_B}{\to} B \,. \end{displaymath} The functor $dom$ remembers the \emph{thickened} algebra \begin{displaymath} dom (\mathbf{B} \to B) = \mathbf{B} \,. \end{displaymath} But notice that we also have the codomain-functor, which is the functor that forgets the thickening \begin{displaymath} cod (\mathbf{B} \to B) = B \end{displaymath} and that the [[adjoint pair]] $(\epsilon \dashv dom)$ does extend to an [[adjoint triple]] \begin{displaymath} CAlg_k^{inf} : \bar A \stackrel{\overset{cod}{\to}}{\stackrel{\overset{\epsilon}{\leftarrow}}{\underset{dom}{\to}}} A \,. \end{displaymath} In the discussion at it is the pair $(cod \dashv \epsilon)$ that appears in the axiomatization, or rather its version on the opposite categories \begin{displaymath} (\epsilon^{op} \dashv cod^{op}) : \bar A^{op} \stackrel{\overset{\epsilon^{op}}{\leftarrow}}{\underset{cod^{op}}{\to}} A^{op} \,, \end{displaymath} not the other adjoint pair $(\epsilon \dashv dom)$ used here. This is the reason for the shift in adjoint triples that is mentioned in over at \emph{[[cohesive (infinity,1)-topos -- infinitesimal cohesion|infinitesimal cohesion]]} . \end{remark} \hypertarget{ASheaves}{}\subsection*{{$\mathbb{A}$-Sheaves}}\label{ASheaves} We discuss the notion of a \emph{$\mathbb{A}$-sheaves} on a Q-category $\mathbb{A}$. \begin{defn} \label{BareSheafCondition}\hypertarget{BareSheafCondition}{} Let $\mathbb{A} = (\bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}}A)$ be a \hyperlink{QCategory}{Q-category}. An object $x \in A$ is called an \textbf{$\mathbb{A}$-sheaf} if for all $y \in \bar A$ the canonical morphism \begin{displaymath} \bar A(y, u^*(x)) \to A(u_*(y), x) \end{displaymath} is an [[isomorphism]] (in [[Set]], hence a [[bijection]]). This morphism is given by \begin{displaymath} g \mapsto \eta_x^{-1} \circ u_*(g) \end{displaymath} where $\eta_x : x \to u_* u^*(x)$ is the $x$-component of the $(u^* \dashv u_*)$-[[unit of an adjunction|counit]] (which is an [[isomorphism]] because $u^*$ is a [[full and faithful functor]]). \end{defn} This appears as (\hyperlink{KontsevichRosenbergSpaces}{Kontsevich-Rosenberg, 3.1.1}). \begin{prop} \label{SheafConditionByExtraRightAdjoint}\hypertarget{SheafConditionByExtraRightAdjoint}{} Let $\mathbb{A}$ be a Q-category with an extra [[right adjoint]] $u^! : A \to \bar A$ \begin{displaymath} \mathbb{A} : \bar A \stackrel{\overset{u^*}{\leftarrow}}{\stackrel{\overset{u_*}{\to}}{\underset{u^!}{\leftarrow}}} A \,. \end{displaymath} Then an object $x \in A$ is an $\mathbb{A}$-sheaf in the sense of def. \ref{BareSheafCondition} precisely if the canonical morphism \begin{displaymath} (u^*(x) \to u^!(x)) := u^*(X) \stackrel{}{\to} u^! u_* u^* (x) \stackrel{\simeq}{\to} u^! (x) \end{displaymath} is an [[isomorphism]] in $\bar A$. \end{prop} This appears as (\hyperlink{KontsevichRosenbergSpaces}{Kontsevich-Rosenberg, 3.1.3}). For more details on the canonical morphism appearing here see the section at [[cohesive topos]]. \begin{proof} Using the $(u_* \dashv u^!)$-[[adjunction]]-[[isomorphism]] we have that the canonical morphism from def. \ref{BareSheafCondition} is isomorphic to \begin{displaymath} \phi : \bar A(y, u^* x) \to A(u_* y, x) \stackrel{\simeq}{\to} \bar A(y, u^! x) \,, \end{displaymath} for all $y \in \bar A$, where the second map sends every morphism to its [[adjunct]]. Using the definition of the first morphism from def. \ref{BareSheafCondition} and the expression of [[adjunct]]s (as discussed there) by composition with (co)-units, we find that the composite map here sends any morphism \begin{displaymath} g : y \to u^* x \end{displaymath} to the composite \begin{displaymath} \itexarray{ y \\ \downarrow \\ u^! u_* y &\stackrel{u^! u_* g}{\to}& u^! u_* u^* x \\ && \downarrow \\ && u^! x } \,. \end{displaymath} Using that [[unit of an adjunction|adjunction units]] are [[natural transformation]]s, we can complete this to a [[commuting diagram]] \begin{displaymath} \itexarray{ y &\stackrel{g}{\to}& u^* x \\ \downarrow && \downarrow \\ u^! u_* y &\stackrel{u^! u_* g}{\to}& u^! u_* u^* x \\ && \downarrow \\ && u^! x } \,. \end{displaymath} This shows that $\phi$ acts on any $g$ by postcomposition with the canonical morphism $u^* x \to u^! x$. By the [[Yoneda lemma]] it follows that $\phi$ is an [[isomorphism]] for all $y$ precisely if $u^* x \to u^!$ is an isomorphism. \end{proof} \begin{def} \label{MonoAndEpiPresheaves}\hypertarget{MonoAndEpiPresheaves}{} Let $\mathbb{A} = (u^* \dashv u_* \dashv u^!): \bar A \to A$ be a Q-category with an extra right adjoint as in prop. \ref{SheafConditionByExtraRightAdjoint}. We say \begin{itemize}% \item an object $x \in A$ is \textbf{$\mathbb{A}$-monopresheaf} if $u^* x \to u^! x$ is a [[monomorphism]] in $\bar A$. \item an object $x \in A$ is \textbf{$\mathbb{A}$-epipresheaf} if $u^* x \to u^! x$ is an [[strict epimorphism]] in $\bar A$. \end{itemize} \end{def} This appears as \hyperlink{KontsevichRosenbergSpaces}{Kontsevich-Rosenberg, 3.1.2, 3.1.4}. \begin{note} \label{PresheafSheafCondition}\hypertarget{PresheafSheafCondition}{} Let $\mathbb{A} : \bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}} A$ be a Q-category where $A$ is a [[small category]], and let $C$ be a category with all small [[limit]]s. Then the Q-category of presheaves from prop \ref{PresheafQCategories} has an extra right adjoint $u^!_C := Ran_{u^*}$ \begin{displaymath} C^{\mathbb{A}} : C^{\bar A} \stackrel{\overset{f^* := C^{u_*}}{\leftarrow}}{\stackrel{\underset{f_* := C^{u^*}}{\to}}{\underset{f^! := u^!_C}{\leftarrow}}} C^{A} \end{displaymath} given by the right [[Kan extension]] along $u^*$, which exists by the assumption that $C$ has all small limits. Therefore by prop. \ref{SheafConditionByExtraRightAdjoint} a presheaf $F \in C^{A}$ is a $C^{\mathbb{A}}$-sheaf, def. \ref{BareSheafCondition}, precisely if the canonical morphism \begin{displaymath} f^* F \to f^! F \end{displaymath} is an [[isomorphism]]. \end{note} This appears as (\hyperlink{KontsevichRosenbergSpaces}{Kontsevich-Rosenberg, 3.5}). \hypertarget{examples_2}{}\subsection*{{Examples}}\label{examples_2} \hypertarget{FormalSmoothness}{}\subsubsection*{{Formal smoothness and $Alg_k^{inf}$-sheaves}}\label{FormalSmoothness} Let \begin{displaymath} CAlg_k^{inf} : \bar A \stackrel{\overset{i}{\leftarrow}}{\underset{p}{\to}} CAlg_k \end{displaymath} be the Q-category of infinitesimal thickenings from prop. \ref{InfinitesimalThickening}. Write \begin{displaymath} [CAlg_k^{inf},Set] : [\bar A,Set] \stackrel{\overset{u^*}{\leftarrow}}{\stackrel{\overset{u_*}{\to}}{\underset{u^!}{\leftarrow}}} [CAlg_k,Set] \end{displaymath} be the corresponding Q-category of copresheaves from prop. \ref{PresheafQCategories}. Notice that $[CAlg_k, Set]$ is the [[presheaf topos]] that contains [[scheme]]s over $k$. Recall the notion of [[formally smooth scheme]], [[formally étale morphism]] and [[formally unramified morphism]] of schemes. \begin{prop} \label{FormalSmoothnessByASheaves}\hypertarget{FormalSmoothnessByASheaves}{} An object $X \in [CAlg_k, Set]$ is \begin{itemize}% \item [[formally étale morphism|formally étale]] precisely if it is an $CAlg_k^{inf}$-sheaf in the sense of def. \ref{BareSheafCondition}, hence precisely if the canonical morphism \begin{displaymath} u^* F \to u^! F \end{displaymath} from prop. \ref{SheafConditionByExtraRightAdjoint} is an [[isomorphism]]; \item [[formally unramified]] precisely if it is a $CAlg_k^{inf}$ monopresheaf, def. \ref{MonoAndEpiPresheaves}, hence precisly if $u^* F \to u^! F$ is a [[monomorphism]]; \item [[formally smooth]] precisely if it is a strict epipresheaf, def. \ref{MonoAndEpiPresheaves}, hence precisely if $u^* F \to u^! F$ is a [[strict epimorphism]]. \end{itemize} \end{prop} This appears as (\hyperlink{KontsevichRosenbergSpaces}{Kontsevich-Rosenberg, 4.1}). This [[category theory|category theoretic]] reformulation of these three properties therefore admits straightforward generalization of these notions to other contexts. See the section at [[cohesive (∞,1)-topos]]. For instance we have the following direct generalization is of interest in [[noncommutative geometry]]. \begin{defn} \label{FormalNCSmoothnessByASheaves}\hypertarget{FormalNCSmoothnessByASheaves}{} Let $Alg_k$ be the full category of [[associative algebra]]s over $k$, not necessarily commutative. Write $Alk_k^{inf} : \bar A \to Alg_k$ for the Q-category of infinitesimal thickenings as in def. \ref{InfinitesimalThickening}. Notice that $[Alg_k, Set]$ is the [[presheaf topos]] that contains [[noncommutative scheme]]s. We then say an object $X \in [Alg_k, Set]$ is \begin{itemize}% \item [[formally étale morphism|formally étale]] precisely if it is an $Alg_k^{inf}$-sheaf in the sense of def. \ref{BareSheafCondition}, hence precisely if the canonical morphism \begin{displaymath} u^* F \to u^! F \end{displaymath} from prop. \ref{SheafConditionByExtraRightAdjoint} is an [[isomorphism]]; \item [[unramified|formally unramified]] precisely if it is a $Alg_k^{inf}$ monopresheaf, hence precisly if $u^* F \to u^! F$ is a [[monomorphism]]; \item [[formally smooth]] precisely if $u^* F \to u^! F$ is a [[strict epimorphism]]. \end{itemize} \end{defn} This appears as (\hyperlink{KontsevichRosenbergSpaces}{Kontsevich-Rosenberg, section 4.2}). \begin{prop} \label{CharacterizationOfFormalNCSmoothness}\hypertarget{CharacterizationOfFormalNCSmoothness}{} Let $R \in Alg_k$ and write $Spec R \in [Alg_k, Set]$ for the corresponding [[representable functor]]. We have that \begin{enumerate}% \item $Spec R$ is an $Alg_k^{inf}$ epipresheaf (formally smooth) precisely if $R$ is Quillen-Cuntz quasi-free: the $R \otimes_k R^{op}$-[[module]] $\Omega^1_{R|k}$, being the [[kernel]] of the multiplication morphism \begin{displaymath} \Omega^1_{R|k} := ker(R \otimes_k R \stackrel{mult}{\to} R) \,, \end{displaymath} is a [[projective object|projective]] in $R \otimes R^{op}$[[Mod]]; \item $Spec R$ is an $Alg_k^{inf}$-monopresheaf (formally unramified) precisely if $\Omega^1_{R|k} = 0$. \end{enumerate} \end{prop} This appears as (\hyperlink{KontsevichRosenbergSpaces}{Kontsevich-Rosenberg, prop. 4.3}). \hypertarget{relation_to_other_concepts}{}\subsection*{{Relation to other concepts}}\label{relation_to_other_concepts} \hypertarget{RelationToCohesiveToposes}{}\subsubsection*{{Relation to cohesive toposes}}\label{RelationToCohesiveToposes} If a Q-category $\mathbb{A}$ has the extra [[right adjoint]] of prop. \ref{SheafConditionByExtraRightAdjoint} and in addition an extra left adjoint to a total of a quadruple of [[adjoint functor]]s \begin{displaymath} \mathbb{A} : \bar A \stackrel{\overset{u_!}{\to}}{\stackrel{\overset{u^*}{\leftarrow}}{\stackrel{\overset{u_*}{\rightarrow}}{\underset{u^!}{\leftarrow}}}} A \end{displaymath} then essential axioms characterizing a [[cohesive topos]] are satisfied, in particular if for instance $\bar A$ and $A$ are [[presheaf topos]]es as in \ref{PresheafQCategories} (this is considered around \hyperlink{KontsevichRosenbergSpaces}{Kontsevich-Rosenberg, 3.5.1}). Notably in this case the canonical [[natural transformation]] \begin{displaymath} u^* \to u^! \end{displaymath} from prop. \ref{SheafConditionByExtraRightAdjoint} is the one appears in the axioms of a [[cohesive topos]]: if this transformation is a [[monomorphism]] in a cohesive topos -- hence if in the language of Q-categories all objects are monopresheaves -- one says that \emph{discrete objects are concrete} in the cohesive topos. Moreover, due to the extra left adjoint $u_!$ there is a canonical dual morphism \begin{displaymath} u_* \to u_! \,. \end{displaymath} In (\hyperlink{Lawvere}{Lawvere}) is the suggestion that it is interesting to consider the full subcategory of $\bar A$ on which $u_* \to u_!$ is an isomorphism. This is dual to the statement of the above section on \hyperlink{ASheaves}{A-Sheaves} which asserts that it is interesting to consider the full subcategory of $A$ on which $u^* \to u^!$ is an isomorphism. More concretely, the axioms of are an abstraction of the situation of prop. \ref{InfinitesimalThickening}. In every cohesive $(\infty,1)$-topos equipped with an \emph{infinitesimal neighbourhood} there is an analog of the characterization of formal smoothness from prop. \ref{FormalSmoothnessByASheaves}. See the section . \hypertarget{sheafification_versus_the_gabriel_localization_}{}\subsubsection*{{Sheafification versus the Gabriel localization $G_{\mathcal{F}} = H^2_{\mathcal{F}}$}}\label{sheafification_versus_the_gabriel_localization_} (\ldots{}) \hypertarget{references}{}\subsection*{{References}}\label{references} The term \emph{$Q$-category} originally was short for \emph{almost quotient category} , presumably because of its role in the theory of [[Gabriel localization]]. The sheaf formalism in terms of $Q$-categories has been introduced in the mimeographed notes \begin{itemize}% \item [[Alexander Rosenberg]], \emph{Q-categories, sheaves and localization}, (in Russian) Seminar on supermanifolds \textbf{25}, Leites ed. Stockholms Universitet 1988 (there is a recent partial English translation) \end{itemize} The formalism is used (and briefly surveyed) in \begin{itemize}% \item [[Maxim Kontsevich]], [[Alexander Rosenberg]], \emph{Noncommutative spaces}, preprint MPI-2004-35 ([[KontsevichRosenbergNCSpaces.pdf:file]], \href{http://www.mpim-bonn.mpg.de/preprints/send?bid=2331}{ps}, \href{http://www.mpim-bonn.mpg.de/preprints/send?bid=2303}{dvi}) \end{itemize} and also used in the general definition of ``noncommutative'' stacks in \begin{itemize}% \item [[Maxim Kontsevich]], [[Alexander Rosenberg]], \emph{Noncommutative stacks}, preprint MPI-2004-37 (\href{http://www.mpim-bonn.mpg.de/preprints/send?bid=2333}{ps}, \href{http://www.mpim-bonn.mpg.de/preprints/send?bid=2305}{dvi}) \end{itemize} The [[epipresheaf]] condition for the Q-category of nilpotent (infinitesimal) thickenings is in the Kontsevich-Rosenberg paper interpreted as [[formally smooth morphism|formal smoothness]] what is further studied in \begin{itemize}% \item [[T. Brzezi?ski]], \emph{Notes on formal smoothness}, \emph{in}: Modules and Comodules (series \emph{Trends in Mathematics}). T Brzeziski, JL Gomez Pardo, I Shestakov, PF Smith (eds), Birkh\"a{}user, Basel, 2008, pp. 113-124 (\href{http://dx.doi.org/10.1007/978-3-7643-8742-6}{doi}, \href{http://arxiv.org/abs/0710.5527}{arXiv:0710.5527}) \end{itemize} The condition that $u_* x \to u_! x$ is an isomorpophism , dual to the condition for $\mathbb{A}$-sheaves considered above, has been considered in \begin{itemize}% \item [[Bill Lawvere]], \emph{Axiomatic cohesion} Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41--49. (\href{http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf}{pdf}) \end{itemize} [[!redirects Q-categories]] \end{document}