Topos Theory

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A QQ-category is nothing but a coreflective subcategory and a Q ∘Q^\circ-category is nothing but a reflective subcategory. Since both of these encode reflective localizations, following Rosenberg the β€œQ” is for 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 β„‚:=(Cβ†ͺR←LCΒ―)\mathbb{C} := (C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}} \bar C) a Q ∘Q^\circ-category there is canonically induced a quadruple of adjoint functors between the corresponding presheaf categories

PSh(AΒ―)←u !β†’u *←u *β†’u !PSh(A) PSh(\bar A) \stackrel{\overset{u_!}{\to}}{\stackrel{}{\stackrel{\overset{u^*}{\leftarrow}}{\stackrel{\overset{u_*}{\to}}{\underset{u^!}{\leftarrow}}}}} PSh(A)

and a presheaf F∈PSh(A)F \in PSh(A) is called an 𝔸\mathbb{A}-sheaf if the canonical morphism

u *F→u !F u^* F \to u^! F

is an isomorphism in PSh(AΒ―)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 CΒ―\bar C is a category of sieves on objects in a small category CC that are regarded as being covering but which do not necessarily satisfy the axioms of a Grothendieck topology and not even of a coverage.


The Yoneda embedding is continuous but not cocontinuous functor. Hence the Grothendieck topologies are used to define smaller image than the category of presheaves such that for that embedding some covering cones will stay covering cones. In one case the cones correspond to the Grothendieck topology but more general families of diagrams may be involved. The important properties of categories of diagrams for doing the sheaf theory can be expressed in terms of certain adjoint pairs of functors. As one application this generalization of sheaf theory can also rephrase categorically properties like formal smoothness and formal etaleness of functors, and as another puts the sheafification in a framework for which another special case is a construction of the Gabriel-type noncommutative localization.



An almost quotient category, or a QQ-category is

  • a pair of adjoint functors

    𝔸:(u *⊣u *):AΒ―β†’u *←u *A, \mathbb{A} : (u^* \dashv u_*) : \bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}} A \,,

    with u *u^* the left adjoint;

  • such that u *u^* is full and faithful

In other words, AA is equipped with an equivalence with a coreflective subcategory of AΒ―\bar{A}.

  • In motivating classes of examples AΒ―\bar A and AA are toposes and 𝔸\mathbb{A} is a geometric morphism between them. Therefore one generally speaks of u *u_* as the direct image and of u *u^* as the inverse image of the Q-category.

  • The definition 1 is nothing but the definition of a coreflective subcategory. However the term QQ-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 DD and contravariant functor to DD are synonyms, but a different word refers to a different context and intuition).

  • One sometimes write the above data as A¯→←uA\bar A \stackrel{\overset{u}{\leftarrow}}{\to} A.


A morphism of Q-categories from 𝔸:(u *⊣u *):AΒ―β†’u *←u *A\mathbb{A} : (u^* \dashv u_*) : \bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}} A to 𝔹:(v *⊣v *):BΒ―β†’v *←v *B\mathbb{B} : (v^* \dashv v_*) : \bar B \stackrel{\overset{v^*}{\leftarrow}}{\underset{v_*}{\to}} B is a triple (Ξ¦,Φ¯,Ο•)(\Phi,\bar{\Phi},\phi) where Ξ¦:Aβ†’B\Phi : A\to B, Φ¯:AΒ―β†’BΒ―\bar{\Phi}:\bar{A}\to\bar{B} are functors and Ο•:Ξ¦u *β‡’v *Φ¯\phi:\Phi u_*\Rightarrow v_*\bar{\Phi} is a natural isomorphism of functors. The composition is given by

(Ξ¦,Φ¯,Ο•)∘(Ξ¦β€²,Φ¯′,Ο•β€²)=(ΦΦ′,Φ¯Φ¯′,Ο•Ξ¦Β―β€²βˆ˜Ξ¦Ο•β€²) (\Phi,\bar{\Phi},\phi)\circ(\Phi',\bar{\Phi}',\phi') = (\Phi\Phi',\bar{\Phi}\bar{\Phi}',\phi\bar{\Phi}' \circ \Phi\phi')

A transformation of morphisms of Q-categories is a pair (Ξ±,Ξ±Β―):(Ξ¦,Φ¯,Ο•)β†’(Ξ¨,Ψ¯,ψ)(\alpha,\bar{\alpha}):(\Phi,\bar{\Phi},\phi)\to (\Psi,\bar{\Psi},\psi) of natural transformations Ξ±:Ξ¦β†’Ξ¨\alpha:\Phi\to\Psi and Ξ±Β―:Φ¯→Ψ¯\bar{\alpha}:\bar{\Phi}\to\bar{\Psi} such that the diagram

Ξ¦u * βŸΆΟ• v *Ψ¯ Ξ±u *↓ ↓v *Ξ±Β― Ξ¨u * ⟢ψ v *Ψ¯\array{ \Phi u_* & \stackrel{\phi}\longrightarrow & v_* \bar{\Psi}\\ \alpha u_*\downarrow && \downarrow v_*\bar{\alpha}\\ \Psi u_* &\stackrel{\psi}\longrightarrow& v_* \bar{\Psi} }


Small Q-categories, morphisms of Q-categories and natural transformations of morphisms form a 2-category of small Q-categories.


A Q ∘Q^\circ-category is a pair of functors Q:A↔AΒ―:IQ: A\leftrightarrow\bar{A}: I, where QQ is fully faithful and right adjoint to II. In other words, AA is equipped with an equivalence with a reflective subcategory of AΒ―\bar{A}.


(Co)Presheaves on a QQ-category

The category of presheaves over any QQ-category canonically inherits itself the structure of a Q-category.


Let 𝔸=AΒ―β†’R←LA\mathbb{A} = \bar A \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} A be a QQ-category. Let CC be any category.

Then forming copresheaves with values in CC yields a Q-category of the form

[𝔸,C]:[AΒ―,C]β†’C L=(βˆ’)∘L←C R=(βˆ’)∘R[A,C]. [\mathbb{A},C] : [\bar A, C] \stackrel{\overset{C^R = (-)\circ R}{\leftarrow}}{\underset{C^L = (-)\circ L}{\to}} [A,C] \,.

The (C R⊣C L)(C^R \dashv C^L)-unit is C Ξ·C^\eta induced by the original unit Ξ·:1 Aβ†’RL\eta: 1_A\to R L

C Ξ·:Id C Aβ†’C L∘C R=C RL C^{\eta} : Id_{C^A} \to C^L \circ C^R = C^{R L}

and the counit C ϡC^\epsilon is induced by the original counit ϡ:LR→1 A¯\epsilon: L R\to 1_{\bar{A}}

C Ο΅:C R∘C L=C LRβ†’Id C AΒ―. C^\epsilon : C^R\circ C^L = C^{L R}\to Id_{C^{\bar{A}}} \,.

The only thing is who is adjoint – now C RC^R is the left adjoint.

The triangle identities for C R⊣C LC^R\dashv C^L can be obtained by expanding. For R:AΒ―β†’AR: \bar{A}\to A, one has C R:C Aβ†’C AΒ―C^R : C^A\to C^{\bar{A}} is given by C R:G↦GRC^R : G\mapsto GR, and for L:Aβ†’AΒ―L:A\to\bar{A} one has C L:F↦FLC^L:F\mapsto F L. Then C Ξ·:Id C Aβ†’C RC L=C LRC^\eta : Id_{C^A}\to C^R C^L = C^{LR} has the components (C Ξ·) G:(Id C A)(G)β‡’C RC L(G)(C^\eta)_G : (Id_{C^A})(G) \Rightarrow C^R C^L (G) given by GΞ·:Gβ†’GLRG \eta : G\to G L R. Thus for each functor G∈C AΒ―G\in C^{\bar{A}}, the composition

GR⟢GηRGRLR⟢GRηGRG R\stackrel{G\eta R}\longrightarrow G R L R \stackrel{G R \eta}\longrightarrow G R

is the identity by the triangle identity for L⊣RL\dashv R, but this is precisely the GG-component of the transformation

C R⟢C RC ηC RC LC R⟢C ϡC RC R. C^R \stackrel{C^R C^\eta}\longrightarrow C^R C^L C^R \stackrel{C^\epsilon C^R}\longrightarrow C^R.

Similarly the FF-component of

C L⟢C ηC LC LC RC L⟢C LC ϡC L, C^{L} \stackrel{C^\eta C^L}\longrightarrow C^L C^R C^L \stackrel{C^L C^\epsilon}\longrightarrow C^L,

for a functor F∈C AF\in C^A reads

FL⟢FLηFLRL⟢FϡLFL F L \stackrel{F L \eta}\longrightarrow F L R L \stackrel{F \epsilon L}\longrightarrow F L

and this composition equals Id FLId_{F L} by another triangle identity for L⊣RL\dashv R.

It is clear that if Ξ·\eta is iso then the composition with Ξ·\eta is also iso. Thus we obtain a QQ-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 LL being full and faithful that C RC^R is full and faithful.

This appears as (Kontsevich-Rosenberg, 2.7).


Assume in the context of prop. 1 that

Then there is an adjoint quadruple

(u !⊣u *⊣u *⊣u !):[AΒ―,C]←u !=RanLβ†’u *=C L←u *=C Rβ†’u !=LanR[A,C], (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] \,,

where LanRLan R denotes the left Kan extension along RR and RanLRan L the right Kan extension along LL.


By general properties of Kan extensions.


Any subcategory BβŠ‚C AΒ―B\subset C^{\bar{A}} containing Im(C R)Im(C^R) determines a Q-subcategory C A↔BC^A\leftrightarrow B.

Domain and codomain fibration


For any category AA, write A¯:=A I\bar A := A^I for its arrow category. This comes equipped with the codomain fibration cod:A¯→Acod : \bar A \to A and the domain cofibration dom:A¯→Adom : \bar A \to A. Both have a joint section ϡ:A→A¯\epsilon : A \to \bar A by a full and faithful functor that assigns identity morphisms. These form a triple of adjoint functors

(codom⊣ϡ⊣dom):A Iβ†’dom←ϡ→codomA. (codom \dashv \epsilon \dashv dom) : A^I \stackrel{\overset{codom}{\to}}{\stackrel{\overset{\epsilon}{\leftarrow}}{\underset{dom}{\to}}} A \,.

Taking this apart, we have that

A dom:A Iβ†’dom←ϡA A^{dom} : A^I \stackrel{\overset{\epsilon}{\leftarrow}}{\underset{dom}{\to}} A

is a QQ-category and

A dom:A I←ϡ→codA A^{dom} : A^I \stackrel{\overset{cod}{\to}}{\underset{\epsilon}{\leftarrow}} A

is a Q oQ^o-category.

This appears as (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 below.


Passing to copresheaf Q-categories as in prop. 1 we have the Q-category

[A I,Set]←u !=(βˆ’)∘codomβ†’u *=(βˆ’)βˆ˜Ο΅β†u *=(βˆ’)∘dom[A,Set] [A^I,Set] \stackrel{\overset{u^* = (-)\circ dom}{\leftarrow}}{\stackrel{\overset{u_* = (-)\circ \epsilon}{\to}}{\overset{u^! = (-)\circ codom}{\leftarrow}}} [A,Set]

where the extra right adjoint is given by precomposition with the domain evaluation.

Q-category of cones

Let CC be a category and LCLC be the category whose objects are cones Ξ±:xβ†’d\alpha: x\to d over (small) diagrams d:Dβ†’Cd: D\to C where DD are variable small categories; and the morphisms from xβ†’Ξ±dx\stackrel{\alpha}\to d to xβ€²β†’Ξ±β€²dβ€²x'\stackrel{\alpha'}\to d' are triples of the form (f,ρ,fΒ―)(f,\rho,\bar{f}) where f:xβ†’xβ€²f:x\to x' is a morphism in CC, ρ:Dβ€²β†’D\rho : D' \to D is a diagram (= functor), and fΒ―:dβˆ˜Οβ†’dβ€²\bar{f}:d \circ\rho \to d' is a morphism of diagrams (= natural transformation) such that

x β†’f xβ€² α⋆ρ↓ ↓α′ d∘ρ β†’fΒ― dβ€²\array{ x & \stackrel{f}\to & x'\\ \alpha \star\rho \downarrow && \downarrow\alpha'\\ d \circ \rho&\stackrel{\bar{f}}\to & d' }

commutes and α⋆ρ\alpha \star \rho denotes the horizontal composition (= Godement product) of natural transformations.

Then one defines composition of morphisms by the formula

(f 1,ρ 1,f 1Β―)∘(f 2,ρ 2,f 2Β―)=def(f 1∘f 2,ρ 2∘ρ 1,f 1¯∘(f 2¯⋆ρ 1)). (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)).

There is a fully faithful functor Q C:Cβ†’LCQ_C:C\to LC that to any x∈Cx\in C assigns the trivial cone id x:xβ†’xid_x :x\to x and to any morphism the corresponding morphism of trivial cones. Its right adjoint is the morphism I C:LCβ†’CI_C:LC\to C defined by sending the cone Ξ±:xβ†’d\alpha: x\to d over a diagram d:Dβ†’Cd:D\to C its vertex xx and to a cone morphism (f,ρ,fΒ―)(f,\rho,\bar{f}), the morphism of vertices ff. Then I C∘Q C=Id CI_C\circ Q_C = Id_C. The identity transformation can be thus taken as the unit of the adjunction. The counit of the adjunction Ο΅:Q C∘I Cβ†’Id LC\epsilon: Q_C\circ I_C \to Id_{LC} is constructed as follows: to a cone Ξ±:xβ†’d\alpha:x\to d assign the morphism (1 x,const,Ξ±)(1_x, const, \alpha) where const:Dβ†’Cconst: D\to C is the constant diagram which is the unique diagram from D=dom(d)D = dom(d) to the final category 1={⋆}1=\{\star\}. One can check that these data indeed define an adjoint pair Q C⊣I CQ_C\dashv I_C of functors. Q C:C↔I C:LCQ_C: C\leftrightarrow I_C: LC is therefore a Q-category, and it is called the Q-category of cones.

If β„’βŠ‚Cat\mathcal{L}\subset Cat is a family of small categories, then one considers the full subcategory L β„’CL_{\mathcal{L}}C of cones whose domains are in β„’\mathcal{L}; the rest of the construction restricts to obtain a Q-category Q C β„’:C↔L β„’C:I C β„’Q^{\mathcal{L}}_C : C\leftrightarrow L_{\mathcal{L}}C : I_C^{\mathcal{L}}.

The most classical case is when β„’\mathcal{L} is the (say skeletal) category DisDis of small discrete categories (=just identity morphisms), one obtains then the Q-category Q C Dis:C↔L DisCQ^{Dis}_C: C\leftrightarrow L_{Dis}C. A semicosite (or semicositus pl. semicositi) is a Q-category of the form C↔CΒ―C\leftrightarrow \bar{C} where CΒ―\bar{C} is a full subcategory of L DisCL_{Dis}C and the adjoint pair is obtained by the restriction. A semicosite is a precosite (=Grothendieck precotopology) if

(i) id x∈Ob(C¯)id_x\in Ob(\bar{C}) whenever x∈Ob(C)x\in Ob(C).

(ii) {xβ†’Ο• ix i} i∈I∈Ob(CΒ―)\{x\stackrel{\phi_i}\to x_i\}_{i\in I}\in Ob(\bar{C}) and {x iβ†’Ο• ijx ij} j∈J i∈Ob(CΒ―)\{x_i\stackrel{\phi_{ij}}\to x_{ij}\}_{j\in J_i}\in Ob(\bar{C}) then {xβ†’Ο• ijβˆ˜Ο• ix ij}∈Ob(CΒ―)\{x\stackrel{\phi_{ij}\circ\phi_i}\to x_{ij}\}\in Ob(\bar{C})

(iii) {xβ†’x i} i∈I∈Ob(CΒ―)\{x\to x_i\}_{i\in I}\in Ob(\bar{C}) and g∈C(x,y)g\in C(x,y), then the family of pushouts {y↦x i∐ xy} i∈I\{y\mapsto x_i\coprod_x y\}_{i\in I} exists and belongs to Ob(CΒ―)Ob(\bar{C}).

An example of a cosite is a cosite of closed sets of a topological space.


The QQ-category of sieves.

The QQ-subcategory of the QQ-category of (all) sieves corresponding to the subcategory of sieves corresponding to the Grothendieck topology…

(needs explanation)

The Q-category factoring a fully faithful factor

Any fully faithful functor among small categories F:Aβ†’BF: A\to B factors canonically into the composition Aβ†’u *AΒ―β†ͺBA\stackrel{u^*}\to \bar{A}\hookrightarrow B where AΒ―βŠ‚B\bar{A}\subset B is the full subcategory of BB whose objects are all bb in ObBOb B such that a↦B(F(a),b)a\mapsto B(F(a),b) is a representable functor A opβ†’SetA^{op}\to Set, and u *u^* is the corestriction of FF to AΒ―\bar{A}. This corestriction makes sense: FF is fully faithful, hence B(F(a),F(a))=B(a,a)B(F(a),F(a)) = B(a,a), i.e. F(a)∈AΒ―F(a)\in \bar{A} for all aa in ObAOb A. For each b∈AΒ―b\in \bar{A}, define now u *(b)u_*(b) as the functor representing B(F(βˆ’),b)B(F(-),b), i.e. by AΒ―(u *(a),b)=B(F(a),b)β‰…B(a,u *(b))\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 *⊣u *u^*\dashv u_* with u *u^* fully faithful.




Let AA be a category and 𝒯\mathcal{T} a map that assigns to every object a collection of cosieves on that object (subfunctors of A(x,βˆ’)A(x,-)), which includes maximal cosieve (functor A(x,βˆ’)A(x,-)).

Write AΒ― 𝒯\bar A_{\mathcal{T}} for the category whose objects are cosieves on AA, and whose morphisms are morphisms in AA that respect the corresponding cosieves. This yields a Q-category

AΒ― 𝒯→←A. \bar A_{\mathcal{T}} \stackrel{\leftarrow}{\to} A \,.

Call this a quasi-cosite if

  1. for any two cosieves in 𝒯\mathcal{T} their intersection is also in 𝒯\mathcal{T};

  2. for any cosive in 𝒯\mathcal{T} any cosieve containing it is also in 𝒯\mathcal{T}.

This is (KontsevichRosenberg, 2.2).


For any Q-category 𝔸\mathbb{A}, the quasi-cosite associated with 𝔸\mathbb{A} is the Q-category 𝒯𝔸\mathcal{T}\mathbb{A} defined by…

The following is supposed to be the standard quasi-cosite for non-commutative geometry.


Let kk be a ring and A:=Alg kA := Alg_k the category of associative algebras over kk. Let AΒ―βŠ‚A I\bar A \subset A^I be the full subcategory of the domain fibration whose objects are the faithfully flat morphisms, i.e. those morphisms Ο•:Rβ†’T\phi : R \to T in Alg kAlg_k such that the induced

ϕ *:RMod→TMod \phi^* : R Mod \to T Mod

is an exact and full and faithful functor. This forms a Q-category.

Write 𝒯Alg k\mathcal{T}Alg_k for the quasi-cosite associated with this Q-category by def. 7. This is the standard quasi-cosite for noncommutative geometry.

This is (KontsevichRosenberg, A.1.9.2).

Infinitesimal thickenings

Generally, infintiesimal thickenings are characterized by coreflective embeddings:

A characteristic property of an infinitesimally thickened point DD is that for any object XX without infinitesimal thickening, there are in general nontrivial morphisms D→XD \to X, but there is only a unique morphism X→DX \to D, reflecting the fact that DD has only a single global point. Thus if by i:C→C¯i : C \to \bar C we denote the inclusion of objects XX without infinitesimal thickening into the collection of possibly infinitesimally thickened objects, and by p:C¯→Cp : \bar C \to C the projection that contracts away the infinitesimal extension, we have indeed an adjunction

CΒ―(i(X),D)≃C(X,p(D))≃C(X,*)≃*. \bar C(i(X), D) \simeq C(X, p(D)) \simeq C(X,*) \simeq * \,.

This general concept is described at infinitesimal neighbourhood site. See also the discussion below at Relation to cohesive toposes.

The following is one realization of this general concept.


Let kk be a field and A:=CAlg kA := CAlg_k be the category of commutative associative algebras over kk.

Let AΒ―βŠ‚A I\bar A \subset A^I be the full subcategory of the codomain fibration Q-category from prop. 5 on those morphisms of commutative algebras which are epimorphisms and whose kernel is nilpotent. Then

CAlg k inf=(ϡ⊣dom):AΒ―β†’dom←ϡA CAlg_k^{inf} = (\epsilon \dashv dom) : \bar A \stackrel{\overset{\epsilon}{\leftarrow}}{\underset{dom}{\to}} A

is a Q-category.

The analogous statement is true with A=Alg kA = Alg_k the category of all associative algebras, not necessarily commutative.

This appears as (Kontsevich-Rosenberg, 2.6).


Here we think of an algebra epimorphism Ο•:Bβ†’B\phi : \mathbf{B} \to B with nilpotent kernel – an infinitesimal ring extension – as the infinitesimal thickening of SpecBSpec B by SpeckerΟ•Spec ker \phi to SpecBSpec \mathbf{B}.

The functor Ο΅\epsilon builds the trivial (empty) infinitesimal thickenings

ϡB:B→id BB. \epsilon B : B \stackrel{id_B}{\to} B \,.

The functor domdom remembers the thickened algebra

dom(B→B)=B. dom (\mathbf{B} \to B) = \mathbf{B} \,.

But notice that we also have the codomain-functor, which is the functor that forgets the thickening

cod(B→B)=B cod (\mathbf{B} \to B) = B

and that the adjoint pair (ϡ⊣dom)(\epsilon \dashv dom) does extend to an adjoint triple

CAlg k inf:AΒ―β†’dom←ϡ→codA. CAlg_k^{inf} : \bar A \stackrel{\overset{cod}{\to}}{\stackrel{\overset{\epsilon}{\leftarrow}}{\underset{dom}{\to}}} A \,.

In the discussion at Infinitesimal cohesion it is the pair (cod⊣ϡ)(cod \dashv \epsilon) that appears in the axiomatization, or rather its version on the opposite categories

(Ο΅ op⊣cod op):AΒ― opβ†’cod op←ϡ opA op, (\epsilon^{op} \dashv cod^{op}) : \bar A^{op} \stackrel{\overset{\epsilon^{op}}{\leftarrow}}{\underset{cod^{op}}{\to}} A^{op} \,,

not the other adjoint pair (ϡ⊣dom)(\epsilon \dashv dom) used here.

This is the reason for the shift in adjoint triples that is mentioned in this remark over at infinitesimal cohesion .


We discuss the notion of a 𝔸\mathbb{A}-sheaves on a Q-category 𝔸\mathbb{A}.


Let 𝔸=(AΒ―β†’u *←u *A)\mathbb{A} = (\bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}}A) be a Q-category. An object x∈Ax \in A is called an 𝔸\mathbb{A}-sheaf if for all y∈AΒ―y \in \bar A the canonical morphism

AΒ―(y,u *(x))β†’A(u *(y),x) \bar A(y, u^*(x)) \to A(u_*(y), x)

is an isomorphism (in Set, hence a bijection).

This morphism is given by

g↦η x βˆ’1∘u *(g) g \mapsto \eta_x^{-1} \circ u_*(g)

where Ξ· x:xβ†’u *u *(x)\eta_x : x \to u_* u^*(x) is the xx-component of the (u *⊣u *)(u^* \dashv u_*)-counit (which is an isomorphism because u *u^* is a full and faithful functor).

This appears as (Kontsevich-Rosenberg, 3.1.1).


Let 𝔸\mathbb{A} be a Q-category with an extra right adjoint u !:Aβ†’AΒ―u^! : A \to \bar A

𝔸:A¯←u !β†’u *←u *A. \mathbb{A} : \bar A \stackrel{\overset{u^*}{\leftarrow}}{\stackrel{\overset{u_*}{\to}}{\underset{u^!}{\leftarrow}}} A \,.

Then an object x∈Ax \in A is an 𝔸\mathbb{A}-sheaf in the sense of def. 9 precisely if the canonical morphism

(u *(x)β†’u !(x)):=u *(X)β†’u !u *u *(x)→≃u !(x) (u^*(x) \to u^!(x)) := u^*(X) \stackrel{}{\to} u^! u_* u^* (x) \stackrel{\simeq}{\to} u^! (x)

is an isomorphism in AΒ―\bar A.

This appears as (Kontsevich-Rosenberg, 3.1.3).

For more details on the canonical morphism appearing here see the section Adjoint quadruples at cohesive topos.


Using the (u *⊣u !)(u_* \dashv u^!)-adjunction-isomorphism we have that the canonical morphism from def. 9 is isomorphic to

Ο•:AΒ―(y,u *x)β†’A(u *y,x)→≃AΒ―(y,u !x), \phi : \bar A(y, u^* x) \to A(u_* y, x) \stackrel{\simeq}{\to} \bar A(y, u^! x) \,,

for all y∈A¯y \in \bar A, where the second map sends every morphism to its adjunct. Using the definition of the first morphism from def. 9 and the expression of adjuncts (as discussed there) by composition with (co)-units, we find that the composite map here sends any morphism

g:y→u *x g : y \to u^* x

to the composite

y ↓ u !u *y β†’u !u *g u !u *u *x ↓ u !x. \array{ y \\ \downarrow \\ u^! u_* y &\stackrel{u^! u_* g}{\to}& u^! u_* u^* x \\ && \downarrow \\ && u^! x } \,.

Using that adjunction units are natural transformations, we can complete this to a commuting diagram

y β†’g u *x ↓ ↓ u !u *y β†’u !u *g u !u *u *x ↓ u !x. \array{ y &\stackrel{g}{\to}& u^* x \\ \downarrow && \downarrow \\ u^! u_* y &\stackrel{u^! u_* g}{\to}& u^! u_* u^* x \\ && \downarrow \\ && u^! x } \,.

This shows that ϕ\phi acts on any gg by postcomposition with the canonical morphism u *x→u !xu^* x \to u^! x. By the Yoneda lemma it follows that ϕ\phi is an isomorphism for all yy precisely if u *x→u !u^* x \to u^! is an isomorphism.


Let 𝔸=(u *⊣u *⊣u !):AΒ―β†’A\mathbb{A} = (u^* \dashv u_* \dashv u^!): \bar A \to A be a Q-category with an extra right adjoint as in prop. 3.

We say

  • an object x∈Ax \in A is 𝔸\mathbb{A}-monopresheaf if u *xβ†’u !xu^* x \to u^! x is a monomorphism in AΒ―\bar A.

  • an object x∈Ax \in A is 𝔸\mathbb{A}-epipresheaf if u *xβ†’u !xu^* x \to u^! x is an strict epimorphism in AΒ―\bar A.

This appears as Kontsevich-Rosenberg, 3.1.2, 3.1.4.


Let 𝔸:AΒ―β†’u *←u *A\mathbb{A} : \bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}} A be a Q-category where AA is a small category, and let CC be a category with all small limits. Then the Q-category of presheaves from prop 1 has an extra right adjoint u C !:=Ran u *u^!_C := Ran_{u^*}

C 𝔸:C A¯←f !:=u C !β†’f *:=C u *←f *:=C u *C A 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}

given by the right Kan extension along u *u^*, which exists by the assumption that CC has all small limits.

Therefore by prop. 3 a presheaf F∈C AF \in C^{A} is a C 𝔸C^{\mathbb{A}}-sheaf, def. 9, precisely if the canonical morphism

f *F→f !F f^* F \to f^! F

is an isomorphism.

This appears as (Kontsevich-Rosenberg, 3.5).


Formal smoothness and Alg k infAlg_k^{inf}-sheaves


CAlg k inf:AΒ―β†’p←iCAlg k CAlg_k^{inf} : \bar A \stackrel{\overset{i}{\leftarrow}}{\underset{p}{\to}} CAlg_k

be the Q-category of infinitesimal thickenings from prop. 2. Write

[CAlg k inf,Set]:[AΒ―,Set]←u !β†’u *←u *[CAlg k,Set] [CAlg_k^{inf},Set] : [\bar A,Set] \stackrel{\overset{u^*}{\leftarrow}}{\stackrel{\overset{u_*}{\to}}{\underset{u^!}{\leftarrow}}} [CAlg_k,Set]

be the corresponding Q-category of copresheaves from prop. 1. Notice that [CAlg k,Set][CAlg_k, Set] is the presheaf topos that contains schemes over kk. Recall the notion of formally smooth scheme, formally Γ©tale morphism and formally unramified morphism of schemes.


An object X∈[CAlg k,Set]X \in [CAlg_k, Set] is

  • formally Γ©tale precisely if it is an CAlg k infCAlg_k^{inf}-sheaf in the sense of def. 9, hence precisely if the canonical morphism

    u *F→u !F u^* F \to u^! F

    from prop. 3 is an isomorphism;

  • formally unramified precisely if it is a CAlg k infCAlg_k^{inf} monopresheaf, def. 1, hence precisly if u *Fβ†’u !Fu^* F \to u^! F is a monomorphism;

  • formally smooth precisely if it is a strict epipresheaf, def. 1, hence precisely if u *Fβ†’u !Fu^* F \to u^! F is a strict epimorphism.

This appears as (Kontsevich-Rosenberg, 4.1).

This category theoretic reformulation of these three properties therefore admits straightforward generalization of these notions to other contexts. See the section Infinitesimal paths at cohesive (∞,1)-topos.

For instance we have the following direct generalization is of interest in noncommutative geometry.


Let Alg kAlg_k be the full category of associative algebras over kk, not necessarily commutative. Write Alk k inf:AΒ―β†’Alg kAlk_k^{inf} : \bar A \to Alg_k for the Q-category of infinitesimal thickenings as in def. 2. Notice that [Alg k,Set][Alg_k, Set] is the presheaf topos that contains noncommutative schemes.

We then say an object X∈[Alg k,Set]X \in [Alg_k, Set] is

This appears as (Kontsevich-Rosenberg, section 4.2).


Let R∈Alg kR \in Alg_k and write SpecR∈[Alg k,Set]Spec R \in [Alg_k, Set] for the corresponding representable functor. We have that

  1. SpecRSpec R is an Alg k infAlg_k^{inf} epipresheaf (formally smooth) precisely if RR is Quillen-Cuntz quasi-free: the RβŠ— kR opR \otimes_k R^{op}-module Ξ© R|k 1\Omega^1_{R|k}, being the kernel of the multiplication morphism

    Ξ© R|k 1:=ker(RβŠ— kRβ†’multR), \Omega^1_{R|k} := ker(R \otimes_k R \stackrel{mult}{\to} R) \,,

    is a projective in RβŠ—R opR \otimes R^{op}Mod;

  2. SpecRSpec R is an Alg k infAlg_k^{inf}-monopresheaf (formally unramified) precisely if Ξ© R|k 1=0\Omega^1_{R|k} = 0.

This appears as (Kontsevich-Rosenberg, prop. 4.3).

Relation to other concepts

Relation to cohesive toposes

If a Q-category 𝔸\mathbb{A} has the extra right adjoint of prop. 3 and in addition an extra left adjoint to a total of a quadruple of adjoint functors

𝔸:A¯←u !β†’u *←u *β†’u !A \mathbb{A} : \bar A \stackrel{\overset{u_!}{\to}}{\stackrel{\overset{u^*}{\leftarrow}}{\stackrel{\overset{u_*}{\rightarrow}}{\underset{u^!}{\leftarrow}}}} A

then essential axioms characterizing a cohesive topos are satisfied, in particular if for instance AΒ―\bar A and AA are presheaf toposes as in 1 (this is considered around Kontsevich-Rosenberg, 3.5.1).

Notably in this case the canonical natural transformation

u *β†’u ! u^* \to u^!

from prop. 3 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 discrete objects are concrete in the cohesive topos.

Moreover, due to the extra left adjoint u !u_! there is a canonical dual morphism

u *β†’u !. u_* \to u_! \,.

In (Lawvere) is the suggestion that it is interesting to consider the full subcategory of AΒ―\bar A on which u *β†’u !u_* \to u_! is an isomorphism. This is dual to the statement of the above section on A-Sheaves which asserts that it is interesting to consider the full subcategory of AA on which u *β†’u !u^* \to u^! is an isomorphism.

More concretely, the axioms of Infinitesimal cohesion are an abstraction of the situation of prop. 2. In every cohesive (∞,1)(\infty,1)-topos equipped with an infinitesimal neighbourhood there is an analog of the characterization of formal smoothness from prop. 4. See the section Infinitesimal paths and de Rham spaces.

Sheafification versus the Gabriel localization G β„±=H β„± 2G_{\mathcal{F}} = H^2_{\mathcal{F}}



The term QQ-category originally was short for almost quotient category , presumably because of its role in the theory of Gabriel localization. The sheaf formalism in terms of QQ-categories has been introduced in the mimeographed notes

  • Alexander Rosenberg, Q-categories, sheaves and localization, (in Russian) Seminar on supermanifolds 25, Leites ed. Stockholms Universitet 1988 (there is a recent partial English translation)

The formalism is used (and briefly surveyed) in

and also used in the general definition of β€œnoncommutative” stacks in

The epipresheaf condition for the Q-category of nilpotent (infinitesimal) thickenings is in the Kontsevich-Rosenberg paper interpreted as formal smoothness what is further studied in

  • T. BrzeziΕ„ski, Notes on formal smoothness, in: Modules and Comodules (series Trends in Mathematics). T BrzeziΕ„ski, JL Gomez Pardo, I Shestakov, PF Smith (eds), BirkhΓ€user, Basel, 2008, pp. 113-124 (doi, arXiv:0710.5527)

The condition that u *xβ†’u !xu_* x \to u_! x is an isomorpophism , dual to the condition for 𝔸\mathbb{A}-sheaves considered above, has been considered in

  • Bill Lawvere, Axiomatic cohesion Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. (pdf)











Revised on February 27, 2014 08:49:01 by Rafael MrΔ‘en? (