A $Q$-category is nothing but a coreflective subcategory and a $Q^\circ$-category is nothing but a reflective subcategory. Since both of these encode reflective localizations, following Rosenberg the “Q” 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 $\mathbb{A} := (A \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}} \bar A)$ a $Q^\circ$-category there is canonically induced a quadruple of adjoint functors between the corresponding presheaf categories
and a presheaf $F \in PSh(A)$ is called an $\mathbb{A}$-sheaf if the canonical morphism
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 sieves 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.
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.
An almost quotient category, or a $Q$-category is
a pair of adjoint functors
with $u^*$ the left adjoint;
such that $u^*$ is full and faithful
In other words, $A$ is equipped with an equivalence with a coreflective subcategory of $\bar{A}$.
In motivating classes of examples $\bar A$ and $A$ are toposes and $\mathbb{A}$ is a geometric morphism between them. Therefore one generally speaks of $u_*$ as the direct image and of $u^*$ as the inverse image of the Q-category.
The definition 1 is nothing but the definition of a coreflective subcategory. However the term $Q$-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).
One sometimes write the above data as $\bar A \stackrel{\overset{u}{\leftarrow}}{\to} A$.
A 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
A transformation of morphisms of Q-categories is a pair $(\alpha,\bar{\alpha}):(\Phi,\bar{\Phi},\phi)\to (\Psi,\bar{\Psi},\psi)$ of natural transformations $\alpha:\Phi\to\Psi$ and $\bar{\alpha}:\bar{\Phi}\to\bar{\Psi}$ such that the diagram
Small Q-categories, morphisms of Q-categories and natural transformations of morphisms form a 2-category of small Q-categories.
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 $\bar{A}$.
The category of presheaves over any $Q$-category canonically inherits itself the structure of a Q-category.
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
The $(C^R \dashv C^L)$-unit is $C^\eta$ induced by the original unit $\eta: 1_A\to R L$
and the counit $C^\epsilon$ is induced by the original counit $\epsilon: L R\to 1_{\bar{A}}$
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
is the identity by the triangle identity for $L\dashv R$, but this is precisely the $G$-component of the transformation
Similarly the $F$-component of
for a functor $F\in C^A$ reads
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.
This appears as (Kontsevich-Rosenberg, 2.7).
Assume in the context of prop. 1 that
$A$ and $\bar A$ are small categories;
$C$ has all small limits and colimits (for instance $C$ = Set).
Then there is an adjoint quadruple
where $Lan R$ denotes the left Kan extension along $R$ and $Ran L$ the right Kan extension along $L$.
By general properties of Kan extensions.
Any subcategory $B\subset C^{\bar{A}}$ containing $Im(C^R)$ determines a Q-subcategory $C^A\leftrightarrow B$.
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 functors
Taking this apart, we have that
is a $Q$-category and
is a $Q^o$-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
where the extra right adjoint is given by precomposition with the domain evaluation.
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
commutes and $\alpha \star \rho$ denotes the horizontal composition (= Godement product) of natural transformations.
Then one defines composition of morphisms by the formula
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 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 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 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.
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…
(needs explanation)
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.
(…)
Let $A$ be a category and $\mathcal{T}$ a map that assigns to every object a collection of cosieves on that object (subfunctors of $A(x,-)$), which includes maximal cosieve (functor $A(x,-)$).
Write $\bar A_{\mathcal{T}}$ for the category whose objects are cosieves on $A$, and whose morphisms are morphisms in $A$ that respect the corresponding cosieves. This yields a Q-category
Call this a quasi-cosite if
for any two cosieves in $\mathcal{T}$ their intersection is also in $\mathcal{T}$;
for any cosive in $\mathcal{T}$ any cosieve containing it is also in $\mathcal{T}$.
This is (KontsevichRosenberg, 2.2).
For any Q-category $\mathbb{A}$, the quasi-cosite associated with $\mathbb{A}$ is the Q-category $\mathcal{T}\mathbb{A}$ defined by…
The following is supposed to be the standard quasi-cosite for non-commutative geometry.
Let $k$ be a ring and $A := Alg_k$ the category of associative algebras over $k$. Let $\bar A \subset A^I$ be the full subcategory of the domain fibration whose objects are the faithfully flat morphisms, i.e. those morphisms $\phi : R \to T$ in $Alg_k$ such that the induced
is an 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. 7. This is the standard quasi-cosite for noncommutative geometry.
This is (KontsevichRosenberg, A.1.9.2).
Generally, infinitesimal thickenings are characterized by 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
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 $k$ be a field and $A := CAlg_k$ be the category of commutative associative algebras over $k$.
Let $\bar A \subset A^I$ 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
is a Q-category.
The analogous statement is true with $A = Alg_k$ the category of all associative algebras, not necessarily commutative.
This appears as (Kontsevich-Rosenberg, 2.6).
Here we think of an algebra epimorphism $\phi : \mathbf{B} \to B$ 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
The functor $dom$ remembers the thickened algebra
But notice that we also have the codomain-functor, which is the functor that forgets the thickening
and that the adjoint pair $(\epsilon \dashv dom)$ does extend to an adjoint triple
In the discussion at Infinitesimal cohesion it is the pair $(cod \dashv \epsilon)$ that appears in the axiomatization, or rather its version on the opposite categories
not the other adjoint pair $(\epsilon \dashv dom)$ 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 $\mathbb{A}$-sheaves on a Q-category $\mathbb{A}$.
Let $\mathbb{A} = (\bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}}A)$ be a Q-category. An object $x \in A$ is called an $\mathbb{A}$-sheaf if for all $y \in \bar A$ the canonical morphism
is an isomorphism (in Set, hence a bijection).
This morphism is given by
where $\eta_x : x \to u_* u^*(x)$ is the $x$-component of the $(u^* \dashv u_*)$-counit (which is an isomorphism because $u^*$ is a full and faithful functor).
This appears as (Kontsevich-Rosenberg, 3.1.1).
Let $\mathbb{A}$ be a Q-category with an extra right adjoint $u^! : A \to \bar A$
Then an object $x \in A$ is an $\mathbb{A}$-sheaf in the sense of def. 9 precisely if the canonical morphism
is an isomorphism in $\bar A$.
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 $(u_* \dashv u^!)$-adjunction-isomorphism we have that the canonical morphism from def. 9 is isomorphic to
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. 9 and the expression of adjuncts (as discussed there) by composition with (co)-units, we find that the composite map here sends any morphism
to the composite
Using that adjunction units are natural transformations, we can complete this to a commuting diagram
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.
Let $\mathbb{A} = (u^* \dashv u_* \dashv u^!): \bar A \to A$ be a Q-category with an extra right adjoint as in prop. 3.
We say
an object $x \in A$ is $\mathbb{A}$-monopresheaf if $u^* x \to u^! x$ is a monomorphism in $\bar A$.
an object $x \in A$ is $\mathbb{A}$-epipresheaf if $u^* x \to u^! x$ is an strict epimorphism in $\bar A$.
This appears as Kontsevich-Rosenberg, 3.1.2, 3.1.4.
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 limits. Then the Q-category of presheaves from prop 1 has an extra right adjoint $u^!_C := Ran_{u^*}$
given by the right Kan extension along $u^*$, which exists by the assumption that $C$ has all small limits.
Therefore by prop. 3 a presheaf $F \in C^{A}$ is a $C^{\mathbb{A}}$-sheaf, def. 9, precisely if the canonical morphism
is an isomorphism.
This appears as (Kontsevich-Rosenberg, 3.5).
Let
be the Q-category of infinitesimal thickenings from prop. 2. Write
be the corresponding Q-category of copresheaves from prop. 1. Notice that $[CAlg_k, Set]$ is the presheaf topos that contains schemes over $k$. Recall the notion of formally smooth scheme, formally étale morphism and formally unramified morphism of schemes.
An object $X \in [CAlg_k, Set]$ is
formally étale precisely if it is an $CAlg_k^{inf}$-sheaf in the sense of def. 9, hence precisely if the canonical morphism
from prop. 3 is an isomorphism;
formally unramified precisely if it is a $CAlg_k^{inf}$ monopresheaf, def. 1, hence precisly if $u^* F \to u^! F$ is a monomorphism;
formally smooth precisely if it is a strict epipresheaf, def. 1, hence precisely if $u^* F \to u^! F$ 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 $Alg_k$ be the full category of associative algebras over $k$, not necessarily commutative. Write $Alk_k^{inf} : \bar A \to Alg_k$ for the Q-category of infinitesimal thickenings as in def. 2. Notice that $[Alg_k, Set]$ is the presheaf topos that contains noncommutative schemes.
We then say an object $X \in [Alg_k, Set]$ is
formally étale precisely if it is an $Alg_k^{inf}$-sheaf in the sense of def. 9, hence precisely if the canonical morphism
from prop. 3 is an isomorphism;
formally unramified precisely if it is a $Alg_k^{inf}$ monopresheaf, hence precisly if $u^* F \to u^! F$ is a monomorphism;
formally smooth precisely if $u^* F \to u^! F$ is a strict epimorphism.
This appears as (Kontsevich-Rosenberg, section 4.2).
Let $R \in Alg_k$ and write $Spec R \in [Alg_k, Set]$ for the corresponding representable functor. We have that
$Spec R$ 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
is a projective in $R \otimes R^{op}$Mod;
$Spec R$ is an $Alg_k^{inf}$-monopresheaf (formally unramified) precisely if $\Omega^1_{R|k} = 0$.
This appears as (Kontsevich-Rosenberg, prop. 4.3).
If a Q-category $\mathbb{A}$ has the extra right adjoint of prop. 3 and in addition an extra left adjoint to a total of a quadruple of adjoint functors
then essential axioms characterizing a cohesive topos are satisfied, in particular if for instance $\bar A$ and $A$ are presheaf toposes as in 1 (this is considered around Kontsevich-Rosenberg, 3.5.1).
Notably in this case the canonical natural transformation
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 $u_!$ there is a canonical dual morphism
In (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 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 Infinitesimal cohesion are an abstraction of the situation of prop. 2. In every cohesive $(\infty,1)$-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.
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The term $Q$-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 $Q$-categories has been introduced in the mimeographed notes
The formalism is used (and briefly surveyed) in
and also used in the general definition of “noncommutative” 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
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