nLab plus construction on presheaves

This is a subentry of sheaf about the plus-construction on presheaves. For other constructions called plus construction, see there.

Idea

The plus construction $(-)^+ : PSh(C) \to PSh(C)$ on presheaves over a site $C$ is an operation that makes a presheaf “closer” to being a sheaf.

For ordinary sheaves of sets, the plus construction replaces a presheaf first by a separated presheaf and then by a sheaf.

$$PSh(C) \stackrel{(-)^+}{\to} SepPSh(C) \stackrel{(-)^+}{\to} Sh(C) \,.$$

(The first of these steps, however, is not a reflection into $SepPsh(C)$.) Notice that in terms of n-truncated morphisms, a presheaf is

• separated precisely if every comparison map $A(U) \to Desc(A,U)$ to the set of descent data is (-1)-truncated, namely a monomorphism;

• a sheaf precisely if every such comparison map is (-2)-truncated, namely an equivalence.

More generally, in (n,1)-topos theory, the plus-construction is applied $(n+1)$-times in a row to an (n,1)-presheaf? (a presheaf of $(n-1)$-groupoids), at each step decreasing the truncatedness of the comparison map until at the last step it becomes an actual (n,1)-sheaf, a.k.a. stack; see Anel and Leena Subramaniam, 2020. When $n=\infty$, even a countable sequence of applications does not suffice, but a transfinite one does; see Lurie, section 6.5.3.

Definition

Definition

Let $C$ be a small site equipped with a Grothendieck topology $J$, let $A:C^{op}\to Set$ be a functor. Then the plus construction is a functor $(-)^+ : PSh(C) \to PSh(C)$ defined by the following equivalent descriptions:

1. $A^+(U) = colim_{(R\hookrightarrow y U)\in J(U)} Hom_{Psh(C)}(R,A)$ where $J(U)$ can be defined in any of the following equivalent ways:

   • The poset of covering sieves on $U$ for the topology. (It is important here that we use covering sieves, or at least families closed under composition and identities.)
   • The class of dense monomorphisms with representable codomain $y U$.
   • The class of monomorphisms with codomain $y U$ that are inverted by the sheafification functor. (Of course, this definition is not available unless we have shown that sheafification exists in some other way.)

2. For $U\in C^{op}$ we define $A^+(U)$ to be the set of equivalence classes of pairs $(R,s)$ where $R\in J(U)$ and $s=(s_f\in A(dom f)|f\in R)$ is a compatible family of elements of $A$ relative to $R$, and $(R,s)\sim (R', s')$ iff there is a $J$-covering sieve $R'' \subseteq R \cap R'$ on which the restrictions of $s$ and $s'$ agree.

3. In the internal language of the presheaf topos $PSh(C)$, we can define $A^+ = \{ K \subseteq A \,|\, j(K\,\text{ is a singleton}) \}/{\sim}$, where $\sim$ is the equivalence relation given by $K \sim L$ if and only if $j(K = L)$ and $j$ is the Lawvere-Tierney topology.

4. $A^+ = Lan_{r^{op}} Ran_{s^{op}} A$, where $r : J \to C$ is the Grothendieck construction of the functor $Cov : C^{op} \to Pos$ sending each $U$ to its set of covering sieves, and $s:C\to J$ sends each $U\in C$ to its maximal sieve $M_U$ (the set of all morphisms with codomain $U$).

The first and last of these definitions also work for higher sheaves on higher sites, i.e. when $C$ is an (infinity,1)-site and $A:C^{op} \to \infty Gpd$ is an (infinity,1)-presheaf, as long as limits, colimits, and Kan extensions are interpreted in the $(\infty,1)$-categorical sense.

Construction of sheafification

The basic property of the plus-construction is that we can construct sheafification (a.k.a. stack completion, in the higher-categorical case) by iterating it. For ordinary presheaves of sets, two iterations suffice: for any presheaf $A$, $A^+$ is separated, while if $A$ is separated then $A^+$ is a sheaf; thus $A^{++}$ is the sheafification of any presheaf $A$.

Remark

Note, though, that $(-)^+ : PSh(C) \to SepPSh(C)$ is not left adjoint to the inclusion $\iota : SepPSh(C) \hookrightarrow PSh(C)$ of the full subcategory of separated presheaves. If it were, it would be a reflector and therefore satisfy $(-)^+ \circ \iota \cong Id$. But this is false, since the plus construction applied to separated presheaves yields their sheafification. See this MathOverflow question for details.

More generally, when working with presheaves of $n$-groupoids for some finite $n$, it takes $n+2$ iterations for the plus-construction to converge to a sheaf/stack. In the case $n=\infty$, no finite number of iterations suffices, and not even a countable number, but a transfinite number does.

We now sketch a proof of these facts, using the implicit infinity-category convention so that our results apply to higher sheaves/stacks as well as ordinary ones.

We use the fourth definition above $A^+ = Lan_{r^{op}} Ran_{s^{op}} A$, which we will write more shortly as $A^+ = r_! s_* A$. There is a canonical map $A\to A^+$, which we can more easily construct and study with the following observation:

Lemma

$s:C\to J$ is right adjoint to $r:J\to C$. Therefore, we have an adjunction $r^* : Psh(C) \rightleftarrows Psh(J) : s^*$, and hence an adjoint quadruple:

$$r_! \dashv r^* \dashv s^* \dashv s_*.$$

Moreover, $r^*$ and $s_*$ are fully faithful and $r_!$ preserves finite limits, so $Psh(J)$ is cohesive and even totally connected over $Psh(C)$.

Proof

Since the maximal sieve $M_U$ is the terminal object of the fiber of $r$, the functor $s$ picks out a fiberwise terminal object in the fibration $r$; thus it is right adjoint to $r$. This implies the existence of the adjoint quadruple as discussed here. Since $r s = 1_C$, we have that $s$ is fully faithful, and hence so are $s_*$ and $s_! \cong r^*$. Finally, the fibers of $r^{op}$ are filtered since covering sieves are closed under intersections, so $r_!$ preserves finite limits since finite limits commute with filtered colimits.

Thus, we can identify $Psh(C)$ with either the discrete objects or codiscrete objects in $Psh(J)$, via the fully faithful functors $r^*$ or $s_*$. If we regard $A\in Psh(C)$ as the discrete object $r^*A$, then the plus-construction is similarly identified with $r^* r_! s_* s^* r^* A$. Thus $r^*(A^+)$ can be written as $ʃ \sharp r^* A$, where $ʃ = r^* r_!$ is the reflector into the discrete objects and $\sharp = s_* s^*$ is the reflector into the codiscrete objects.

The rest of the proof can actually be carried out “synthetically” in the internal homotopy type theory of the cohesive topos $Psh(J)$. This argument is formalized in the HoTT/Coq library. But here we formulate the argument externally and categorically.

Lemma

The plus-construction is a well-pointed endofunctor. That is, we have a natural transformation $\eta : A \to A^+$ and a homotopy (an equality in the case $n=0$) $\theta : \eta_{A^+} \sim (\eta_A)^+$. Moreover, the whiskered homotopy $\theta \eta_A$ is homotopic to the naturality square $\eta_{A^+} \circ \eta_A \sim (\eta_A)^+ \circ \eta_A$.

Proof

This follows fairly formally from its identification with $ʃ \sharp$, the composite of two reflectors.

Lemma

The plus-construction preserves finite limits.

Proof

Both functors $r_!$ and $s_*$ preserve finite limits.

Lemma

For $A\in Psh(C)$, the following are equivalent:

1. $A$ is a sheaf.
2. The canonical map $r^* A \to s_* A$ is an equivalence.
3. $r^* A$ is codiscrete.
4. $\eta_A : A \to A^+$ is an equivalence.
5. $\eta_A : A \to A^+$ has a retraction.

Proof

To see that the first two are equivalent, note that for a covering sieve $R$ of $U$ we have $(r^* A)(R) = A(U)$ while $(s_* A)(R)$ is the space of descent data for $R$. Now we have $s_* A \cong s_* s^* r^* A = \sharp r^* A$, so the second and third statements are equivalent.

For the fourth, if $r^*A$ is codiscrete, then it is equivalent to $\sharp r^* A$; hence the latter is also discrete, and thus equivalent to $ʃ \sharp r^* A$. Of course if $\eta_A$ is an equivalence, it has a retraction. Finally, if it has a retraction, then the composite $\sharp r^* A \to ʃ \sharp r^* A = r^* A^+$ is a retraction for $r^* A \to \sharp r^* A$, so that $r^*A$ is codiscrete.

Theorem

If some (perhaps transfinite) iteration of the plus-construction on $A\in Psh(C)$ stabilizes, it is a sheaf and the sheafification of $A$.

Proof

This follows formally from the fact that the plus-construction is a well-pointed endofunctor whose fixed points are the sheaves.

Now since the plus-construction is an accessible endofunctor of a locally presentable category, its iteration always converges at some transfinite stage. It follows that sheafification exists. (Roughly this argument appears in Lurie, section 6.5.3.)

We now want to prove that if $A$ is a presheaf of $n$-groupoids for some finite $n$, the iteration in fact converges already after $n+2$ steps. Though long expected, it seems that this was first actually proven by Anel and Leena Subramaniam, 2020; what follows is inspired by their proof. We will actually prove something slightly more general.

Definition

A presheaf $A\in Psh(C)$ is $n$-separated if the map $\eta_A : A\to A^+$ is n-truncated.

Thus $A$ is a sheaf if and only if it is $(-2)$-separated. Moreover, since the plus-construction preserves finite limits, it preserves truncated objects; thus if $A$ is $n$-truncated (i.e. a presheaf of $n$-groupoids) then so is $A^+$ and hence so is $\eta_A$. Thus, any $n$-truncated presheaf is $n$-separated.

We will prove that if $A$ is $n$-separated, then its plus-construction converges after $n+2$ steps. The central auxiliary definition is:

Definition

A map $f:A\to B$ in $Psh(C)$ is plus-connected if there is a lift in its $\eta$-naturality square. That is, there is a map $h:B \to A^+$ and homotopies $f^+ h \sim \eta_B$ and $h f \sim \eta_A$ that paste to the naturality square $f^+ \eta_A \sim \eta_B f$.

Lemma

For any $A\in Psh(C)$, the map $\eta_A : A\to A^+$ is plus-connected.

Proof

Lemma says precisely that $1_{A^+}$ is a lift in the $\eta$-naturality square for $\eta_A$.

Lemma

If $f:A\to B$ is plus-connected, so is its diagonal $\triangle f : A \to A\times_B A$.

Proof

Since the plus-construction preserves finite limits, it suffices to observe that if a given commutative square has a lift, so does the induced square between its diagonals.

So far everything has been very formal. The concrete input from the fact that we are talking about a Grothendieck topology comes here:

Lemma

If $f:A\to B$ is a monomorphism that is plus-connected, then $f^+ : A^+ \to B^+$ is an equivalence.

Proof

Since the plus-construction preserves finite limits, it preserves monomorphisms; thus it suffices to show that $f^+$ is surjective. An object $x\in B^+(U)$ is determined by a matching family over a covering sieve $R\hookrightarrow y U$, consisting of $x_V \in B(V)$ for all $V\to U$ in $R$. Since $f$ is plus-connected, we have a map $h:B\to A^+$ assigning to each $x_V$ an object $h(x_V) \in A^+(V)$, determined by a matching family over a covering sieve $S_V \hookrightarrow y V$. Now we can compose the sieves $S_V$ and $R$ to obtain a covering sieve $T$ on $U$, such that the restriction of $x$ to $T$ (which is equal to $x$) is in the image of $f^+$.

Lemma

For $n\ge -1$, if $f:A\to B$ is $n$-truncated and plus-connected, then $f^+ : A^+ \to B^+$ is $(n-1)$-truncated.

Proof

By induction. The base case $n=-1$ is the previous lemma. For the inductive step, $f$ is $(n+1)$-truncated and plus-connected, then its diagonal $\triangle f$ is $n$-truncated and plus-connected. Thus, by induction $(\triangle f)^+$ is $(n-1)$-truncated. But since the plus-construction preserves finite limits, $(\triangle f)^+$ is equivalent to $\triangle (f^+)$, so $f^+$ is $n$-truncated.

Theorem

If $A$ is $n$-separated, then its plus-construction converges after $n+2$ steps.

Proof

If $A$ is $n$-separated, then by definition $\eta_A : A\to A^+$ is $n$-truncated, and we have shown it is always plus-connected. Thus, $(\eta_A)^+$ is $(n-1)$-truncated. Hence so is $\eta_{A^+}$, in other words $A^+$ is $(n-1)$-separated.

Thus, by induction, if $A$ is $n$-separated, the $(n+2)$-th iteration of the plus-construction is $(n-(n+2))$-separated, i.e. $(-2)$-separated, i.e. a sheaf.

Remark

Note that we do not need to know that the category $J$ is obtained as the category of covering sieves for a topology on $C$. We only need that there is a projection $r:J\to C$ that is a fibration with cofiltered fibers and a fully faithful right adjoint, plus the result of Lemma . In particular, $J$ could be a “mono-saturated lex modulator” in the sense of ALS, generating a non-topological localization of $Psh(C)$.

Remark

We can also construct left exact localizations of non-presheaf toposes in this way. If $E$ is a topos that is itself a left exact localization of some presheaf topos $Psh(C)$, then a lex modulator on $E$ induces a $J$ as above, and by composing the reflectors $ʃ$ and $\sharp$ with the reflection into $E$ we can make the same argument work to construct a left exact localization of $E$.

Related concepts

• sheafification

• (∞,1)-sheafification / ∞-stackification

References

Related entries: sheafification

A standard textbook reference in the context of 1-topos theory is:

• Peter Johnstone, Sketches of an elephant - a topos theory compendium, Section C.2.2, proof of proposition 2.2.6, p.551

Remarks on the plus-construction in (infinity,1)-topos theory is in section 6.5.3 of

• Jacob Lurie, Higher Topos Theory

The plus-construction for presheaves in values in abelian categories is also called the Heller-Rowe construction:

• Alex Heller, K. A. Rowe, On the category of sheaves

  Amer. J. Math. 84 1962 205–216, MR144341, doi

The plus-construction is studied for higher sheaves, and shown to converge after $n+2$ steps for $n$-presheaves, in

• Mathieu Anel, Chaitanya Leena Subramaniam. Small object arguments, plus-construction, and left-exact localizations. arxiv:2004.00731, 2020.

The above argument is formalized in the internal logic of $Psh(J)$ here:

• Canonical binary meets of subuniverses, https://github.com/HoTT/HoTT/blob/master/theories/Modalities/Meet.v