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plus construction on presheaves

Context

Topos Theory

topos theory

Background

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Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

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

Contents

Idea

The plus construction () +:PSh(C)PSh(C)(-)^+ : PSh(C) \to PSh(C) on presheaves over a site CC is an operation that replaces a presheaf via local isomorphisms first by a separated presheaf and then by a sheaf.

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

Notice that in terms of n-truncated morphisms, a presheaf is

In the context of (n,1)-topos theory, therefore, the plus-construction is applied (n+1)(n+1)-times in a row. The second but last step makes an (n,1)-presheaf into a separated infinity-stack and then the last step into an actual (n,1)-sheaf. (See Lurie, section 6.5.3.)

Definition

Definition

Let CC be a small site equipped with a Grothendieck topology JJ, let A:C opSetA:C^{op}\to Set be a functor. Then the plus construction (functor) () +:PSh(C)PSh(C)(-)^+ : PSh(C) \to PSh(C), resp. the plus construction A +A^+ of APSh(C)A \in PSh(C) is defined by one of following equivalent descriptions:

  1. A +:Ucolim (RU)J(U)A(R)A^+:U\mapsto colim_{(R\to U)\in J(U)}A(R) where J(U)J(U) denotes the poset of JJ-covering sieves on UU.

  2. Let A:C opSetA:C^{op}\to Set be a functor. Then for UC opU\in C^{op} we define A +(U)A^+(U) to be an equivalence class of pairs (R,s)(R,s) where RJ(U)R\in J(U) and s=(s fA(domf)|fR)s=(s_f\in A(dom f)|f\in R) is a compatible family of elements of AA relative to RR, and (R,s)(R ,s )(R,s)\sim (R^\prime,s^\prime) iff there is a JJ-covering sieve R RR \R^{\prime \prime}\subseteq R\cap R^\prime on which the restrictions of ss and s s^\prime agree.

  3. A +:Ucolim (VU)WA(V)A^+:U\mapsto colim_{(V\hookrightarrow U)\in W}A(V) where WW denotes the class W:=(f *) 1Core(Sh(C) 1)W:=(f^*)^{-1}Core(Sh(C)_1) of those morphisms in PSh(C)PSh(C) which are sent to isomorphisms by the sheafification functor f *f^* and the colimit is taken over all dense monomorphisms only.

Properties

Remark
  1. () +:AA +(-)^+:A\mapsto A^+ is a functor.

  2. A +A^+ is a functor.

  3. A +A^+ is a separated presheaf.

  4. If AA is separated then A +A^+ is a sheaf.

Note that () +:PSh(C)SepPSh(C)(-)^+ : PSh(C) \to SepPSh(C) is not left adjoint to the inclusion ι:SepPSh(C)PSh(C)\iota : SepPSh(C) \hookrightarrow PSh(C) of the full subcategory of separated presheaves. If it were, it would be a reflector and therefore satisfy () +ιId(-)^+ \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.

Internal description

The plus construction can be described in the internal language of the presheaf topos PSh(C)PSh(C). For a presheaf AA, seen as a set from the internal point of view, the separated presheaf A +A^+ is given by the internal expression

A +={KA|j(K is a singleton)}/, A^+ = \{ K \subseteq A \,|\, j(K\,\text{ is a singleton}) \}/{\sim},

where \sim is the equivalence relation given by KLK \sim L if and only if j(K=L)j(K = L) and jj is the Lawvere–Tierney topology describing the subtopos Sh(C)PSh(C)Sh(C) \hookrightarrow PSh(C).

With this internal description, the verification of the properties of the plus construction becomes an exercise with sets and subsets (instead of colimits).

References

Related entries: sheafification

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

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

Plus construction for presheaves in values in abelian categories is also called Heller-Rowe construction:

  • Alex Heller, K. A. Rowe, On the category of sheaves Amer. J. Math. 84 1962 205–216, MR144341, doi
Revised on December 2, 2016 07:26:04 by Ingo Blechschmidt (137.250.162.16)