plus construction on presheaves


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



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.)



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.


  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.


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 May 8, 2015 06:32:10 by Ingo Blechschmidt (