Simplicial presheaf

<h2 id="homotopy-sheaves-of-a-simplicial-presheaf">Homotopy sheaves of a simplicial presheaf</h2>
<p>Let <i>F</i> be a simplicial presheaf on a site. The homotopy sheaves 
  
    
      
        
          π
          
            ∗
          
        
        F
      
    
    {\displaystyle \pi _{*}F}
  
 of <i>F</i> is defined as follows. For any 
  
    
      
        f
        :
        X
        →
        Y
      
    
    {\displaystyle f:X\to Y}
  
 in the site and a 0-simplex <i>s</i> in <i>F</i>(<i>X</i>), set 
  
    
      
        (
        
          π
          
            0
          
          
            pr
          
        
        F
        )
        (
        X
        )
        =
        
          π
          
            0
          
        
        (
        F
        (
        X
        )
        )
      
    
    {\displaystyle (\pi _{0}^{\text{pr}}F)(X)=\pi _{0}(F(X))}
  
 and 
  
    
      
        (
        
          π
          
            i
          
          
            pr
          
        
        (
        F
        ,
        s
        )
        )
        (
        f
        )
        =
        
          π
          
            i
          
        
        (
        F
        (
        Y
        )
        ,
        
          f
          
            ∗
          
        
        (
        s
        )
        )
      
    
    {\displaystyle (\pi _{i}^{\text{pr}}(F,s))(f)=\pi _{i}(F(Y),f^{*}(s))}
  
. We then set 
  
    
      
        
          π
          
            i
          
        
        F
      
    
    {\displaystyle \pi _{i}F}
  
 to be the sheaf associated with the pre-sheaf 
  
    
      
        
          π
          
            i
          
          
            pr
          
        
        F
      
    
    {\displaystyle \pi _{i}^{\text{pr}}F}
  
.
</p>
<h2 id="model-structures">Model structures</h2>
<p>The category of simplicial presheaves on a site admits many different <a href="/facts/Model_category/KcxFUgY8">model structures</a>.
</p><p>Some of them are obtained by viewing simplicial presheaves as functors 
</p>

S
          
            o
            p
          
        
        →
        
          Δ
          
            o
            p
          
        
        S
        e
        t
        s
      
    
    {\displaystyle S^{op}\to \Delta ^{op}Sets}

<p>The category of such functors is endowed with (at least) three model structures, namely the projective, the Reedy, and the injective model structure. The weak equivalences / fibrations in the first are maps
</p>

F
          
        
        →
        
          
            G
          
        
      
    
    {\displaystyle {\mathcal {F}}\to {\mathcal {G}}}

F
          
        
        (
        U
        )
        →
        
          
            G
          
        
        (
        U
        )
      
    
    {\displaystyle {\mathcal {F}}(U)\to {\mathcal {G}}(U)}

<p>is a weak equivalence / fibration of simplicial sets, for all <i>U</i> in the site <i>S</i>. The injective model structure is similar, but with weak equivalences and cofibrations instead.
</p>
<h2 id="stack">Stack</h2>
<p class="note">Main article: <a href="/facts/Stack_(mathematics)/MkASN1rA">Stack (mathematics)</a></p>
<p>A simplicial presheaf <i>F</i> on a site is called a stack if, for any <i>X</i> and any <a href="/facts/Hypercovering/zSvm5JzP">hypercovering</a> <i>H</i> →<i>X</i>, the canonical map
</p>

F
        (
        X
        )
        →
        holim
        ⁡
        F
        (
        
          H
          
            n
          
        
        )
      
    
    {\displaystyle F(X)\to \operatorname {holim} F(H_{n})}

<p>is a <a href="/facts/Weak_equivalence_(homotopy_theory)/KUWfV7rc">weak equivalence</a> as simplicial sets, where the right is the <a href="/facts/Homotopy_limit/wYTpWPCV">homotopy limit</a> of
</p>

[
        n
        ]
        =
        {
        0
        ,
        1
        ,
        …
        ,
        n
        }
        ↦
        F
        (
        
          H
          
            n
          
        
        )
      
    
    {\displaystyle [n]=\{0,1,\dots ,n\}\mapsto F(H_{n})}
  
.
<p>Any sheaf <i>F</i> on the site can be considered as a stack by viewing 
  
    
      
        F
        (
        X
        )
      
    
    {\displaystyle F(X)}
  
 as a constant simplicial set; this way, the category of sheaves on the site is included as a subcategory to the homotopy category of simplicial presheaves on the site. The inclusion functor has a left adjoint and that is exactly 
  
    
      
        F
        ↦
        
          π
          
            0
          
        
        F
      
    
    {\displaystyle F\mapsto \pi _{0}F}
  
.
</p><p>If <i>A</i> is a sheaf of abelian group (on the same site), then we define 
  
    
      
        K
        (
        A
        ,
        1
        )
      
    
    {\displaystyle K(A,1)}
  
 by doing classifying space construction levelwise (the notion comes from the <a href="/facts/Obstruction_theory/h5T62I8z">obstruction theory</a>)  and set 
  
    
      
        K
        (
        A
        ,
        i
        )
        =
        K
        (
        K
        (
        A
        ,
        i
        −
        1
        )
        ,
        1
        )
      
    
    {\displaystyle K(A,i)=K(K(A,i-1),1)}
  
. One can show (by induction): for any <i>X</i> in the site,
</p>

H
          
            i
          
        
        ⁡
        (
        X
        ;
        A
        )
        =
        [
        X
        ,
        K
        (
        A
        ,
        i
        )
        ]
      
    
    {\displaystyle \operatorname {H} ^{i}(X;A)=[X,K(A,i)]}

<p>where the left denotes a sheaf cohomology and the right the homotopy class of maps.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Cubical_set/bE2tTh6N">cubical set</a></li>
<li><a href="/facts/N-group_(category_theory)/h2XalQ26">N-group (category theory)</a></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="further-reading">Further reading</h2>
<ul><li>Konrad Voelkel, <a href="http://blog.konradvoelkel.de/2012/11/simplicial-presheaves-model/">Model structures on simplicial presheaves</a></li></ul>

<ul><li>Jardine, J.F. (2004). "Generalised sheaf cohomology theories". In Greenlees, J. P. C. (ed.). <i>Axiomatic, enriched and motivic homotopy theory. Proceedings of the NATO Advanced Study Institute, Cambridge, UK, 9--20 September 2002</i>. NATO Science Series II: Mathematics, Physics and Chemistry. Vol. 131. Dordrecht: Kluwer Academic. pp. 29–68. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-4020-1833-9. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1063.55004">1063.55004</a>.</li>
<li>Jardine, J.F. (2007). <a href="http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf">"Simplicial presheaves"</a> (PDF).</li>
<li>B. Toën, <a href="https://wayback.archive-it.org/all/20090625184038/http://www.math.univ-toulouse.fr/~toen/crm-2008.pdf">Simplicial presheaves and derived algebraic geometry</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://uwo.ca/math/faculty/jardine/">J.F. Jardine's homepage</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Toën, Bertrand (2002), "Stacks and Non-abelian cohomology" (PDF), Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory, MSRI <a href="https://perso.math.univ-toulouse.fr/btoen/files/2015/02/msri2002.pdf" target="_blank">https://perso.math.univ-toulouse.fr/btoen/files/2015/02/msri2002.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Jardine 2007, §1 - Jardine, J.F. (2007). "Simplicial presheaves" (PDF). <a href="http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf" target="_blank">http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Simplicial presheaf open-in-new

Simplicial presheaf