User - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T13:24:27Z http://mathoverflow.net/feeds/user/9109 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43934/equivalence-of-two-definitions-of-sheaves-on-a-site Equivalence of two definitions of sheaves on a site shirker 2010-10-28T04:42:52Z 2010-10-31T11:06:01Z <p>I'd like to prove that two definitions of sheaves on a site are equivalent, but I'm having trouble proving one direction.</p> <p>Let $C$ be a category with pullbacks. Let $(C,T)$ be a site defined through a collection $\Phi$ of covering families. (So that a subfunctor of $Hom(-,U)$ is a covering sieve in $T$ if and only if it contains a covering family in $\Phi$).</p> <p>Let $F$ be a presheaf on $C$, taking values in some complete category.</p> <p>Then it should be the case that the following are equivalent characterisations of $F$ being a sheaf.</p> <p>1) For each $U\in C$, and each covering sieve $R \subset Hom(-,U)$, the map $F(U) \to {{\lim \atop \leftarrow} \atop {V\to U \in R}} F(V)$ is an isomorphism.</p> <p>2) The following diagram is an equalizer for each $U$ in $C$, and covering family {$U_\alpha \to U$} in $\Phi$: $$F(U) \to \prod_\gamma F(U_\gamma) {\rightarrow \atop \rightarrow} \prod_{(\alpha,\beta)} F(U_\alpha \times_U U_\beta)$$</p> <p><strong>Question:</strong> How do you prove $2) \implies 1)$?. (I've proved $1) \implies 2)$)</p> <p>Here's my proof attempt of $2) \implies 1)$ so far: </p> <p>$\bullet$ Suppose we have a covering family {$U_\alpha \to U$} in $\Phi$,and that it generates a sieve $S$. I proved that the following is an equalizer:</p> <p>$${{\lim \atop \leftarrow} \atop {V\to U \in S}} F(V) \to \prod_\gamma F(U_\gamma) {\rightarrow \atop \rightarrow} \prod_{(\alpha,\beta)} F(U_\alpha \times_U U_\beta)$$</p> <p>(This is also what I used to prove $1) \implies 2)$). </p> <p>$\bullet$ Let $R$ be an arbitrary covering sieve, which is a subfunctor of $Hom(-,U)$. Then it contains a covering family {$U_\alpha \to U$}, and the diagram of $2)$ is an equalizer by assumption. If $S$ is the covering sieve generated by the covering family (so $S \subset R$), the diagram above is also an equalizer. By an isomorphism of equalizers, the map $F(U) \to {{\lim \atop \leftarrow} \atop {V\to U \in S}} F(V)$ is an isomorphism. </p> <p>Because of the commutative diagram</p> <p>$F(U) \to {{\lim \atop \leftarrow} \atop {V\to U \in R}} F(V)$</p> <p>$\ \ \ \ \ \ \cong \searrow \ \ \ \ \ \downarrow$</p> <p>$\ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\lim \atop \leftarrow} \atop {V\to U \in S}} F(V)$</p> <p>we get that the map $F(U) \to {{\lim \atop \leftarrow} \atop {V\to U \in R}} F(V)$ is a split monic. From here, I'm not sure how to show that it's also epi.</p> <p>So the crux of my issue is that I can prove the isomorphism in $1)$ only for covering sieves generated by covering families, but not for arbitrary covering sieves.</p> http://mathoverflow.net/questions/38748/commutativity-of-a-diagram-of-boundary-morphisms-from-the-long-exact-sequence-of Commutativity of a diagram of boundary morphisms from the long exact sequence of homotopy groups of a fibration and its loop spaces shirker 2010-09-14T23:43:30Z 2010-09-15T01:59:20Z <p>Let $f: X \to Y$ be a fibration of pointed Kan complexes, and let $F$ be the fiber. </p> <p>Question: How do you prove that the following diagram of homotopy groups commutes?:</p> <p>$\pi_n(Y) \to \pi_{n-1}(\Omega Y)$</p> <p>$\ \ \downarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \downarrow$</p> <p>$\pi_{n-1}(F) \to \pi_{n-2}(\Omega F)$</p> <p>Admittedly, I don't know for certain that it commutes, but it looks like it should.</p> <p>All the arrows are boundary maps ($\delta$) from long exact sequences of homotopy groups for a fibration. The horizontal maps are isomorphisms.</p> <p>The definition of $\delta$ that I'm using is that for for a fibration $X \to Y$, $\alpha \in \pi_n(Y)$, $\delta ([\alpha]) = [\beta d^0]$, where $\beta:\Delta^n \to X$ fits into the following diagram:</p> <p>$\Lambda^n_0 \to^* X$</p> <p>$\ \downarrow \ \ \ \ \ \downarrow$</p> <p>$\Delta^n \to^\alpha Y$</p> <p>So far, I've attempted to chase elements around this diagram and use prismatic arguments, but I haven't found one that works.</p> <p>$\Omega F \to \Omega X \to \Omega Y$</p> <p>$\ \ \downarrow\ \ \ \ \ \ \downarrow\ \ \ \ \ \ \ \downarrow$</p> <p>$PF \to PX \to PY$</p> <p>$\ \ \downarrow\ \ \ \ \ \ \downarrow\ \ \ \ \ \ \ \downarrow$</p> <p>$\ \ F \to \ \ X \to \ \ Y$</p> <p>It feels like a useful fact that $PX \to PY\times_Y X$ is a fibration, which is true since pointed simplicial sets form a simplicial model category.</p> <p>Edit: The path and loop spaces I'm using are defined by the following: For pointed simplicial sets X and Y, define the simplicial set $hom_*(X,Y)$ to have n- simplices $hom_{sSet_*}(X \wedge \Delta^n_+,Y)$, where $\Delta^n_+$ is the standard n-simplex with a disjoint basepoint.</p> <p>Then I'm using $PX=hom_*(\Delta^1 , X)$</p> <p>and $\Omega X=hom_*(\Delta^1/\partial \Delta^1,X)$</p> http://mathoverflow.net/questions/43934/equivalence-of-two-definitions-of-sheaves-on-a-site Comment by 2010-10-29T05:46:29Z 2010-10-29T05:46:29Z Thanks, it was in MacLane (p.124). For some reason, I missed it before. http://mathoverflow.net/questions/38748/commutativity-of-a-diagram-of-boundary-morphisms-from-the-long-exact-sequence-of/38761#38761 Comment by 2010-09-15T14:03:42Z 2010-09-15T14:03:42Z Thanks, I found maps of spaces $\Omega Y \to PB\times_B E ^{\simeq\atop \leftarrow} F$, which cut my diagram in half and makes both parts commute. http://mathoverflow.net/questions/38748/commutativity-of-a-diagram-of-boundary-morphisms-from-the-long-exact-sequence-of Comment by 2010-09-15T01:55:31Z 2010-09-15T01:55:31Z Thanks, I've added that now.