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Equivalence of two definitions of sheaves on a site

2010-10-28T04:42:52Z

I'd like to prove that two definitions of sheaves on a site are equivalent, but I'm having trouble proving one direction.

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 $\Phi$).

Let $F$ be a presheaf on $C$, taking values in some complete category.

Then it should be the case that the following are equivalent characterisations of $F$ being a sheaf.

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.

2) The following diagram is an equalizer for each $U$ in $C$, and covering family {$U_\alpha \to U$} in $\Phi$: $$ F(U) \to \prod_\gamma F(U_\gamma) {\rightarrow \atop \rightarrow} \prod_{(\alpha,\beta)} F(U_\alpha \times_U U_\beta)$$

Question: How do you prove $2) \implies 1)$?. (I've proved $1) \implies 2)$)

Here's my proof attempt of $2) \implies 1)$ so far:

$\bullet$ 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:

$$ {{\lim \atop \leftarrow} \atop {V\to U \in S}} F(V) \to \prod_\gamma F(U_\gamma) {\rightarrow \atop \rightarrow} \prod_{(\alpha,\beta)} F(U_\alpha \times_U U_\beta)$$

(This is also what I used to prove $1) \implies 2)$).

$\bullet$ 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.

Because of the commutative diagram

$F(U) \to {{\lim \atop \leftarrow} \atop {V\to U \in R}} F(V)$

$\ \ \ \ \ \ \cong \searrow \ \ \ \ \ \downarrow$

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\lim \atop \leftarrow} \atop {V\to U \in S}} F(V)$

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.

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.

Commutativity of a diagram of boundary morphisms from the long exact sequence of homotopy groups of a fibration and its loop spaces

2010-09-14T23:43:30Z

Let $f: X \to Y$ be a fibration of pointed Kan complexes, and let $F$ be the fiber.

Question: How do you prove that the following diagram of homotopy groups commutes?:

$\pi_n(Y) \to \pi_{n-1}(\Omega Y)$

$\ \ \downarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \downarrow$

$\pi_{n-1}(F) \to \pi_{n-2}(\Omega F)$

Admittedly, I don't know for certain that it commutes, but it looks like it should.

All the arrows are boundary maps ($\delta$) from long exact sequences of homotopy groups for a fibration. The horizontal maps are isomorphisms.

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:

$ \Lambda^n_0 \to^* X$

$\ \downarrow \ \ \ \ \ \downarrow$

$\Delta^n \to^\alpha Y$

So far, I've attempted to chase elements around this diagram and use prismatic arguments, but I haven't found one that works.

$\Omega F \to \Omega X \to \Omega Y$

$\ \ \downarrow\ \ \ \ \ \ \downarrow\ \ \ \ \ \ \ \downarrow$

$PF \to PX \to PY$

$\ \ \downarrow\ \ \ \ \ \ \downarrow\ \ \ \ \ \ \ \downarrow$

$\ \ F \to \ \ X \to \ \ Y$

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.

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.

Then I'm using $PX=hom_*(\Delta^1 , X)$

and $\Omega X=hom_*(\Delta^1/\partial \Delta^1,X)$

Comment by

2010-10-29T05:46:29Z

Thanks, it was in MacLane (p.124). For some reason, I missed it before.

Comment by

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.

Comment by

2010-09-15T01:55:31Z

Thanks, I've added that now.