Sh(X).yaml

id: Sh(X)
name: category of sheaves
notation: $\Sh(X)$
objects: sheaves of sets on a topological space $X$
morphisms: morphisms of sheaves
description: Here, we assume that the topological space $X$ is neither discrete nor indiscrete, since otherwise this category is just a product of copies of $\Set$. Another valid notation is $\Sh(X,\Set)$.
nlab_link: https://ncatlab.org/nlab/show/category+of+sheaves

tags:
  - algebraic geometry
  - topology

related_categories:
  - Set
  - SetxSet
  - Sh(X,Ab)
  - SepPsh(X)

comments:
  - It is likely that none of the currently remaining unknown properties (locally ℵ₁-presentable, exact filtered colimits, etc.) are satisfied for a <i>generic</i> space $X$, but we need to make this precise by adding additional requirements to $X$. Maybe we need to create separate entries for specific spaces $X$.

satisfied_properties:
  - property: locally small
    proof: This is easy.

  - property: Grothendieck topos
    proof: This holds by definition of a Grothendieck topos.

unsatisfied_properties:
  - property: skeletal
    proof: Consider constant sheaves for isomorphic but non-equal sets.

  - property: generator
    proof: >-
      Suppose $\Sh(X)$ had a generator. Then by <a href="/content/topos-with-generator">this result</a>, every subterminal object would be either initial or terminal. On the other hand, the subterminal objects of $\Sh(X)$ are of the form $y_U \coloneqq \Hom({-}, U)$ for $U$ open. (Alternative descriptions of $y_U$ include:
      $$y_U(V) = \begin{cases} 1, & V \subseteq U; \\ \varnothing & \mathrm{otherwise} \end{cases}$$
      with the unique restriction maps; and $y_U$ represents the functor $\Gamma(U, {-})$ of sections over $U$.) Therefore, the lemma implies that every open set of $X$ is either $\varnothing$ or $X$, contradicting our assumption that $X$ does not have the indiscrete topology.

special_objects:
  initial object:
    description: constant sheaf with value $\varnothing$, sending all non-empty open sets to $\varnothing$ and the empty set to a singleton
  terminal object:
    description: constant sheaf with value a singleton
  coproducts:
    description: associated sheaf to the section-wise disjoint union
  products:
    description: section-wise defined direct product

special_morphisms:
  isomorphisms:
    description: morphisms of sheaves that are bijective on every open set
    proof: This is easy.
  monomorphisms:
    description: morphisms of sheaves that are injective on every open subset
    proof: 'For the non-trivial direction, assume that $f : F \to G$ is a monomorphism of sheaves on $X$. Then the diagonal $F \to F \times_G F$ is an isomorphism. Since pullbacks of sheaves are constructed section-wise, it follows that the diagonal $F(U) \to F(U) \times_{G(U)} F(U)$ is an isomorphism for every open set $U \subseteq X$. But this means that $f(U) : F(U) \to G(U)$ is injective.'
  epimorphisms:
    description: 'morphisms of sheaves $f : F \to G$ that are "locally surjective": for every local section $g \in G(U)$ there is an open covering $U = \bigcup_{i \in I} U_i$ such that each $g|_{U_i} \in G(U_i)$ is contained in the image of $f(U_i) : F(U_i) \to G(U_i)$.'
    proof: 'The one direction is easy. For the other one, assume that $f : F \to G$ is an epimorphism of sheaves. For every $x \in X$ the map on stalks $f_x : F_x \to G_x$ is an epimorphism because the stalk functor $\Sh(X) \to \Set$ admits a right adjoint: take skyscraper sheaves. For $x \in U$ then $g_x \in G_x$ has a preimage in $F_x$, say represented by some $f \in F(V_x)$ for some $x \in V_x \subseteq U$. By construction of the stalk $G_x$, there is some $x \in U_x \subseteq V_x$ with $f(U_x)(f|_{U_x}) = g|_{U_x}$. Hence, the sets $(U_x)$ provide the open covering.'
  regular monomorphisms:
    description: same as monomorphisms
    proof: This is because the category is mono-regular.
  regular epimorphisms:
    description: same as epimorphisms
    proof: This is because the category is epi-regular.