SepPsh(X).yaml

id: SepPsh(X)
name: category of separated presheaves
notation: $\SepPsh(X)$
objects: separated presheaves of sets on a topological space $X$
morphisms: morphisms of presheaves
description: Here, we assume that the topological space $X$ is such that there is a non-empty family of open subsets whose union is not in the family, since otherwise this category is almost the category of all presheaves.
nlab_link: https://ncatlab.org/nlab/show/separated+presheaf

tags:
  - algebraic geometry
  - topology

related_categories:
  - Sh(X)

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

  - property: Grothendieck quasitopos
    proof: It is equivalent to $\BiSep(\Open(X), J, K)$ where $J$ is the trivial Grothendieck topology and $K$ is the open covering topology.

  - property: cocartesian cofiltered limits
    proof: For non-empty $U$, both sides of $X \sqcup \lim_{i\in I} Y_i \to \lim_{i\in I} (X \sqcup Y_i)$ can be calculated component-wise. Therefore, for those $U$, the conclusion follows from the corresponding fact in $\Set$. For $U = \varnothing$, we can see that both sides are empty if and only if $X(\varnothing) = \varnothing$ and $Y_i(\varnothing) = \varnothing$ for some $i$, and otherwise both sides are a singleton.

unsatisfied_properties:
  - property: skeletal
    proof: Consider the constant presheaves for two non-equal singleton sets.

  - property: disjoint finite coproducts
    proof: The equalizer of the two coprojections $1 \rightrightarrows 1 + 1$ has value $1$ at $\varnothing$.

  - property: generator
    proof: 'The subcategory $\Sh(X)$ of $\SepPsh(X)$ is reflective by <a href="https://ncatlab.org/nlab/show/Sketches+of+an+Elephant" target="_blank">Johnstone</a> Prop 2.6.12 and A4.4. Therefore, if $\SepPsh(X)$ had a generator then so would $\Sh(X)$, which we know is not the case.'

  - property: effective congruences
    proof: 'Let $\{ U_i : i \in I \}$ be a non-empty family of open sets whose union $U$ is not in the family. We then consider the relation $E$ on $X \coloneqq y_U + y_U$ where for $x_1, x_2 \in X(V)$, $(x_1, x_2) \in E(V)$ if and only if either $x_1 = x_2$ or $V \subseteq U_i$ for some $i \in I$. It is easy to see that $E$ is a congruence. However, $E \hookrightarrow X \times X$ is not a regular monomorphism, whereas any effective congruence would necessarily be an equalizer.'

special_objects:
  initial object:
    description: empty presheaf sending every open set to $\varnothing$
  terminal object:
    description: constant presheaf with value a singleton
  coproducts:
    description: take the section-wise disjoint union, and then collapse the value at $\varnothing$ to a singleton if it is non-empty
  products:
    description: section-wise defined direct product

special_morphisms:
  isomorphisms:
    description: morphisms of separated presheaves that are bijective on every open set
    proof: This is easy.
  monomorphisms:
    description: morphisms of separated presheaves that are injective on every open set
    proof: ...
  epimorphisms:
    description: 'morphisms of separated presheaves $\varphi : F \to G$ which are "locally surjective": for every local section $g \in G(U)$ there is an open covering $\bigcup_{i\in I} U_i = U$ such that each $g|_{U_i} \in G(U_i)$ is contained in the image of $\varphi(U_i) : F(U_i) \to G(U_i)$'
    proof: ...
  regular monomorphisms:
    description: 'morphisms of separated presheaves $\varphi : F \hookrightarrow G$ that are injective on every open set, and such that if a local section $g \in G(U)$ has an open covering $\bigcup_{i\in I} U_i = U$ such that each $g|_{U_i} \in G(U_i)$ is contained in the image of $\varphi(U_i) : F(U_i) \to G(U_i)$, then $g$ is contained in the image of $\varphi(U) : F(U) \to G(U)$'
    proof: ...
  regular epimorphisms:
    description: morphisms of separated presheaves that are surjective on every open set
    proof: ...