CatDat

Unknown properties

There are 7 properties for which the database doesn't have an answer if they are satisfied or not. Please help to contribute the data!

• has exact filtered colimits
• is finitary algebraic
• has a generator
• is locally finitely presentable
• is locally strongly finitely presentable
• is locally ℵ₁-presentable
• is strongly connected

Special objects

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

Special morphisms

• isomorphisms: morphisms of sheaves that are bijective on every open set
• monomorphisms: morphisms of sheaves that are injective on every open subset
• epimorphisms: 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 $f(U_i) : F(U_i) \to G(U_i)$.
• regular monomorphisms: same as monomorphisms
• regular epimorphisms: same as epimorphisms

Comments

• It is likely that neither of the currently remaining unknown properties (finitary algebraic, locally ℵ₁-presentable, exact filtered colimits, etc.) are satisfied for a generic 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 $X$.