CatDat

Categories with unknown properties

There are 13 categories where at least one property is unknown. In total, there are 73 unknown (category, property)-pairs.

• category of abelian sheaves (12)
• category of Banach spaces with linear contractions (2)
• category of commutative monoids (1)
• category of free abelian groups (4)
• category of locally ringed spaces (10)
• category of measurable spaces (1)
• category of metric spaces with continuous maps (2)
• category of metric spaces with non-expansive maps (2)
• category of metric spaces with ∞ allowed (2)
• category of pseudo-metric spaces with non-expansive maps (7)
• category of schemes (11)
• category of sheaves (9)
• category of Z-functors (10)

Functors with unknown properties

There are 0 functors where at least one property is unknown. 🎉

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Morphisms with unknown properties

There are 0 morphisms where at least one property is unknown. 🎉

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Categories with unknown special morphisms

There are 18 categories where at least one type of special morphism is unknown.

• category of algebras (2)
• category of commutative algebras (1)
• category of commutative monoids (1)
• category of commutative rings (1)
• category of locally ringed spaces (3)
• category of metric spaces with continuous maps (1)
• category of metric spaces with non-expansive maps (2)
• category of metric spaces with ∞ allowed (2)
• category of monoids (1)
• category of partially ordered sets (1)
• category of preordered sets (1)
• category of pseudo-metric spaces with non-expansive maps (1)
• category of rings (2)
• category of rngs (2)
• category of schemes (4)
• category of semigroups (1)
• category of small categories (2)
• category of smooth manifolds (2)

Indistinguishable category pairs

There are 5 pairs of categories that cannot be distinguished by the properties currently recorded in the database. This indicates that the data may be incomplete or that a distinguishing property may be missing.

• category of abelian groups ≈ category of left modules over a ring
• category of algebras ≈ category of rings
• category of commutative algebras ≈ category of commutative rings
• category of filtered vector spaces ≈ category of torsion-free abelian groups
• category of left modules over a division ring ≈ category of vector spaces

Indistinguishable functor pairs

There are 8 pairs of functors that cannot be distinguished by the properties currently recorded in the database. This indicates that the data may be incomplete or that a distinguishing property may be missing.

• abelianization functor for groups ≈ enveloping group functor
• discrete topology functor ≈ forgetful functor from torsion abelian groups to abelian groups
• empty functor to the category of sets ≈ span endpoints inclusion
• forgetful functor from abelian groups to groups ≈ forgetful functor from commutative rings to rings
• forgetful functor from abelian groups to groups ≈ indiscrete topology functor
• forgetful functor from commutative rings to rings ≈ indiscrete topology functor
• forgetful functor for groups ≈ forgetful functor for vector spaces
• opposite category functor ≈ opposite monoid functor

Indistinguishable morphism pairs

There are 3 pairs of morphisms that cannot be distinguished by the properties currently recorded in the database. This indicates that the data may be incomplete or that a distinguishing property may be missing.

• embedding of integer into rational numbers ≈ multiplication with 2
• embedding of integer into rational numbers ≈ universal morphism
• multiplication with 2 ≈ universal morphism

Missing category combinations

Among the consistent category property combinations of the form p ∧ ¬q, the following are not yet witnessed by a category in the database or its dual. If some of these combinations are inconsistent, this indicates that some implication is missing.

Missing functor combinations

Every consistent functor property combination of the form p ∧ ¬q is witnessed by a functor in the database or its dual. 🎉

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Missing morphism combinations

Every consistent morphism property combination of the form p ∧ ¬q is witnessed by a morphism in the database or its dual. 🎉

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