CatDat

preserves coequalizers

A functor $F$ preserves coequalizers when for every parallel pair of morphisms $f,g : A \rightrightarrows B$ whose coequalizer $p : B \to Q$ exists, also $F(p) : F(B) \to F(Q)$ is a coequalizer of $F(f),F(g) : F(A) \rightrightarrows F(B)$.

• Dual property: preserves equalizers
• Related properties: cocontinuous, preserves reflexive coequalizers, right exact
• nLab Link

Relevant implications

• conservative andpreserves coequalizers () implies faithful
• preserves coequalizers andpreserves coproducts () implies cocontinuous
• preserves coequalizers andpreserves finite coproducts () implies right exact
• preserves coequalizers () implies preserves epimorphisms
• preserves coequalizers implies preserves reflexive coequalizers
• preserves finite coproducts andpreserves reflexive coequalizers () implies preserves coequalizers
• right exact implies preserves coequalizers

Examples

There are 14 functors with this property.

• abelianization functor for groups
• binary coproduct functor on sets
• binary diagonal functor on the category of sets
• countable copower functor on sets
• discrete topology functor
• doubling functor on sets
• enveloping group functor
• forgetful functor for topological spaces
• forgetful functor from abelian groups to groups
• forgetful functor from groups to monoids
• free group functor
• identity functor on the category of sets
• modulo p functor
• monoid ring functor

Counterexamples

There are 13 functors without this property.

• binary product functor on sets
• contravariant power set functor
• covariant power set functor
• forgetful functor for groups
• forgetful functor for rings
• forgetful functor for vector spaces
• forgetful functor from rings to monoids
• functor of continuous functions
• group of units functor
• p-torsion functor
• sequences functor on sets
• squaring functor on sets
• torsion functor

Unknown

There are 0 functors for which the database has no information on whether they satisfy this property.

—