CatDat

equalizer-preserving

A functor $F$ preserves equalizers when for every parallel pair of morphisms $f,g : A \to B$ whose equalizer $i : E \to A$ exists, also $F(i) : F(E) \to F(A)$ is an equalizer of $F(f),F(g) : F(A) \to F(B)$.

• Dual property: coequalizer-preserving

Relevant implications

• conservative andequalizer-preserving implies faithful *
• continuous implies cofinitary andequalizer-preserving andproduct-preserving
• equalizer-preserving andfinite-product-preserving implies left exact *
• equalizer-preserving andproduct-preserving implies continuous *

*Those implications also require assumptions on the source or target category.

Examples

There are 3 functors with this property.

• contravariant power set functor
• forgetful functor for vector spaces
• identity functor on the category of sets

Counterexamples

There are 3 functors without this property.

• abelianization functor for groups
• covariant power set functor
• free group functor

Unknown

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

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