CatDat

preserves finite coproducts

A functor $F$ preserves finite coproducts when for every family of objects $(A_i)$ in the source whose coproduct $\prod_i A_i$ exists, also the coproduct $\coprod_i F(A_i)$ exists in the target and such that the canonical morphism $\coprod_i F(A_i) \to F(\coprod_i A_i)$ is an isomorphism.

• Dual property: preserves finite products
• Related properties: cocontinuous, preserves binary coproducts, preserves coproducts, preserves initial objects
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Relevant implications

• finitary andpreserves finite coproducts () implies preserves coproducts
• preserves binary coproducts andpreserves initial objects () is equivalent to preserves finite coproducts
• preserves coequalizers andpreserves finite coproducts () implies right exact
• preserves coproducts implies preserves finite coproducts
• preserves finite coproducts implies preserves binary coproducts andpreserves initial objects
• preserves finite coproducts () implies preserves finite products
• preserves finite products () implies preserves finite coproducts
• right exact implies preserves finite coproducts

Examples

There are 22 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 groups to monoids
• free group functor
• group of units functor
• identity functor on the category of sets
• modulo p functor
• monoid ring functor
• opposite category functor
• opposite monoid functor
• p-torsion functor
• path components functor
• torsion functor
• trivial functor from the category of groups
• trivial functor from the category of sets
• walking isomorphism object inclusion

Counterexamples

There are 14 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 abelian groups to groups
• forgetful functor from groups to pointed sets
• forgetful functor from rings to monoids
• functor of continuous functions
• fundamental group functor
• indiscrete topology functor
• sequences functor on sets
• squaring functor on sets

Unknown

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

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