CatDat

preserves coproducts

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

• Dual property: preserves products
• Related properties: cocontinuous, preserves finite coproducts, preserves initial objects
• nLab Link

Relevant implications

• cocontinuous implies preserves coproducts
• finitary andpreserves finite coproducts () implies preserves coproducts
• preserves coequalizers andpreserves coproducts () implies cocontinuous
• preserves coproducts implies preserves finite coproducts

Examples

There are 37 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
• empty functor to the category of sets
• enveloping group functor
• forgetful functor for topological spaces
• forgetful functor from finite abelian groups to abelian groups
• forgetful functor from finite groups to groups
• forgetful functor from finite sets to sets
• forgetful functor from groups to monoids
• forgetful functor from Hausdorff spaces to topological spaces
• forgetful functor from torsion abelian groups to abelian groups
• forgetful functor from torsion-free abelian groups to abelian groups
• free group functor
• group of units functor
• identity functor on the category of sets
• inclusion functor from extended natural numbers to ordinal numbers
• modulo p functor
• monoid ring functor
• morphism endpoints inclusion
• nerve functor
• opposite category functor
• opposite monoid functor
• p-torsion functor
• path components functor
• rational product functor
• span endpoints inclusion
• Stone-Čech compactification functor
• torsion functor
• trivial functor from the category of groups
• trivial functor from the category of sets
• trivial functor from the delooping
• trivial functor from the walking idempotent
• walking isomorphism object inclusion

Counterexamples

There are 19 functors without this property.

• binary product functor on sets
• Brauer group functor
• 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 commutative rings to rings
• forgetful functor from groups to pointed sets
• forgetful functor from rings to monoids
• functor of continuous functions
• fundamental group functor
• indiscrete topology functor
• ring idempotents functor
• sequences functor on sets
• simple-group probing functor
• squaring functor on sets
• walking morphism representation

Unknown

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

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