CatDat

preserves products

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

• Dual property: preserves coproducts
• Related properties: continuous, preserves finite products, preserves terminal objects
• nLab Link

Relevant implications

• cofinitary andpreserves finite products () implies preserves products
• continuous implies preserves products
• preserves equalizers andpreserves products () implies continuous
• preserves products implies preserves finite products

Examples

There are 41 functors with this property.

• binary diagonal functor on the category of sets
• binary product functor on sets
• contravariant power set functor
• empty functor to the category of sets
• forgetful functor for groups
• forgetful functor for rings
• forgetful functor for topological spaces
• forgetful functor for vector spaces
• forgetful functor from abelian groups to groups
• forgetful functor from commutative rings to rings
• 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 groups to pointed sets
• forgetful functor from Hausdorff spaces to topological spaces
• forgetful functor from rings to monoids
• forgetful functor from torsion-free abelian groups to abelian groups
• functor of continuous functions
• fundamental group functor
• group of units functor
• identity functor on the category of sets
• indiscrete topology functor
• modulo p functor
• morphism endpoints inclusion
• nerve functor
• opposite category functor
• opposite monoid functor
• p-torsion functor
• path components functor
• ring idempotents functor
• sequences functor on sets
• simple-group probing functor
• span endpoints inclusion
• squaring functor on sets
• 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
• walking morphism representation

Counterexamples

There are 15 functors without this property.

• abelianization functor for groups
• binary coproduct functor on sets
• Brauer group functor
• countable copower functor on sets
• covariant power set functor
• discrete topology functor
• doubling functor on sets
• enveloping group functor
• forgetful functor from torsion abelian groups to abelian groups
• free group functor
• inclusion functor from extended natural numbers to ordinal numbers
• monoid ring functor
• rational product functor
• Stone-Čech compactification functor
• torsion functor

Unknown

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

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