CatDat

preserves finite products

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

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

Relevant implications

• cofinitary andpreserves finite products () implies preserves products
• left exact implies preserves finite products
• preserves coreflexive equalizers andpreserves finite products () implies preserves equalizers
• preserves equalizers andpreserves finite products () implies left exact
• preserves finite products implies preserves terminal objects
• preserves products implies preserves finite products

Examples

There are 21 functors with this property.

• abelianization functor for groups
• binary diagonal functor on the category of sets
• binary product functor on sets
• contravariant power set functor
• discrete topology functor
• enveloping group functor
• 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 groups to monoids
• forgetful functor from rings to monoids
• functor of continuous functions
• group of units functor
• identity functor on the category of sets
• modulo p functor
• p-torsion functor
• sequences functor on sets
• squaring functor on sets
• torsion functor

Counterexamples

There are 6 functors without this property.

• binary coproduct functor on sets
• countable copower functor on sets
• covariant power set functor
• doubling functor on sets
• free group functor
• monoid ring functor

Unknown

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

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