CatDat

continuous

A functor is continuous when it preserves all small limits.

• Dual property: cocontinuous
• Related properties: cofinitary, left exact, preserves equalizers, preserves products, right adjoint
• nLab Link

Relevant implications

• cofinitary andleft exact () implies continuous
• continuous implies cofinitary andleft exact andpreserves products
• continuous () implies right adjoint
• preserves equalizers andpreserves products () implies continuous
• representable implies continuous
• right adjoint implies continuous

Examples

There are 16 functors with this property.

• binary diagonal functor on the category of sets
• binary product functor on sets
• contravariant power set 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
• p-torsion functor
• sequences functor on sets
• squaring functor on sets

Counterexamples

There are 11 functors without this property.

• abelianization functor for groups
• binary coproduct functor on sets
• countable copower functor on sets
• covariant power set functor
• discrete topology functor
• doubling functor on sets
• enveloping group functor
• free group functor
• modulo p functor
• monoid ring 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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