exact
A functor is exact when it is left exact and right exact.
- Dual property: exact (self-dual)
- Related properties: cocontinuous, continuous, left exact, right exact
- nLab Link
Relevant implications
Examples
There are 13 functors with this property.
- binary diagonal functor on the category of sets
- discrete topology functor
- forgetful functor for topological spaces
- forgetful functor from groups to monoids
- identity functor on the category of sets
- opposite category functor
- opposite monoid functor
- span endpoints inclusion
- 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 27 functors without this property.
- abelianization functor for groups
- binary coproduct functor on sets
- binary product functor on sets
- contravariant power set functor
- countable copower functor on sets
- covariant power set functor
- doubling functor on sets
- enveloping group 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
- free group functor
- functor of continuous functions
- fundamental group functor
- group of units functor
- indiscrete topology functor
- modulo p functor
- monoid ring functor
- p-torsion functor
- path components functor
- sequences functor on sets
- simple-group probing functor
- squaring functor on sets
- torsion functor
Unknown
There are 0 functors for which the database has no information on whether they satisfy this property.
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