Implication Details
Claim: If a category is one-sorted finitary algebraic, then it has an extremal generator.
Proof: Pick an algebraic theory that represents the category. The free algebra on one generator is an extremal generator since it represents the underlying set functor, which is faithful and conservative.
Show 52 categories using this implication
- empty category
- category of abelian groups
- category of algebras
- category of finite sets and bijections
- category of sets equipped with a binary relation
- category of sets equipped with an irreflexive binary relation
- category of sets equipped with a symmetric binary relation
- category of sets equipped with a symmetric irreflexive binary relation
- category of commutative algebras
- category of commutative monoids
- category of commutative rings
- category of F(I)-sets
- category of finite sets and surjections
- category of finite abelian groups
- category of finite groups
- category of fields
- category of fields of characteristic zero
- category of Jónsson-Tarski algebras
- category of M-sets
- category of monoids
- category of sets with a distinguished subset
- partially ordered set of natural numbers
- partially ordered set of extended natural numbers
- category of quivers
- category of quivers with finite components
- category of left modules over a ring
- category of left modules over a division ring
- category of left modules over a non-semisimple ring
- category of rings
- category of rngs
- category of semigroups
- category of sequences of sets
- category of pointed sets
- category of set functions and commutative squares
- category of sets and bijections
- category of large families of sets and bijections
- category of large families of sets which are mostly singletons
- category of pairs of sets
- category of empty-or-finite pairs of sets
- category of sheaves
- category of combinatorial species
- category of uniform spaces
- category of large families of vector spaces which are mostly zero
- category of finite Z-sets
- category of Z-sets
- preordered set of integers w.r.t. divisibility
- forked commutative square
- real interval [0,1]
- walking commutative square
- walking composable pair
- walking morphism
- walking span