Implication details
Claim: If a category is total, then it is cocomplete.
Proof: If the category is locally small and total, then the Yoneda embedding makes into a reflective subcategory of the presheaf category , where the latter is cocomplete. Now use that a reflective subcategory of a cocomplete category is cocomplete. For a general total category , use the equivalence to a locally small, total category.
This implication has a dual.
Show 52 categories using this implication
- empty category
- discrete category on two objects
- category of finite sets and bijections
- delooping of a group
- delooping of an infinite countable group
- delooping of a non-trivial finite group
- delooping of an infinite uncountable group
- delooping of the additive monoid of natural numbers
- category of sets equipped with an irreflexive binary relation
- category of sets equipped with a symmetric irreflexive binary relation
- category of coproducts of Euclidean spaces
- category of finite sets and injections
- category of finite sets and surjections
- category of finite abelian groups
- category of finite groups
- category of finite ordered sets
- category of finite sets
- category of finite sets of even cardinality
- category of finite sets of odd cardinality
- category of finite sets of cardinality a power of 3
- category of finite-dimensional vector spaces
- category of finite-dimensional vector spaces [countable field]
- category of finite-dimensional vector spaces [finite field]
- category of finite-dimensional vector spaces [uncountable field]
- category of fields of characteristic zero
- category of free abelian groups
- category of finitely generated free abelian groups
- category of finitely generated free modules over Z x Z
- category of groups
- category of smooth manifolds
- category of metric spaces with non-expansive maps
- category of metric spaces with continuous maps
- partially ordered set of natural numbers
- category of pseudo-metric spaces with non-expansive maps
- category of partially ordered sets without isolated points
- category of quivers with finite components
- category of sets and relations
- category of schemes
- category of connected sequences of sets
- category of sets and bijections
- category of countable sets
- discrete category of sets
- category of large families of sets which are mostly singletons
- category of sets with finite-to-one maps
- category of empty-or-finite pairs of sets
- category of combinatorial species
- category of countable-dimensional vector spaces
- category of finite Z-sets
- walking fork
- walking idempotent
- walking span
- walking splitting