CatDat

complete

A category is complete when every small diagram in the category has a limit.

• Dual property: cocomplete
• nLab Link

Relevant implications

• complete implies   finitely complete and  filtered limits and  wide pullbacks and  connected limits

  trivial
• complete is equivalent to   products and  equalizers

  Mac Lane, V.2, Cor. 2
• wide pullbacks and  terminal object implies   complete

  See the nLab.
• essentially small and  thin and  complete implies   cocomplete

  The supremum of a subset in a (small) partial order is the infimum of the set of upper bounds.
• essentially small and  thin and  complete and  infinitary distributive implies   cartesian closed

  This is an application of the adjoint functor theorem. Specifically, if $P$ is a complete lattice in which $\sup_i \inf(t,x_i) = \inf(t, \sup_i y_i)$ always holds, then the functor $\int(t,-)$ is a left adjoint because it preserves all suprema.
• locally presentable implies   locally essentially small and  well-powered and  well-copowered and  complete and  cocomplete and  generator

  For the non-trivial conclusions see Adamek-Rosicky, Thm. 1.20, Cor. 1.28, Rem. 1.56, Thm. 1.58.
• essentially small and  thin and  cocomplete implies   complete

  [dualized] The supremum of a subset in a (small) partial order is the infimum of the set of upper bounds.
• self-dual and  complete implies   cocomplete

  trivial by self-duality
• self-dual and  cocomplete implies   complete

  trivial by self-duality

Examples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of commutative rings
• category of groups
• category of left R-modules
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with ∞ allowed
• category of monoids
• category of pointed sets
• category of posets
• category of rings
• category of rngs
• category of sets
• category of simplicial sets
• category of small categories
• category of topological spaces
• category of vector spaces
• category of Z-functors
• partial order [0,1]
• partial order of extended natural numbers
• preorder of integers w.r.t. divisiblity
• trivial category
• walking isomorphism
• walking morphism

Counterexamples

• category of combinatorial species
• category of fields
• category of finite abelian groups
• category of finite orders
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of finitely generated abelian groups
• category of free abelian groups
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of non-empty sets
• category of schemes
• category of sets and relations
• category of smooth manifolds
• delooping of a non-trivial finite group
• delooping of an infinite group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• discrete category on two objects
• empty category
• partial order of natural numbers
• partial order of ordinal numbers
• walking parallel pair of morphisms

Unknown

For these categories the database has no info if they satisfy this property or not.

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