CatDat

finitely complete

A category is finitely complete when every finite diagram has a limit.

• Dual property: finitely cocomplete
• Related properties: complete
• nLab Link

Relevant implications

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

  trivial
• finitely complete is equivalent to   finite products and  equalizers

  Mac Lane, V.2, Cor. 1
• exact filtered colimits implies   filtered colimits and  finitely complete

  by definition
• elementary topos is equivalent to   cartesian closed and  finitely complete and  subobject classifier

  by definition
• subobject classifier implies   finitely complete and  mono-regular

  The first part holds by convention, and the second part: any monomorphism $U \to X$ is the equalizer of $$\chi_U,\chi_X : X \to \Omega$.
• Malcev implies   finitely complete

  by definition
• thin and  finitely complete implies   Malcev

  In a thin category, every subobject of $X^2 = X$ containing $X$ is already $X$.
• self-dual and  finitely complete implies   finitely cocomplete

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

  trivial by self-duality

Examples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative rings
• category of finite abelian groups
• category of finite orders
• category of finite sets
• category of finitely generated abelian groups
• category of free abelian groups
• 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 metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of monoids
• category of pointed sets
• category of posets
• category of rings
• category of rngs
• category of schemes
• 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 fields
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of non-empty sets
• 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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