Math 2.0

Category theory; homological algebra Definition

Category of posets

Builds on CategoryPartial orderMonotone functionFunction composition

Definition

The category of posets 𝐏𝐨𝐬\mathbf{Pos} is the category given by the following data.

  1. Its objects are the partially ordered sets.
  2. A morphism from (P,≤P)(P, \le_P) to (Q,≤Q)(Q, \le_Q) is a monotone function f ⁣:P→Qf \colon P \to Q, that is, x≤Pyx \le_P y implies f(x)≤Qf(y)f(x) \le_Q f(y).
  3. Composition is the usual composition of functions, and the identity morphism of PP is the identity map of PP.

Remarks

The data form a category. The identity map is monotone, and if ff and gg are monotone, then x≤yx \le y gives f(x)≤f(y)f(x) \le f(y) and then g(f(x))≤g(f(y))g(f(x)) \le g(f(y)). Associativity and the identity laws hold because they hold for functions. Posets with order-reversing maps do not form a category in the same way: the composite of two order-reversing maps preserves order. A morphism of 𝐏𝐨𝐬\mathbf{Pos} is an isomorphism exactly when it is an order isomorphism. A monotone bijection need not be one: the identity from a two-element antichain to a two-element chain is a monotone bijection whose inverse is not monotone.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

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Category, concept, no evidence levelCategoryconceptElement, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionFunction, definition, no evidence levelFunctiondefinitionPreorder, definition, no evidence levelPreorderdefinitionFunction composition, definition, no evidence levelFunction compositiondefinitionPartial order, definition, no evidence levelPartial orderdefinitionMonotone function, definition, no evidence levelMonotone functiondefinitionCategory of posets, definition, no evidence levelCategory of posetsdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Category concept
  2. Element definition
  3. Proposition concept
  4. Set definition
  5. Implication definition
  6. Predicate definition
  7. Ordered pair definition
  8. Existential quantifier definition
  9. Universal quantifier definition
  10. Cartesian product definition
  11. Subset definition
  12. Uniqueness quantifier definition
  13. Binary relation definition
  14. Antisymmetric relation definition
  15. Reflexive relation definition
  16. Transitive relation definition
  17. Function definition
  18. Preorder definition
  19. Function composition definition
  20. Partial order definition
  21. Monotone function definition
  22. Category of posets definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Category_of_preordered_sets
About this page
TagCNGSMAB60A
URIm2://CNGSMAB60A
Kinddefinition
WritingW0 stub
MSC18B35, 06A06
DefinesCategory of posets
UsesCategory, Partial order, Monotone function, Function composition

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