Partially ordered set: Difference between revisions

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== In category theory ==

{{main|Posetal category}}

Every poset (and every [[Preorder|preordered set]]) may be considered as a [[Category (mathematics)|category]] where, for objects <math>x</math> and <math>y,</math> there is at most one [[morphism]] from <math>x</math> to <math>y.</math> More explicitly, let {{nowrap|1=hom(''x'', ''y'') = {{mset|(''x'', ''y'')}}}} if {{nowrap|''x'' ≤ ''y''}} (and otherwise the [[empty set]]) and <math>(y, z) \circ (x, y) = (x, z).</math> Such categories are sometimes called ''[[Posetal category|posetal]]''. In differential topology, homology theory (HT) is used for classifying equivalent smooth manifolds M, related to the geometrical shapes of M.

Posets are [[Equivalence of categories|equivalent]] to one another if and only if they are [[Isomorphism of categories|isomorphic]]. In a poset, the smallest element, if it exists, is an [[initial object]], and the largest element, if it exists, is a [[terminal object]]. Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset is [[isomorphism-closed]]. In differential topology, homology theory (HT) is used for classifying equivalent smooth manifolds M, related to the geometrical shapes of M. In homology theory is given an axiomatic HT approach, especially to singular homology.{{clarify|date=May 2023}} The HT members are algebraic invariants under diffeomorphisms. The axiomatic HT category is taken in G. Kalmbach from the book Eilenberg–Steenrod (see the references) in order to show that the set theoretical topological concept for the HT definition can be extended to partial ordered sets P. Important are chains and filters in P (replacing shapes of M) for defining HT classifications, available for many P applications not related to set theory.

== Partial orders in topological spaces ==