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- Posetal_category abstract "In mathematics, a posetal category, or thin category[1], is a category whose homsets each contain at most one morphism. As such a posetal category amounts to a preordered class (or a preordered set, if its objects form a set). As suggested by the name, the further requirement that the category be skeletal is often assumed for the definition of "posetal"; in the case of a category that is posetal, being skeletal is equivalent to the requirement that the only isomorphisms are the identity morphisms, equivalently that the preordered set satisfies antisymmetry and hence is a poset.All diagrams commute in a posetal category. When the commutative diagrams of a category are interpreted as a typed equational theory whose objects are the types, a posetal category corresponds to an inconsistent theory understood as one satisfying the axiom x = y at all types.Viewing a 2-category as an enriched category whose homobjects are categories, the homobjects of any extension of a posetal category to a 2-category having the same 1-cells are monoids.Some lattice theoretic structures are definable as posetal categories of a certain kind, usually with the stronger assumption of being skeletal. For example a poset may be defined as a posetal category, a distributive lattice as a posetal distributive category, a Heyting algebra as a posetal finitely cocomplete cartesian closed category, and a Boolean algebra as a posetal finitely cocomplete *-autonomous category. Conversely, categories, distributive categories, finitely cocomplete cartesian closed categories, and finitely cocomplete *-autonomous categories can be considered the respective categorifications of posets, distributive lattices, Heyting algebras, and Boolean algebras.".
- Posetal_category wikiPageExternalLink thin+category.
- Posetal_category wikiPageID "31353487".
- Posetal_category wikiPageLength "2040".
- Posetal_category wikiPageOutDegree "22".
- Posetal_category wikiPageRevisionID "631477661".
- Posetal_category wikiPageWikiLink *-autonomous_category.
- Posetal_category wikiPageWikiLink 2-category.
- Posetal_category wikiPageWikiLink Antisymmetry.
- Posetal_category wikiPageWikiLink Boolean_algebra.
- Posetal_category wikiPageWikiLink Cartesian_closed_category.
- Posetal_category wikiPageWikiLink Categorification.
- Posetal_category wikiPageWikiLink Category:Category_theory.
- Posetal_category wikiPageWikiLink Category_(mathematics).
- Posetal_category wikiPageWikiLink Cocomplete.
- Posetal_category wikiPageWikiLink Commutative_diagram.
- Posetal_category wikiPageWikiLink Complete_category.
- Posetal_category wikiPageWikiLink Distributive_category.
- Posetal_category wikiPageWikiLink Distributive_lattice.
- Posetal_category wikiPageWikiLink Enriched_category.
- Posetal_category wikiPageWikiLink Heyting_algebra.
- Posetal_category wikiPageWikiLink Mathematics.
- Posetal_category wikiPageWikiLink Monoid.
- Posetal_category wikiPageWikiLink Partially_ordered_set.
- Posetal_category wikiPageWikiLink Poset.
- Posetal_category wikiPageWikiLink Preorder.
- Posetal_category wikiPageWikiLink Preordered_class.
- Posetal_category wikiPageWikiLink Preordered_set.
- Posetal_category wikiPageWikiLink Skeleton_(category_theory).
- Posetal_category wikiPageWikiLinkText "Posetal category".
- Posetal_category wikiPageWikiLinkText "posetal category".
- Posetal_category wikiPageWikiLinkText "posetal".
- Posetal_category hasPhotoCollection Posetal_category.
- Posetal_category subject Category:Category_theory.
- Posetal_category hypernym Category.
- Posetal_category type TelevisionStation.
- Posetal_category type Function.
- Posetal_category comment "In mathematics, a posetal category, or thin category[1], is a category whose homsets each contain at most one morphism. As such a posetal category amounts to a preordered class (or a preordered set, if its objects form a set).".
- Posetal_category label "Posetal category".
- Posetal_category sameAs m.0gkzn2b.
- Posetal_category sameAs Q7233031.
- Posetal_category sameAs Q7233031.
- Posetal_category wasDerivedFrom Posetal_category?oldid=631477661.
- Posetal_category isPrimaryTopicOf Posetal_category.