About: Posetal category

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In mathematics, specifically category theory, a posetal category, or thin category, 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 class satisfies antisymmetry and hence, if a set, is a poset.

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rdf:type	television station
rdfs:label	Posetal category (en)
rdfs:comment	In mathematics, specifically category theory, a posetal category, or thin category, 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 class satisfies antisymmetry and hence, if a set, is a poset. (en)
dcterms:subject	Category theory
Wikipage page ID	31353487 (xsd:integer)
Wikipage revision ID	930189398 (xsd:integer)
Link from a Wikipage to another Wikipage	Enriched category Indiscrete category *-autonomous category Mathematics Cocomplete Monoid Antisymmetric relation Commutative diagram 2-category Distributive category Distributive lattice Lattice (order) Diagram (category theory) Heyting algebra Category theory Boolean algebra Poset Cartesian closed category Categorification Category (mathematics) Category theory Set (mathematics) Skeleton (category theory) Preordered class Preordered set
sameAs	Posetal category Posetal category Posetal category
dbp:wikiPageUsesTemplate	dbt:Refimprove dbt:Reflist
has abstract	In mathematics, specifically category theory, a posetal category, or thin category, 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 class satisfies antisymmetry and hence, if a set, 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 codiscrete 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 hom-objects are categories, the hom-objects 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, under this assumption, a poset may be defined as a small posetal category, a distributive lattice as a small posetal distributive category, a Heyting algebra as a small posetal finitely cocomplete cartesian closed category, and a Boolean algebra as a small 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. (en)
gold:hypernym	Category
prov:wasDerivedFrom	wikipedia-en:Posetal_category?oldid=930189398&ns=0
page length (characters) of wiki page	2335 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Posetal_category
is Link from a Wikipage to another Wikipage of	Complete category T-norm Partially ordered set DisCoCat Category of preordered sets Sheaf (mathematics) Thin category
is Wikipage redirect of	Thin category
is foaf:primaryTopic of	wikipedia-en:Posetal_category

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