Small, locally small, and large categories

Depends on

• Category, object, morphism, domain, codomain, identity, composition, and hom-collection
• Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $\mathbf{CAT}$ is not formed

Used by

• A colimit of a set-valued functor is the set of connected components of its category of elements Corollary
• A limit weighted by a Set-valued weight on a small index category exists in a complete target, and the weighted colimit in a cocomplete one Corollary
• Ends exist over a small index category in a complete target, and coends in a cocomplete one Corollary
• The end of the hom-bifunctor is the commutative monoid of natural endomorphisms of the identity functor Corollary
• The hom-functor turns a coend into an end and carries an end to an end Corollary
• A locally small category that is not well-powered: one object admits no set of representative monomorphisms Counterexample
• Assuming Choice, cardinality of a small category and κ-small diagrams Definition
• Dense subcategory Definition
• Finite, small, and large limits and colimits; complete and cocomplete categories Definition
• Global Kan extensions as adjoints to restriction Definition
• Locally finite k-linear abelian categories Definition
• Mutually left and mutually right adjoint contravariant functors Definition
• Pointwise Kan extensions as those preserved by representables Definition
• Separating and coseparating sets of objects Definition
• Set-weighted limits and colimits Definition
• The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category Definition
• The power and the copower of an object by a set Definition
• The solution-set condition for a functor, stated object by object Definition
• Weakly initial object and jointly weakly initial set Definition
• Well-powered and co-well-powered categories, and supplied well-powerings Definition
• The coend of the hom-bifunctor Example
• The free word monoid on X represents M mapstoSet(X,U(M)) Example
• Every category is locally small False statement
• FALSE: every functor on CᵒᵖtimesC has an end False statement
• FALSE: the free-cocompletion theorem holds for an arbitrary large locally small source category with no change in meaning False statement
• The hom-set form of an adjunction needs no size hypothesis False statement
• The Yoneda lemma requires its category to be small False statement
• Under dependent choice, algebras for a finitary monad on a complete cocomplete locally small category have coequalizers Lemma
• Under the Axiom of Choice, essential surjectivity onto a small category admits a splitting Lemma
• A small product of preadditive categories is preadditive Proposition
• Groups and group homomorphisms form the large locally small category Grp Proposition
• If C is small and D is locally small then [C,D] is locally small; if both are small it is small Proposition
• Left modules over a fixed ring and module homomorphisms form the large locally small category R-Mod Proposition
• Posets and monotone maps form the large locally small category Poset Proposition
• Sets and functions form the large locally small category Set Proposition
• Topological spaces and continuous maps form the large locally small category Top Proposition
• Under the Axiom of Choice, a connected small groupoid is equivalent to the automorphism group of any one of its objects Proposition
• Under the Axiom of Choice, every small category has a skeleton Proposition
• Unital rings and unit-preserving ring homomorphisms form the large locally small category Ring Proposition
• Vector spaces over a fixed field and linear maps form the large locally small category Vect_F Proposition

…and 26 more results.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)