Sets and functions form the large locally small category $\mathbf{S}\mathbf{e}\mathbf{t}$

Depends on

• Category, object, morphism, domain, codomain, identity, composition, and hom-collection
• Small, locally small, and large categories
• A function is a relation $f$ with $(a,b) \in f$ and $(a,c) \in f$ implying $b = c$; $f : A \to B$, the value $f(a)$, domain and codomain
• The set $B^{A}$ of all functions $A \to B$
• Burali-Forti: there is no set of all ordinals

Used by

• A colimit of a set-valued functor is the set of connected components of its category of elements Corollary
• Set, Cat, and every complete category are cartesian monoidal Corollary
• A componentwise family between functors need not be a natural transformation Counterexample
• A componentwise family of morphisms need not be a natural transformation and hence need not be a unit Counterexample
• A functor need not preserve monomorphisms Counterexample
• A reflexive coequalizer of sets not preserved by Set(ℕ,-) Counterexample
• The functor D(X)=X⊔ X on Set is not covariantly representable Counterexample
• A presheaf on a topological space Definition
• Presheaves, covariantly and contravariantly representable functors, and representations 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 tensor product of a presheaf and a covariant set-valued functor Definition
• A Cartesian product represents X mapstoSet(X,A)timesSet(X,B) Example
• A Kan extension computing the free-group functor Example
• A left Kan extension along a full subcategory inclusion of preorders Example
• A left Kan extension along the inclusion of the rationals in the reals Example
• A split coequalizer on a two-element set Example
• A tagged disjoint union represents X mapstoSet(A,X)timesSet(B,X) Example
• A weighted limit computing a kernel pair Example
• Actions of a group G on sets are functors BG toSet Example
• Density computed for a presheaf on a two-object discrete category Example
• Equalizers in Set are agreement subsets and coequalizers are quotients by the generated equivalence relation Example
• Evaluation of functions is dinatural in its argument set Example
• For a group G the monad G×(-) on sets has the G-sets as its algebras Example
• For a monoid action, Yoneda says that an equivariant map from the regular action is determined by the identity element Example
• Fubini checked by hand on a product of two walking arrows Example
• Pointed sets are equivalent to sets and partial functions but not isomorphic as categories Example
• Powers and copowers of a set by a set Example
• Products in Set are Cartesian products and coproducts are tagged disjoint unions Example
• Pullbacks in Set are fibre products and pushouts are quotients of tagged disjoint unions Example
• Quivers and quiver homomorphisms form a functor category of set-valued diagrams Example
• Singletons define a natural transformation from the identity functor on sets to the covariant power-set functor Example
• The arrow category Set^→: functions as objects and commuting squares as morphisms Example
• The distributive and exponential laws of sets are natural isomorphisms Example
• The end formula checked by hand against natural transformations on the walking arrow Example
• The free-group functor F:Set toGrp and free-module functor R⁽⁻⁾:Set→ R-Mod Example
• The function set B^A represents X mapstoSet(X× A,B) Example
• The orbit-set and fixed-point constructions as Kan extensions Example
• The singleton set and trivial group are terminal, while the empty set and trivial group are initial Example

…and 30 more results.

Dependency tree · two levels

15 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)