Presheaves, covariantly and contravariantly representable functors, and representations

Depends on

• The assignments $\mathcal C(a,-)$ and $\mathcal C(-,a)$ are functors to $\mathbf{Set}$
• Natural transformation and its components
• Natural isomorphism
• Opposite category $\mathcal C^{\mathrm{op}}$
• Sets and functions form the large locally small category $\mathbf{Set}$

Used by

• For a presheaf P, Nat(C(-,a),P)≅ P(a) naturally in a and P Corollary
• With the objectwise SAFT universal arrows supplied, a continuous Set-valued functor from a chosen-well-powered SAFT category is representable Corollary
• The functor D(X)=X⊔ X on Set is not covariantly representable Counterexample
• Set-weighted limits and colimits Definition
• The tensor product of a presheaf and a covariant set-valued functor Definition
• The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding Definition
• Universal elements of covariant functors and presheaves Definition
• A Cartesian product represents X mapstoSet(X,A)timesSet(X,B) Example
• A representable presheaf on a poset is the indicator of a principal down-set Example
• A tagged disjoint union represents X mapstoSet(A,X)timesSet(B,X) Example
• The free group on X represents G mapstoSet(X,U(G)) Example
• The free word monoid on X represents M mapstoSet(X,U(M)) Example
• The function set B^A represents X mapstoSet(X× A,B) Example
• The one-point space represents the underlying-set functor on Top Example
• Two singleton sets give canonically isomorphic representations of the identity functor on Set Example
• ℤ[x] represents the underlying-set functor on unital rings Example
• Initial and terminal objects are exactly the representations of the constant singleton functor Proposition
• A coend is a colimit weighted by the hom-bifunctor, and an end a limit weighted by it Theorem
• A representable functor carries a weighted limit to the weighted limit of the composed diagram Theorem
• A supplied pointwise right adjoint extends uniquely to a functor Theorem
• A weighted limit and a weighted colimit are unique up to a unique compatible isomorphism Theorem
• A weighted limit is an end of powers and a weighted colimit a coend of copowers Theorem
• Every covariantly representable functor to Set preserves all existing small limits Theorem
• Freyd's representability theorem for continuous Set-valued functors satisfying a solution set condition Theorem
• Weighting by a representable evaluates the diagram Theorem
• With a supplied well-powering, a subobject classifier represents the subobject functor Theorem

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, Definition 2.1.4 (standard reference, not scraped)
• Tom Leinster, Basic Category Theory, Definitions 4.1.3 and 4.1.17 (standard reference, not scraped)