Chapter 2. Representability and the Yoneda Lemma (pp. 49–72)
FC03 — Riehl, Category Theory in Context.
This chapter contains 6 definitions, 1 theorem, 5 propositions, 1 lemma, 4 corollaries, 17 examples, 2 remarks, 3 conventions and 1 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Example: Discrete dynamical systems
Discrete dynamical systems: (N,s,0) is universal—given (X,f,x₀), there is a unique r:N→X with r(0)=x₀ and r∘s=f∘r.
Definition: Representable characterization
Representable characterization: c is initial iff C(c,-) is naturally the constant singleton functor; c is terminal iff C(-,c) is.
Definition: A set-valued covariant/contravariant functor is representable when naturally…
A set-valued covariant/contravariant functor is representable when naturally isomorphic to C(c,-)/C(-,c); a representation includes the chosen c and chosen natural isomorphism.
Example: Covariant representables
Covariant representables: identity Set by 1; underlying Group/Mod_R/Ring by Z/R/Z[x]; tuples by free groups/modules; units of rings by Z[x,x^{-1}]; points/objects/arrows/isomorphisms/paths by walking objects; based-set underlying functor; path and loop functors.
Terminology
Covariant representable universal constructions are commonly called free; the dual term is cofree.
Example: Contravariant representables
Contravariant representables: power set by Ω={⊤,⊥}; opens/closed subsets by the Sierpiński space; Hom(-×A,B) by B^A; vector dual by k; cohomology by K(A,n); principal G-bundles by BG.
Example: For ordinal ω and F:ω→Set, natural transformations ω(k,-)⇒F are in bijection…
For ordinal ω and F:ω→Set, natural transformations ω(k,-)⇒F are in bijection with Fk, determined by the image of id_k.
Example: For BG, the covariant and contravariant representables are the regular…
For BG, the covariant and contravariant representables are the regular left/right G-set G; equivariant maps G→X are determined by the identity element.
Proposition: For a left G-set X, G-equivariant maps G→X correspond bijectively to…
For a left G-set X, G-equivariant maps G→X correspond bijectively to elements of X via evaluation at e∈G.
Theorem: Yoneda lemma
Yoneda lemma: for locally small C, F:C→Set, c∈C, Nat(C(c,-),F)≅Fc via α↦α_c(id_c), naturally in c and F; dually for presheaves.
Remark: Yoneda is a natural isomorphism between the natural-transformation bifunctor…
Yoneda is a natural isomorphism between the natural-transformation bifunctor Nat(y(-),-) and evaluation; it also shows the relevant transformation collection is a set.
Corollary: The covariant and contravariant Yoneda embeddings Cop→SetC and…
The covariant and contravariant Yoneda embeddings Cop→SetC and C→Set{Cop} are full and faithful.
Convention: Common convention calls both Theorem 2.2.4 and its full-faithfulness…
Common convention calls both Theorem 2.2.4 and its full-faithfulness Corollary 2.2.8 “the Yoneda lemma.”
Corollary: Every row operation on matrices with n rows is left multiplication by a…
Every row operation on matrices with n rows is left multiplication by a unique n×n matrix, namely its value on the identity matrix.
Corollary: Cayley theorem
Cayley theorem: every group is isomorphic to a subgroup of a permutation group.
Proposition: Objects x,y are isomorphic iff their covariant representables are naturally…
Objects x,y are isomorphic iff their covariant representables are naturally isomorphic (equivalently for contravariant representables); hence any two representing objects of one functor are isomorphic.
Convention: A representing object is unique up to the unique isomorphism compatible with…
A representing object is unique up to the unique isomorphism compatible with the chosen representations; “the” is used in this canonical-isomorphism sense.
Corollary: The full subcategory of terminal objects is empty or a contractible groupoid
The full subcategory of terminal objects is empty or a contractible groupoid; in particular any two terminal objects are uniquely isomorphic.
Definition: A contractible groupoid is equivalent to 1
A contractible groupoid is equivalent to 1; explicitly it is nonempty with exactly one morphism in every hom-set.
Definition: A universal property of c is a representable functor F together with a…
A universal property of c is a representable functor F together with a universal element x∈Fc that, via Yoneda, classifies the representing natural isomorphism.
Example: Z[x] represents U:Ring→Set with universal element x
Z[x] represents U:Ring→Set with universal element x; Ring(Z[x],R)≅UR by evaluation at x.
Example: A left G-set is representable iff it is a nonempty free transitive G-set, i.e
A left G-set is representable iff it is a nonempty free transitive G-set, i.e. a G-torsor; a representation is selected by a universal element.
Example/comparison
Affine space A^n is an R^n-torsor; choosing an origin is exactly choosing a representation Rn≅An.
Example: Tensor product V⊗W represents U↦Bilin(V,W;U)
Tensor product V⊗W represents U↦Bilin(V,W;U); its universal element is the universal bilinear map, through which every bilinear map factors uniquely.
Proposition: For vector spaces V,W, there is a canonical isomorphism V⊗W≅W⊗V, induced by…
For vector spaces V,W, there is a canonical isomorphism V⊗W≅W⊗V, induced by the natural symmetry Bilin(V,W;-)≅Bilin(W,V;-).
Remark: The tensor symmetry is the unique map induced by the universal bilinear map…
The tensor symmetry is the unique map induced by the universal bilinear map (v,w)↦w⊗v; its inverse is induced by swapping back.
Example: The group ring Z[G] represents R↦Group(G,R^×)
The group ring Z[G] represents R↦Group(G,R^×); its universal element is the canonical homomorphism G→Z[G]^×.
Definition: For F:C→Set, ∫F has objects (c,x), x∈Fc, and arrows f:(c,x)→(c′,x′) with…
For F:C→Set, ∫F has objects (c,x), x∈Fc, and arrows f:(c,x)→(c′,x′) with Ff(x)=x′; Π:∫F→C forgets x.
Convention: For contravariant F, define ∫F with arrow directions chosen so that the…
For contravariant F, define ∫F with arrow directions chosen so that the canonical projection Π:∫F→C still points to C.
Definition: For F:C^op→Set, ∫F has objects (c,x) and arrows f:c→c′ satisfying Ff(x′)=x.
For F:C^op→Set, ∫F has objects (c,x) and arrows f:c→c′ satisfying Ff(x′)=x.
Example: For n-Color:Graph^op→Set, ∫n-Color is the category of n-colored graphs and…
For n-Color:Graph^op→Set, ∫n-Color is the category of n-colored graphs and color-preserving graph homomorphisms.
Example: For an underlying-set functor U:C→Set, ∫U is the category C of based objects…
For an underlying-set functor U:C→Set, ∫U is the category C_* of based objects and maps preserving the chosen element.
Example: For a discrete indexing category, ∫F is the dependent sum ⨿c Fc
For a discrete indexing category, ∫F is the dependent sum ⨿_c Fc; the dependent product ∏_c Fc is the set of sections of Π:∫F→C.
Example: ∫C(c,-) is the slice c/C and ∫C(-,c) is C/c
∫C(c,-) is the slice c/C and ∫C(-,c) is C/c; their arrows are commuting triangles under/over c.
Lemma: For F:C^op→Set, ∫F is isomorphic to the comma category y↓F relative to Yoneda
For F:C^op→Set, ∫F is isomorphic to the comma category y↓F relative to Yoneda; dually in the covariant case.
Proposition: A covariant set-valued functor is representable iff its category of elements…
A covariant set-valued functor is representable iff its category of elements has an initial object; dually a contravariant one is representable iff ∫F has a terminal object.
Proposition: For F:C→Set, the full subcategory of ∫F spanned by representations is empty…
For F:C→Set, the full subcategory of ∫F spanned by representations is empty or a contractible groupoid.
Example: For a G-set X, ∫X is the translation groupoid TGX; X is representable iff…
For a G-set X, ∫X is the translation groupoid T_GX; X is representable iff this groupoid is contractible, equivalently iff X is nonempty, free, and transitive.
Example: Discrete dynamical systems form ∫U for U:End→Set
Discrete dynamical systems form ∫U for U:End→Set; (N,s,0) is the initial/universal element, recovering recursion from Proposition 2.4.8.
Example: Universal-element examples via ∫F: Ω with a chosen singleton for subsets
Universal-element examples via ∫F: Ω with a chosen singleton for subsets; pointed vectors represented by (k,1); tensor products by universal bilinear maps; free groups by universal generators; dual vectors by a terminal slice object; rings with an element by (Z[x],x).