A relative Yoneda Lemma (manuscript)

M. Künzer

March 14, 2002

Abstract

We construct set-valued right Kan extensions via a relative Yoneda Lemma.

A remark of the referee

As the referee pointed out, (2.1) ‘can essentially be found in much greater generality’ in

[Ke 82] G. M. Kelly, Basic concepts of enriched category theory, LMS Lecture Notes 64, Cambridge University Press, 1982.

He continued to explain that to this end, one reformulates the formula [Ke 82, 4.6 (ii), p. 113], given in terms of weighted limits, by means of [Ke 82, 3.10, p. 99] and [Ke 82, 2.2, p. 58].

Therefore, we withdraw this note as a preprint. Since (2.1, 3.1) might be of some use for the working mathematician, we leave it accessible as a manuscript. When using (2.1), the reader is asked to cite [Ke 82], when using (3.1), he is asked to cite [K 58].

0 Introduction

0.1 A relative Yoneda Lemma

The category of set-valued presheaves on a category C shall be denoted by C^{∧}. The Yoneda embedding, sending x to _{C}(−, x), shall be denoted by C ^{\overline{}^{y_{C}}} C^{∧}. The presheaf category construction being contravariantly functorial, we obtain the Proposition (2.1, the relative Yoneda Lemma). Given a functor C ^{\overline{}^{f}} D, we have

f^{∧} ⊣ y_{D}^{∧} ∘ f^{∧∧} ∘ y_{C^{∧}}.

There is a set theoretical caveat. In particular, this formula is correct only after some additional comments, see (2.1).

The right hand side is also known as the set-valued right Kan extension functor along C^{o} ^{\overline{}^{f^{o}}} D^{o}. There are several formulas for right Kan extensions in the literature, for instance using ends [ML 71, X.4], all of them necessarily yielding the same result up to natural equivalence, by uniqueness of the adjoint. In particular, (2.1) is merely still another such a formula.

Letting f = 1_{C}, we recover the (absolute) Yoneda Lemma, thus giving a solution to the exercise [ML 71, X.7, ex. 2]. Concerning its origin, Mac Lane recalled the following incident, taking place in around 1954/55 [ML 98].

Mac Lane, then visiting Paris, was anxious to learn from Yoneda, and commenced an interview with Yoneda in a café at the Gare du Nord. The interview was continued on Yoneda’s train until its departure. In its course, Mac Lane learned about the lemma and subsequently baptized it.

0.2 A relative co-Yoneda Lemma

Suppose given a functor C ^{\overline{}^{k}} Z. Kan constructed in [K 58, Th. 14.1] the left adjoint to the functor from Z to C^{∧} that sends z ∈ Z to _{Z}(k(−), z) ∈ C^{∧}. Now given a functor C ^{\overline{}^{f}} D, we can specialize to Z = D^{∧} and to a functor k that sends c ∈ C to _{D}(−, fc) ∈ D^{∧}, thus obtaining a left adjoint to C^{∧} ^{\overline{}^{f^{∧}}} D^{∧}. (Kan’s notation is as follows. Identify Z = Z, V = C, 𝔐^{V} = C^{∧}. The functor H^{V}(Z_{V}, Z) maps from Z_{V}, Z to 𝔐^{V}, i.e. from (the category of covariant functors from V to Z) × Z to the category of presheaves on V. Given V ^{\overline{}^{k}} Z and an object z ∈ Z, the pair (k, z) is mapped to _{Z}(−, z) ∘ k.) We shall rephrase this special case of Kan’s formula as follows, in order to be able to compare the left and the right adjoint of f^{∧}.

Let C^{∨} := (C^{o})^{∧}. The co-Yoneda embedding, sending x to _{C}(x, −), shall be denoted by C ^{\overline{}^{y′_{C}}} C^{∨}. There is a tensor product C^{∧} × C^{∨} \overline{}(Set), sending v × v′ to v ⊗_{C} v′. The according univalent functor that sends v to v ⊗_{C} − shall be denoted by C^{∧} ^{\overline{}^{z_{C}}} C^{∨∨}.

Proposition ([K 58], cf. 3.1, the relative co-Yoneda Lemma). Given a functor C ^{\overline{}^{f}} D, we have

y′_{D}^{∨} ∘ f^{∨∨} ∘ z_{C} ⊣ f^{∧}.

Again, there is a set-theoretical caveat.

0.3 Acknowledgements

I would like to thank F. Borceux for pointing out set theoretical difficulties. I would like to thank R. Rentschler for kind hospitality. The author received support from the EU TMR-network ‘Algebraic Lie Representations’, grant no. ERB FMRX-CT97-0100.

0.4 Notation

(i) We write composition of morphisms on the right, ^{\overline{}^{a}} ^{\overline{}^{b}} = ^{\overline{}^{ab}} . However, we write composition of functors on the left, ^{\overline{}^{f}} ^{\overline{}^{g}} = ^{\overline{}^{g∘f}} .

(ii) Given a functor C ^{\overline{}^{f}} D, the opposite functor between the opposite categories is denoted by C^{o} ^{\overline{}^{f^{o}}} D^{o}.

(iii) Given functors C [figure: f; f′] D [figure: g; g′] E and natural transformations f ^{\overline{}^{α}} f′ and g ^{\overline{}^{β}} g′, we denote

by g ∘ f ^{\overline{}^{β∘f}} g′ ∘ f the natural transformation defined by (β ∘ f)c = β(fc) for c ∈ C and by g ∘ f ^{\overline{}^{g∘α}} g ∘ f′ the natural transformation defined by (g ∘ α)c = g(αc) for c ∈ C. More generally,

(g ∘ f ^{\overline{}^{β∘α}} g′ ∘ f′) := (g ∘ f ^{\overline{}^{β∘f}} g′ ∘ f ^{\overline{}^{g′∘α}} g′ ∘ f′) = (g ∘ f ^{\overline{}^{g∘α}} g ∘ f′ ^{\overline{}^{β∘f′}} g′ ∘ f′).

1 Universes

Since we will iterate the construction ‘forming the presheaf category over a category’ once, we shall work in the setting of universes, which enables us to do such ‘large’ constructions when keeping track of the universe needed. Therefore, we start with a preliminary section to recall this well-known technique from [SGA 4 I, App.] and to fix some notation.

Definition 1.1 (N. Bourbaki, [SGA 4 I, App., 1, Déf. 1]) A universe is a set 𝔘 that satisfies the conditions (U1-4).

(U1) x ∈ y ∈ 𝔘 implies x ∈ 𝔘.

(U2) x, y ∈ 𝔘 implies {x, y} ∈ 𝔘.

(U3) x ∈ 𝔘 implies 𝔓(x) ∈ 𝔘.

(U4) Given I ∈ 𝔘 and a map I \overline{}𝔘, i \overline{}x_{i}, the union ⋃_{i∈I} x_{i} is in 𝔘.

Here 𝔓(x) denotes the power set of x.

Remark 1.2

(i) x ∈ 𝔘 implies {x} = {x, x} ∈ 𝔘.

(ii) x ⊆ y ∈ 𝔘 implies x ∈ 𝔘. In particular, if y surjects onto some set z, then z is in bijection to an element of 𝔘.

(iii) Let (x, y) := {x, {x, y}}. The element x is the unique element of

{a ∈ (x, y) | for all b ∈ (x, y)\{a} we have a ∈ b},

since {x, y} ∈ x would contradict von Neumann’s axiom, asserting that any nonempty set S contains an element that has an empty intersection with S. Now {x, y} is the unique element of (x, y)\{x}, and y is the unique element of {x, y}\{x}.

(iv) X ∈ 𝔘 and Y ∈ 𝔘 implies X × Y := ⋃_{x∈X} ⋃_{y∈Y} {(x, y)} ∈ 𝔘.

(v) Given I ∈ 𝔘 and a map I \overline{}𝔘, i \overline{}x_{i}, the disjoint union ∐_{i∈I} x_{i} := ⋃_{i∈I} X_{i} × \{i\} is in 𝔘.

(vi) Given I ∈ 𝔘 and a map I \overline{}𝔘, i \overline{}x_{i}, the product ∏_{i∈I} x_{i} ⊆ 𝔓(∐_{i∈I} x_{i}) is in 𝔘.

Assume given universes 𝔘, 𝔙 and 𝔚 such that

∅ ≠ 𝔘 ∈ 𝔙 ∈ 𝔚.

This may be achieved by means of Bourbaki’s axiom (A.6) in [SGA 4 I, App., 4], which says that any set is element of some universe. Note that 𝔘 ⊆ 𝔙 ⊆ 𝔚.

For set theoretical purposes the following definition is convenient.

Definition 1.3 (cf. [M 65, I.2]) A small category C is a tuple (M, K ⊆ M × M × M), M being a set, subject to the conditions (i-iii). We write M =: Mor C, the set of morphisms of C. K is the composition law of C.

(i) For every (a, b) ∈ M × M there exists at most one c ∈ M such that (a, b, c) ∈ K. In this case, we write c = ab and say that ‘(the composite) ab exists’.

Let

Ob C := { a ∈ M | for all b for which ba exists, we have ba = b, and

for all c for which ac exists, we have ac = c.} ⊆ Mor C

be the set of objects of C. If a ∈ Ob C, we also write a =: 1_{a}.

(ii) For every a ∈ M there exists a unique source element s(a) ∈ Ob C such that s(a)a exists, and a unique target element t(a) ∈ Ob C such that at(a) exists.

(iii) For a, b ∈ M such that the composite ab exists, the composites t(a)b and as(b) exist.

(iv) Given a, b, c ∈ M such that ab and bc exist. Then (ab)c and a(bc) exist and equal each other.

The set of morphisms with start x and target y shall be denoted by _{C}(x, y).

Suppose given a, b ∈ M. Note that if ab exists, we have s(ab) = s(a) and t(ab) = t(b). Note that s(s(a)) = s(a), t(t(a)) = t(a). Note that ab exists iff t(a) = s(b).

Definition 1.4 A set X is said to be 𝔘-small if there exists a bijection from X to an element of 𝔘. The category of 𝔘-small sets is denoted by (Set_{𝔘}). Note that by a skeleton argument, (Set_{𝔘}) is equivalent to the category of sets contained as elements in 𝔘, denoted by (Set_{𝔘}^{0}).

A small category C is said to be 𝔘-small if Mor C is 𝔘-small. A category C is said to be essentially 𝔘-small if it is equivalent to a 𝔘-small category, or, equivalently, if it has a 𝔘-small skeleton.

Remark 1.5 The category (Set_{𝔘}^{0}) is 𝔙-small since ∐_{(x,y)∈𝔘×𝔘} _{(Set_{𝔘}^{0})}(x, y) is an element of 𝔙.

Hence the category (Set_{𝔘}) is essentially 𝔙-small.

Given categories C and D, we denote the category of functors mapping from C to D by C, D .

Lemma 1.6 If C and D are 𝔘-small, then C, D is 𝔘-small. If C and D are essentially 𝔘-small, then C, D is essentially 𝔘-small.

Using induced equivalences, it suffices to prove the first assertion. But then

Ob C, D ⊆ 𝔓(Mor C × Mor D)

is 𝔘-small. Moreover, given f, g ∈ C, D , the set

C,D (f, g) ⊆ ∐_{x∈Ob C} _{D}(fx, gx)

is 𝔘-small. Hence Mor C, D is 𝔘-small.

Lemma 1.7 Given an essentially 𝔘-small category C, the category

C^{∧_{𝔘}} := C^{o}, (Set_{𝔘})

of (Set_{𝔘})-valued presheaves over C is essentially 𝔙-small. Sometimes we write C^{∧} = C^{∧_{𝔘}} if the universe is unambiguous. Likewise, the category

C^{∨_{𝔘}} := C, (Set_{𝔘})

of (Set_{𝔘})-valued copresheaves over C is essentially 𝔙-small. Sometimes we write C^{∨} = C^{∨_{𝔘}}.

This follows from (1.5, 1.6).

Definition 1.8 Given an essentially 𝔘-small category C, we have the Yoneda embedding

[figure: C; y_{C}; C^{∧_{𝔘}}; x; _{C}(−, x)]

and the co-Yoneda embedding

[figure: C; y′_{C}; C^{∨_{𝔘}}; x; _{C}(x, −).]

For a functor C ^{\overline{}^{f}} D, we denote

\begin{matrix}C^{∧_{𝔘}} & ^{\overline{}^{f^{∧}}} & D^{∧_{𝔘}} \\ (u ∘ f^{o} ^{\overline{}^{β∘f^{o}}} u′ ∘ f^{o}) & \overline{} & (u ^{\overline{}^{β}} u′). \end{matrix}

Given functors C [figure: f; g] D and a natural transformation f ^{\overline{}^{α}} g, we denote by f^{∧} ^{\overline{}^{α^{∧}}} g^{∧} the natural transformation that is given at u ∈ D^{∧_{𝔘}} by the morphism u ∘ f^{o} ^{\overline{}^{u∘α^{o}}}u ∘ g^{o} in C^{∧_{𝔘}}, that evaluated at c ∈ C^{o} in turn yields ufc^{\overline{}^{uαc}} ugc. Analogously for C^{∨_{𝔘}} ^{\overline{}^{f^{∨}}} D^{∨_{𝔘}} and f^{∨} ^{\overline{}^{α^{∨}}} g^{∨}.

2 The right Kan extension

Proposition 2.1 (the relative Yoneda Lemma) Given essentially 𝔘-small categories C and D, and a functor C ^{\overline{}^{f}} D. Then the right adjoint C^{∧_{𝔘}} ^{\overline{}^{φ}} D^{∧_{𝔘}} of C^{∧_{𝔘}} ^{\overline{}^{f^{∧}}} D^{∧_{𝔘}} is given by

[figure: C^{∧_{𝔘}∧_{𝔙}}; f^{∧∧}; D^{∧_{𝔘}∧_{𝔙}}; y_{C^{∧}}; D^{∧_{𝔙}}; y_{D}^{∧}; C^{∧_{𝔘}}; φ; D^{∧_{𝔘}}]

Keeping the name of the functor after restricting the image to D^{∧_{𝔘}}, we write shorthand

f^{∧} ⊣ y_{D}^{∧} ∘ f^{∧∧} ∘ y_{C^{∧}}.

The unit of this adjunction

1_{D^{∧}} ^{\overline{}^{ε}} φ ∘ f^{∧}

at u ∈ D^{∧_{𝔘}}, i.e.

u ^{\overline{}^{εu}} _{C^{∧}}(f^{∧,o} ∘ y_{D}^{o}(−), u ∘ f^{o}),

applied to d ∈ D, is given by

[figure: ud; εud; _{C^{∧}}(_{D}(f^{o}(−), d), u ∘ f^{o}); x; (x)εud,]

where the natural transformation (x)εud sends at c ∈ C

[figure: _{D}(fc, d); (x)εud; ufc; a; (a) [((x)εud)c] = (x)(ua^{o}).]

The counit of this adjunction

f^{∧} ∘ φ ^{\overline{}^{η}} 1_{C^{∧}}

at v ∈ C^{∧}, i.e.

C^{∧}(f^{∧,o} ∘ y_{D}^{o} ∘ f^{o}(−), v) ^{\overline{}^{ηv}} v,

applied to c ∈ C, is given by

[figure: _{C^{∧}}(_{D}(f^{o}(−), fc), v); ηvc; vc; ξ; (ξ)ηvc = (1_{fc})ξc]

Since the set

C^{∧}(_{D}(f(−), d), v) ⊆ ∏_{c∈Ob C} _{(Set_{𝔘})}(_{D}(fc, d), vc)

is 𝔘-small, φ exists as the factorization of y_{D}^{∧} ∘ f^{∧∧} ∘ y_{C^{∧}} over the inclusion D^{∧_{𝔘}} ⊆ D^{∧_{𝔙}}.

Various compatibilities need to be verified to ensure the well-definedness of ε and η (^{1}).

1 Given u ∈ D^{∧_{𝔘}}, we need to see that u ^{\overline{}^{εu}} φ ∘ f^{∧}(u) is a natural transformation. Suppose given

d′ ^{\overline{}^{a}} d in D. We have to show that for any x ∈ ud

(x)(εud)((φ ∘ f^{∧})ua^{o}) = (x)(ua^{o})(εud′)

is an equality of natural transformations from _{D}(f^{o}(−), d′) to u ∘ f^{o}. At c ∈ C, the element fc ^{\overline{}^{a′}} d′ is mapped by the left hand side to (x)(u(a′a)^{o}), and by the right hand side to (x(ua^{o}))(ua^{′o}).

We need to see that 1_{D^{∧_{𝔘}}} ^{\overline{}^{ε}} φ ∘ f^{∧} is a natural transformation. Suppose given u ^{\overline{}^{s}} u′ in D^{∧_{𝔘}}. We have to show that for any d ∈ D and any x ∈ ud

(x)(sd)(εu′d) = (x)(εud)((φ ∘ f^{∧}(s))d)

is an equality of natural transformations from _{D}(f^{o}(−), d) to u′ ∘ f. At c ∈ C, the element fc ^{\overline{}^{a}} d is mapped by the left hand side to (x)(sd)(u′a^{o}), and by the right hand side to (x)(ua^{o})(s(fc)).

We have to show that (f^{∧} ∘ ε)(η ∘ f^{∧}) = 1_{f^{∧}}. Suppose given u ∈ D^{∧}, c ∈ C and x ∈ (f^{∧}u)c = u(fc). We obtain

(x)((f^{∧} ∘ ε)uc)((η ∘ f^{∧})uc) = (x)(εu(fc))(η(u ∘ f^{o})c)
= (1_{fc}) [((x)(εu(fc)))c]
= (x)(u1_{fc}^{o})
= x.

We have to show that (ε ∘ φ)(φ ∘ η) = 1_{φ}. Suppose given v ∈ C^{∧_{𝔘}} and d ∈ D. Note that φv = _{C^{∧}}(f^{∧,o} ∘ y_{D}^{o}(−), v) and therefore φvd = _{C^{∧}}(_{D}(f^{o}(−), d), v). The application ((ε ∘ φ)vd)((φ ∘ η)vd) writes

[figure: C; (_{D}(f^{o}(−), d), v); ε(φv)d; C^{∧}; (_{D}(f^{o}(−), d), _{C^{∧}}(f^{∧,o} ∘ y_{D}^{o} ∘ f^{o}(−), v)); (−)ηv; C^{∧}; (_{D}(f^{o}(−), d), v)]

An element ξ ∈ φvd, i.e. _{D}(f^{o}(−), d) ^{\overline{}^{ξ}} v, is thus mapped to the composite

D(f^{o}(−), d) ^{\overline{}^{(ξ)ε(φv)d}} _{C^{∧}}(f^{∧,o} ∘ y_{D}^{o} ∘ f^{o}(−), v) ^{\overline{}^{ηv}} v.

Now suppose given c ∈ C and fc ^{\overline{}^{a}} d. We have

[figure: _{D}(fc, d); ((ξ)ε(φv)d)c; _{C^{∧}}(_{D}(f^{o}(−), fc), v); ηvc; vc]
a  \overline{}(ξ)((φv)a^{o})
= (ξ)_{C^{∧}}(_{D}(f^{o}(−), a), v)
= _{D}(f^{o}(−), a)ξ  \overline{}(1_{fc})_{D}(fc, a)(ξc)
= (a)(ξc),

whence ξ((ε ∘ φ)vd)((φ ∘ η)vd) = ξ.

Remark 2.2 (the absolute Yoneda Lemma) In case f = 1_{C}, we obtain 1 ⊣ y_{C}^{∧} ∘y_{C^{∧}}, which by uniqueness of the right adjoint yields the comparison isomorphism

[figure: _{C^{∧}}(_{C}(−, c), v); ηvc; ∼; vc; ξ; (1_{c})ξc,]

Given v ∈ C^{∧_{𝔘}}, we need to see that f^{∧} ∘φ(v) ^{\overline{}^{ηv}} v is a natural transformation. Suppose given c′ ^{\overline{}^{b}} c in C. We have to show that for any ξ ∈ (f^{∧} ∘ φ(v))c = _{C^{∧}}(_{D}(f^{o}(−), fc), v) we have

(x)(ηvc)(vb^{o}) = (x)((f^{∧} ∘ φ(v))b^{o})(ηvc′).

The left hand side yields (1_{fc})(ξc)(vb^{o}), the right hand side yields (fb)ξc′.

We need to see that f^{∧} ∘ φ ^{\overline{}^{η}} 1_{C^{∧_{𝔘}}} is a natural transformation. Suppose given v ^{\overline{}^{t}} v′ in C^{∧_{𝔘}}. We have to show that for any c ∈ C and any ξ ∈ (f^{∧} ∘ φ(v))c = _{C^{∧}}(_{D}(f^{o}(−), fc), v) we have

(ξ)(ηvc)(tc) = (ξ)((f^{∧} ∘ φ(t))c)(ηv′c).

The left hand side yields (1_{fc})(ξc)(tc). The right hand side yields (1_{fc})((ξt)c).

at v ∈ C^{∧_{𝔘}} and c ∈ C, with inverse given by εvc.

Corollary 2.3 If f is full, then the counit η of the adjunction f^{∧} ⊣ y_{D}^{∧} ∘ f^{∧∧} ∘ y_{C^{∧}} is a monomorphism. If f is full and faithful, then η is an isomorphism.

3 The left Kan extension

For sake of comparison to (2.1), we rephrase the pertinent case of Kan’s formula in our setting.

Let C be a 𝔘-small category. Let v ∈ C^{∧_{𝔘}}, let w ∈ C^{∨_{𝔘}}. We define the set v ⊗_{C} w as the quotient of the disjoint union

v ×_{C} w := ∐_{c∈C}vc × wc

modulo the equivalence relation generated by the following relation ∼_{C}. The equivalence class of (p, q) ∈ vc × wc, c ∈ C, shall be denoted by p ⊗ q.

Given (p, q) ∈ vc × wc, (p′, q′) ∈ vc′ × wc′, we say that (p, q) ∼_{C} (p′, q′) if there exists a morphism c ^{\overline{}^{a}} c′ such that

\begin{aligned}(p′)va^{o} & = & p \\ (q)wa & = & q′. \end{aligned}

Thus the quotient map v ×_{C} w ^{\overline{}^{ν}} v ⊗_{C} w has the following universal property. Given a map v ×_{C} w ^{\overline{}^{ν′}} X such that for any morphism c ^{\overline{}^{a}} c′, any p′ ∈ vc′ and any q ∈ wc we have

((p′)va^{o}, q)ν′ = (p′, (q)wa)ν′,

there exists a unique map v ⊗_{C} w ^{\overline{}^{ν̃′}} X such that ν′ = νν̃′.

In particular, given morphisms v ^{\overline{}^{m}} v′ and w ^{\overline{}^{n}} w′, we obtain a map m ⊗_{C} n that maps an element represented by (p, q) ∈ vc × wc, c ∈ C, as follows.

[figure: v; ⊗_{C}; w; m⊗_{C}n; v′; ⊗_{C}; w′; p; ⊗; q; (p)mc; ⊗; (q)nc]

Thus the tensor product defines a functor C^{∧_{𝔘}} × C^{∨_{𝔘}} ^{^{= ⊗_{C} −} \overline{}}(Set_{𝔘}). We denote the univalent tensor product functor by

[figure: C^{∧_{𝔘}}; z_{C}; C^{∨_{𝔘}∨_{𝔘}}; v; v ⊗_{C} −.]

Proposition 3.1 (the relative co-Yoneda Lemma, Kan [K 58, Th. 14.1]) Given 𝔘small categories C and D, and a functor C ^{\overline{}^{f}} D. The left adjoint C^{∧_{𝔘}} ^{\overline{}^{ψ}} D^{∧_{𝔘}} of C^{∧_{𝔘}} ^{\overline{}^{f^{∧}}} D^{∧_{𝔘}} is given by

[figure: C^{∨_{𝔘}∨_{𝔘}}; f^{∨∨}; D^{∨_{𝔘}∨_{𝔘}}; z_{C}; y′_{D}^{∧}; C^{∧_{𝔘}}; ψ; (D^{o})^{∨_{𝔘}} = D^{∧_{𝔘}}]

For short,

y′_{D}^{∨} ∘ f^{∨∨} ∘ z_{C} ⊣ f^{∧}.

The unit of this adjunction

1_{C^{∧}} ^{\overline{}^{ε}} f^{∧} ∘ ψ

at v ∈ C^{∧}, i.e.

v ^{\overline{}^{εv}} v ⊗_{C} f^{∨} ∘ y′_{D} ∘ f^{o}(−),

applied to c ∈ C, is given by

[figure: vc; εvc; v ⊗_{C} _{D}(fc, f(−)); x; x ⊗ 1_{fc} .]

The counit of this adjunction

ψ ∘ f^{∧} ^{\overline{}^{η}} 1_{D^{∧}}

at u ∈ D^{∧}, i.e.

u ∘ f^{o} ⊗_{C} f^{∨} ∘ y′_{D}(−) ^{\overline{}^{ηu}} u,

applied to d ∈ D, is given by

[figure: u ∘ f^{o}; ⊗_{C}; _{D}(d, f(−)); ηud; ud; p; ⊗; q; (p)uq^{o},]

where p ⊗ q is represented by (p, q) ∈ ufc × _{D}(d, fc) for some c ∈ C. Various compatibilities have to be verified to ensure the well-definedness of ε and η (^{2}).

2 Given v ∈ C^{∧}, we need to see that v ^{\overline{}^{εv}} f^{∧} ∘ ψ(v) is a natural transformation. Suppose given c′ ^{\overline{}^{b}} c. We have to show that for any x ∈ vc

(x)(εvc)((f^{∨} ∘ ψ(v))b^{o}) = (x)(vb^{o})(εvc′).

The left hand side yields x ⊗ fb. The right hand side yields x(vb^{o}) ⊗ 1_{fc′}.

We need to see that 1_{C^{∧}} ^{\overline{}^{ε}} f^{∧} ∘ ψ is a natural transformation. Suppose given v ^{\overline{}^{t}} v′ in C^{∧}. For any c ∈ C and any x ∈ vc we have

(x)(tc)(εv′c) = (x)(εvc)(tc ⊗ _{D}(fc, f(−))) = (x)tc ⊗ 1_{fc}.

Given u ∈ D^{∧} and d ∈ D, we need to see that ηud is a well-defined map. Suppose given c, c′ ∈ C,

We have to show that (ε∘f^{∧})(f^{∧}∘η) = 1_{f^{∧}}. Suppose given u ∈ D^{∧}, c ∈ C and x ∈ (f^{∧}u)c. We obtain

(x)((ε ∘ f^{∧})uc)((f^{∧} ∘ η)uc) = (x)(ε(u ∘ f^{o})c)(ηu(fc))
= (x ⊗ 1_{fc})(ηu(fc))
= (x)u1_{fc}^{o}.
= x.

We have to show that (ψ ∘ ε)(η ∘ ψ) = 1_{ψ}. Suppose given v ∈ C^{∧}, d ∈ D, c ∈ C, s ∈ vc, t ∈ _{D}(d, fc), so that s ⊗ t ∈ ψvd = v ⊗_{C} _{D}(d, f(−)). We obtain

(s ⊗ t)((ψ ∘ ε)vd)((η ∘ ψ)vd) = (s ⊗ t)(εv ⊗ _{D}(d, f(−)))(η(v ⊗_{C} f^{∨} ∘ y′_{D})d)
= ((s ⊗ 1_{fc}) ⊗ t)(η(v ⊗_{C} f^{∨} ∘ y′_{D})d)
= (s ⊗ 1_{fc})(v ⊗_{C} _{D}(t, f(−)))
= (s ⊗ t).

Remark 3.2 (the absolute co-Yoneda Lemma) In case f = 1_{C}, we obtain y′_{D}^{∨}∘z_{C} ⊣ 1_{C^{∧}}, which by uniqueness of the left adjoint yields the comparison isomorphism

[figure: v; ⊗_{C}; _{C}(c, −); ηvc; ∼; vc; s; ⊗; t; (s)vt^{o}]

at v ∈ C^{∧_{𝔘}} and c ∈ C, with inverse given by εvc.

Corollary 3.3 If f is full, then the unit ε of the adjunction y′_{D}^{∨} ∘ f^{∨∨} ∘ z_{C} ⊣ f^{∧} is an epimorphism. If f is full and faithful, then ε is an isomorphism.

c ^{\overline{}^{b}} c′ in C and p′ ∈ ufc′, q ∈ _{D}(d, fc). Since

((p′)uf^{o}b^{o})uq^{o} = (p′)u(q(fb))^{o}, the universal property applies.

Given u ∈ D^{∧}, we need to see that ψ∘f^{∧}(u) ^{\overline{}^{ηu}} u is a natural transformation. Suppose given d′ ^{\overline{}^{a}} d.

For all c ∈ C, p ∈ ufc and q ∈ _{D}(d, fc) we obtain

(p ⊗ q)(ηud)(ua^{o}) = (p ⊗ q)((ψ ∘ f^{∧}(u))a^{o})(ηud′) = (p)u(aq)^{o}.

We need to see that ψ ∘ f^{∧} ^{\overline{}^{η}} 1_{D^{∧}} is a natural transformation. Suppose given u′ ^{\overline{}^{s}} u in D^{∧}. We have to show that for all d ∈ D, c ∈ C, p ∈ ufc and q ∈ _{D}(d, fc), we have

(p ⊗ q)(ηud)(sd) = (p ⊗ q)((ψ ∘ f^{∧}(s))d)(ηu′d).

The left hand side yields (p)(uq^{o})(sd). The right hand side yields (p)(s(fc))(u′q^{o}).

4 References

Bourbaki, N.

[SGA 4] Univers, appendice de l’exposé I de SGA 4, tome 1, SLN 269, 1972.

Kan, D. M.

[K 58] Adjoint functors, Trans. Am. Math. Soc. 87, pp. 294-329, 1958.

Mac Lane, S.

[ML 71] Categories for the Working Mathematician, GTM 5, 1971.

[ML 98] The Yoneda Lemma, Math. Japon. 47 (1), p. 156, 1998.

Mitchell, B.

[M 65] Theory of categories, Academic Press, 1965.

Matthias Künzer Universität Ulm Abt. Reine Mathematik D-89069 Ulm Germany kuenzer@mathematik.uni-ulm.de