📐 Category Theory V: Kan Extensions

Table of Contents

• #1. Motivation: Extending Functors Along a Third Category|1. Motivation: Extending Functors Along a Third Category
• #2. Definition of Kan Extensions|2. Definition of Kan Extensions
  • #2.1 Left Kan Extension|2.1 Left Kan Extension
  • #2.2 Right Kan Extension|2.2 Right Kan Extension
  • #2.3 Kan Extensions as Adjoints to Precomposition|2.3 Kan Extensions as Adjoints to Precomposition
• #3. The Pointwise Formula|3. The Pointwise Formula
  • #3.1 Comma Categories Recalled|3.1 Comma Categories Recalled
  • #3.2 The Colimit Formula for Left Kan Extensions|3.2 The Colimit Formula for Left Kan Extensions
  • #3.3 The Limit Formula for Right Kan Extensions|3.3 The Limit Formula for Right Kan Extensions
  • #3.4 The Coend Formula|3.4 The Coend Formula
  • #3.5 Proof Sketch: The Colimit Formula Satisfies the Universal Property|3.5 Proof Sketch: The Colimit Formula Satisfies the Universal Property
• #4. Pointwise Kan Extensions|4. Pointwise Kan Extensions
  • #4.1 Definition and Characterisation|4.1 Definition and Characterisation
  • #4.2 Preservation by Right Adjoints|4.2 Preservation by Right Adjoints
• #5. All Concepts are Kan Extensions|5. All Concepts are Kan Extensions
  • #5.1 Limits and Colimits as Kan Extensions|5.1 Limits and Colimits as Kan Extensions
  • #5.2 Adjoints as Kan Extensions|5.2 Adjoints as Kan Extensions
  • #5.3 The Yoneda Lemma and the Density Theorem|5.3 The Yoneda Lemma and the Density Theorem
• #6. Derived Functors as Kan Extensions|6. Derived Functors as Kan Extensions
• #7. Ends, Coends, and the Ninja Yoneda Lemma|7. Ends, Coends, and the Ninja Yoneda Lemma
  • #7.1 Ends|7.1 Ends
  • #7.2 Coends|7.2 Coends
  • #7.3 The Ninja Yoneda Lemma|7.3 The Ninja Yoneda Lemma
  • #7.4 Natural Transformations as Ends|7.4 Natural Transformations as Ends
• #References|References

1. Motivation: Extending Functors Along a Third Category 🔍

Suppose \(F: \mathcal{C} \to \mathcal{E}\) is a functor and \(K: \mathcal{C} \to \mathcal{D}\) is another functor. The extension problem asks: does there exist a functor \(\mathcal{D} \to \mathcal{E}\) that, when precomposed with \(K\), agrees with \(F\) (or is as close to agreeing as possible)?

\[\begin{array}{c} \mathcal{C} \xrightarrow{K} \mathcal{D} \\ \searrow^{F} \quad \downarrow^{?} \\ \mathcal{E} \end{array}\]

More precisely: if \(K\) is injective on objects (e.g. a full inclusion), can we extend \(F\) to a functor defined on all of \(\mathcal{D}\)? If \(K\) is not injective, the question is subtler: we ask for the best approximation to an extension, in a universal sense. This is the Kan extension problem.

Motivating example: geometric realization. The standard \(n\)-simplex \(\Delta^n \subset \mathbb{R}^{n+1}\) defines a functor \(\Delta^\bullet: \mathbf{\Delta} \to \mathbf{Top}\) from the simplex category. The Yoneda embedding \(y: \mathbf{\Delta} \to [\mathbf{\Delta}^{op}, \mathbf{Set}]\) sends \([n]\) to the representable presheaf \(\mathbf{\Delta}(-,[n])\). Geometric realization \(|{-}|: [\mathbf{\Delta}^{op}, \mathbf{Set}] \to \mathbf{Top}\) is the left Kan extension of \(\Delta^\bullet\) along \(y\): it extends the functor \(\Delta^\bullet\) from the simplex category to all simplicial sets in the “most efficient” way compatible with colimits.

Notation for this file. Throughout: - \(\mathcal{C}, \mathcal{D}, \mathcal{E}\) denote categories (locally small unless stated otherwise). - \([\mathcal{C}, \mathcal{E}]\) denotes the functor category with objects the functors \(\mathcal{C} \to \mathcal{E}\) and morphisms the natural transformations. - \(K^*: [\mathcal{D}, \mathcal{E}] \to [\mathcal{C}, \mathcal{E}]\) denotes the precomposition functor sending \(G \mapsto G \circ K\). - For objects \(d \in \mathcal{D}\), \((K \downarrow d)\) denotes the comma category of \(K\) over \(d\); see §3.1.

2. Definition of Kan Extensions 📐

2.1 Left Kan Extension

Definition (Left Kan Extension). Let \(K: \mathcal{C} \to \mathcal{D}\) and \(F: \mathcal{C} \to \mathcal{E}\) be functors. A left Kan extension of \(F\) along \(K\) is a functor \(\mathrm{Lan}_K F: \mathcal{D} \to \mathcal{E}\) together with a natural transformation

\[\eta: F \Rightarrow (\mathrm{Lan}_K F) \circ K,\]

called the unit, satisfying the following universal property: for any functor \(G: \mathcal{D} \to \mathcal{E}\) and any natural transformation \(\alpha: F \Rightarrow G \circ K\), there exists a unique natural transformation \(\bar{\alpha}: \mathrm{Lan}_K F \Rightarrow G\) such that \((\bar{\alpha} \star K) \circ \eta = \alpha\), i.e. the composite

\[F \xRightarrow{\eta} (\mathrm{Lan}_K F) \circ K \xRightarrow{\bar{\alpha} \star K} G \circ K\]

equals \(\alpha\). Here \(\bar{\alpha} \star K\) denotes the whiskering of \(\bar{\alpha}\) by \(K\).

In other words, \(\mathrm{Lan}_K F\) is the initial functor under \(K\) equipped with a natural transformation from \(F\): it is initial in the category of pairs \((G, \alpha: F \Rightarrow G \circ K)\).

The situation is depicted by the following diagram, in which the dashed arrow is the unique fill-in:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
\mathcal{C} \arrow[r, "K"] \arrow[dr, "F"'] & \mathcal{D} \arrow[d, "{\mathrm{Lan}_K F}", dashed] \arrow[d, bend left=50, "G" description] \\
& \mathcal{E}
\end{tikzcd}
\end{document}

The natural transformation \(\eta\) fills the triangle on the left (from \(F\) to \((\mathrm{Lan}_K F) \circ K\)), and any other filling \(\alpha\) factors uniquely through \(\eta\) via \(\bar{\alpha}\).

2.2 Right Kan Extension

Definition (Right Kan Extension). A right Kan extension of \(F\) along \(K\) is a functor \(\mathrm{Ran}_K F: \mathcal{D} \to \mathcal{E}\) together with a natural transformation

\[\varepsilon: (\mathrm{Ran}_K F) \circ K \Rightarrow F,\]

called the counit, satisfying the dual universal property: for any functor \(G: \mathcal{D} \to \mathcal{E}\) and any natural transformation \(\beta: G \circ K \Rightarrow F\), there exists a unique natural transformation \(\bar{\beta}: G \Rightarrow \mathrm{Ran}_K F\) such that \(\varepsilon \circ (\bar{\beta} \star K) = \beta\).

That is, \(\mathrm{Ran}_K F\) is the terminal functor over \(K\) equipped with a natural transformation to \(F\): it is terminal in the category of pairs \((G, \beta: G \circ K \Rightarrow F)\).

2.3 Kan Extensions as Adjoints to Precomposition

The defining universal properties state precisely that \(\mathrm{Lan}_K\) and \(\mathrm{Ran}_K\) are adjoints to the precomposition functor.

Proposition (Kan Extensions as Adjoints). If \(\mathrm{Lan}_K F\) exists for every \(F: \mathcal{C} \to \mathcal{E}\), then \(\mathrm{Lan}_K: [\mathcal{C}, \mathcal{E}] \to [\mathcal{D}, \mathcal{E}]\) is left adjoint to the precomposition functor \(K^*: [\mathcal{D}, \mathcal{E}] \to [\mathcal{C}, \mathcal{E}]\):

\[\mathrm{Lan}_K \dashv K^*.\]

More precisely, the universal property of \(\mathrm{Lan}_K F\) gives a natural bijection:

\[[\mathcal{D}, \mathcal{E}](\mathrm{Lan}_K F,\, G) \;\cong\; [\mathcal{C}, \mathcal{E}](F,\, G \circ K) = [\mathcal{C}, \mathcal{E}](F,\, K^* G).\]

Dually, if \(\mathrm{Ran}_K F\) exists for every \(F\), then \(\mathrm{Ran}_K \dashv\) no — rather, \(K^* \dashv \mathrm{Ran}_K\):

\[K^* \dashv \mathrm{Ran}_K,\]

with the natural bijection \([\mathcal{C}, \mathcal{E}](K^* G,\, F) \cong [\mathcal{D}, \mathcal{E}](G,\, \mathrm{Ran}_K F)\).

Proof sketch. The unit of the adjunction \(\mathrm{Lan}_K \dashv K^*\) at \(F\) is exactly \(\eta: F \Rightarrow K^*(\mathrm{Lan}_K F)\), and the counit at \(G\) is the unique map \(\overline{\mathrm{id}}: \mathrm{Lan}_K(G \circ K) \Rightarrow G\) induced by the identity \(\mathrm{id}: G \circ K \Rightarrow G \circ K\) and the universal property of \(\mathrm{Lan}_K(G \circ K)\). One checks the triangle identities. \(\square\)

3. The Pointwise Formula ⚙️

The abstract universal property tells us what \(\mathrm{Lan}_K F\) is, but not how to compute it. When the target category \(\mathcal{E}\) has enough colimits, there is an explicit formula.

3.1 Comma Categories Recalled

The comma category \((K \downarrow d)\) for a fixed \(d \in \mathcal{D}\) has: - Objects: pairs \((c, f)\) with \(c \in \mathcal{C}\) and \(f: Kc \to d\) a morphism in \(\mathcal{D}\). - Morphisms: a morphism \((c, f) \to (c', f')\) is a morphism \(g: c \to c'\) in \(\mathcal{C}\) such that \(f' \circ Kg = f\), i.e. the triangle

\[Kc \xrightarrow{Kg} Kc' \xrightarrow{f'} d \quad \text{equals} \quad Kc \xrightarrow{f} d.\]

There is a canonical projection functor \(\Pi_d: (K \downarrow d) \to \mathcal{C}\) sending \((c, f) \mapsto c\) and \(g \mapsto g\). The composite \(F \circ \Pi_d: (K \downarrow d) \to \mathcal{E}\) is the diagram whose colimit will compute \((\mathrm{Lan}_K F)(d)\).

Dually, the comma category \((d \downarrow K)\) has objects \((c, f)\) with \(f: d \to Kc\), and the composite \(F \circ \Pi_d^{op}: (d \downarrow K) \to \mathcal{E}\) is the diagram whose limit computes \((\mathrm{Ran}_K F)(d)\).

3.2 The Colimit Formula for Left Kan Extensions

Theorem (Pointwise Left Kan Extension). Let \(K: \mathcal{C} \to \mathcal{D}\) and \(F: \mathcal{C} \to \mathcal{E}\). If \(\mathcal{C}\) is small and \(\mathcal{E}\) has all colimits of shape \((K \downarrow d)\) for each \(d \in \mathcal{D}\), then the left Kan extension \(\mathrm{Lan}_K F\) exists and is computed pointwise by:

\[(\mathrm{Lan}_K F)(d) \;=\; \mathrm{colim}\bigl(F \circ \Pi_d: (K \downarrow d) \to \mathcal{E}\bigr).\]

More explicitly, \((\mathrm{Lan}_K F)(d)\) is the colimit of the diagram that sends each pair \((c, f: Kc \to d)\) to \(Fc \in \mathcal{E}\).

Functoriality: For a morphism \(g: d \to d'\) in \(\mathcal{D}\), the map \((\mathrm{Lan}_K F)(g): (\mathrm{Lan}_K F)(d) \to (\mathrm{Lan}_K F)(d')\) is induced by the functor \((K \downarrow d) \to (K \downarrow d')\) sending \((c, f: Kc \to d) \mapsto (c, g \circ f: Kc \to d')\), which gives a morphism of diagrams, hence a unique map between their colimits by universality.

3.3 The Limit Formula for Right Kan Extensions

Theorem (Pointwise Right Kan Extension). Dually, if \(\mathcal{C}\) is small and \(\mathcal{E}\) has all limits of shape \((d \downarrow K)\) for each \(d \in \mathcal{D}\), then:

\[(\mathrm{Ran}_K F)(d) \;=\; \mathrm{lim}\bigl(F \circ \Pi_d^{op}: (d \downarrow K) \to \mathcal{E}\bigr).\]

The limit is over all pairs \((c, f: d \to Kc)\) with value \(Fc\).

3.4 The Coend Formula

The pointwise formula has a cleaner reformulation in terms of coends (to be defined in §7). When \(\mathcal{E}\) is tensored over \(\mathbf{Set}\) — meaning for each set \(S\) and object \(e \in \mathcal{E}\) there is an object \(S \otimes e = \coprod_{s \in S} e\) (the copower or tensor of \(S\) with \(e\)) — the left Kan extension formula takes the form:

\[(\mathrm{Lan}_K F)(d) \;=\; \int^{c \in \mathcal{C}} \mathcal{D}(Kc, d) \otimes Fc.\]

Here \(\mathcal{D}(Kc, d)\) is a set, \(\otimes\) is the copower, and \(\int^c\) denotes the coend (§7.2). In the special case \(\mathcal{E} = \mathbf{Set}\):

\[(\mathrm{Lan}_K F)(d) \;=\; \int^{c \in \mathcal{C}} \mathcal{D}(Kc, d) \times Fc.\]

This is the density formula for left Kan extensions in \(\mathbf{Set}\).

Dually, the right Kan extension in a cotensored category uses an end:

\[(\mathrm{Ran}_K F)(d) \;=\; \int_{c \in \mathcal{C}} Fc^{\mathcal{D}(d, Kc)},\]

where \(e^S = \prod_{s \in S} e\) denotes the power (cotensor) of \(e\) with \(S\).

3.5 Proof Sketch: The Colimit Formula Satisfies the Universal Property

Claim: Setting \(L(d) := \mathrm{colim}_{(K \downarrow d)} (F \circ \Pi_d)\) defines a functor \(L: \mathcal{D} \to \mathcal{E}\), and there is a natural transformation \(\eta: F \Rightarrow L \circ K\) such that \((L, \eta)\) is the left Kan extension.

Proof sketch (in three steps):

Step 1 (Unit construction). For each \(c \in \mathcal{C}\), the pair \((c, \mathrm{id}_{Kc}: Kc \to Kc)\) is an object of \((K \downarrow Kc)\). The colimit cocone at \(d = Kc\) therefore includes a canonical leg \(\lambda^{Kc}_{(c, \mathrm{id}_{Kc})}: Fc \to L(Kc)\). Set \(\eta_c := \lambda^{Kc}_{(c, \mathrm{id}_{Kc})}\). Naturality of \(\eta\) in \(c\) follows from the naturality of the colimit construction.

Step 2 (Universality). Given any \(G: \mathcal{D} \to \mathcal{E}\) and \(\alpha: F \Rightarrow G \circ K\), we must produce a unique \(\bar{\alpha}: L \Rightarrow G\). For each \(d \in \mathcal{D}\), a cocone from the diagram \(F \circ \Pi_d\) to \(Gd\) is given by the family indexed by \((c, f: Kc \to d)\):

\[Fc \xrightarrow{\alpha_c} G(Kc) \xrightarrow{G(f)} Gd.\]

This is indeed a cocone (check: for \(g: (c,f) \to (c', f')\) in \((K \downarrow d)\), the composites agree by naturality of \(\alpha\) and functoriality of \(G\)). By the universal property of \(L(d) = \mathrm{colim}_{(K \downarrow d)} F \circ \Pi_d\), there is a unique map \(\bar{\alpha}_d: L(d) \to Gd\) such that \(\bar{\alpha}_d \circ \lambda^d_{(c,f)} = G(f) \circ \alpha_c\) for all \((c,f)\).

Step 3 (Uniqueness). Any \(\bar{\alpha}': L \Rightarrow G\) with \((\bar{\alpha}' \star K) \circ \eta = \alpha\) satisfies \(\bar{\alpha}'_d \circ \lambda^d_{(c,f)} = G(f) \circ \alpha_c\) for all \((c,f)\) (by a short diagram chase using the cocone property). Hence \(\bar{\alpha}' = \bar{\alpha}\) by uniqueness of colimit maps. \(\square\)

4. Pointwise Kan Extensions 🎯

4.1 Definition and Characterisation

Definition (Pointwise Kan Extension). A left Kan extension \((L, \eta)\) of \(F\) along \(K\) is pointwise if it is computed by the colimit formula of §3.2: \(L(d) = \mathrm{colim}_{(K \downarrow d)} F \circ \Pi_d\) for every \(d \in \mathcal{D}\).

An equivalent intrinsic characterisation: \((L, \eta)\) is pointwise if and only if it is preserved by all representable functors: for every object \(e \in \mathcal{E}\), the functor \(\mathcal{E}(e, -): \mathcal{E} \to \mathbf{Set}\) sends \((L, \eta)\) to the left Kan extension of \(\mathcal{E}(e, F-): \mathcal{C} \to \mathbf{Set}\) along \(K\). That is, the natural map

\[\mathcal{E}(e, L(d)) \;\longrightarrow\; \mathrm{Lan}_K\bigl[\mathcal{E}(e, F-)\bigr](d)\]

is an isomorphism for all \(e \in \mathcal{E}\) and \(d \in \mathcal{D}\).

Theorem (Existence of Pointwise Left Kan Extensions). If \(\mathcal{C}\) is small and \(\mathcal{E}\) is cocomplete, then for any \(K: \mathcal{C} \to \mathcal{D}\) and \(F: \mathcal{C} \to \mathcal{E}\), the pointwise left Kan extension \(\mathrm{Lan}_K F\) exists.

4.2 Preservation by Right Adjoints

Proposition (Pointwise Kan Extensions are Preserved by Right Adjoints). If \((L, \eta)\) is a pointwise left Kan extension of \(F: \mathcal{C} \to \mathcal{E}\) along \(K: \mathcal{C} \to \mathcal{D}\), and \(H: \mathcal{E} \to \mathcal{E}'\) is a functor that preserves the relevant colimits (e.g. \(H\) has a right adjoint), then \((H \circ L, H \star \eta)\) is the pointwise left Kan extension of \(H \circ F\) along \(K\):

\[H \circ \mathrm{Lan}_K F \;\cong\; \mathrm{Lan}_K(H \circ F).\]

Proof sketch. Since right adjoints preserve limits (by concepts/category-theory/03-limits-colimits|File 3, §5), left adjoints preserve colimits by duality. The formula \((H \circ L)(d) = H(\mathrm{colim}_{(K \downarrow d)} F \circ \Pi_d) \cong \mathrm{colim}_{(K \downarrow d)} H \circ F \circ \Pi_d\) follows. \(\square\)

5. All Concepts are Kan Extensions 🔑

Mac Lane’s slogan becomes a theorem once we identify the correct choice of \(K\), \(F\), and \(\mathcal{E}\) in each case.

5.1 Limits and Colimits as Kan Extensions

Let \(\mathbf{1}\) denote the terminal category with one object \(*\) and one (identity) morphism. For any category \(\mathcal{I}\), there is a unique functor \(!: \mathcal{I} \to \mathbf{1}\).

Proposition (Limits and Colimits as Kan Extensions). Let \(F: \mathcal{I} \to \mathcal{C}\) be a diagram.

1. Colimits as left Kan extensions: \(\mathrm{colim}\, F = (\mathrm{Lan}_! F)(*)\). That is, the left Kan extension of \(F\) along \(!: \mathcal{I} \to \mathbf{1}\), evaluated at the unique object \(*\), is the colimit of \(F\).

2. Limits as right Kan extensions: \(\mathrm{lim}\, F = (\mathrm{Ran}_! F)(*)\).

Proof sketch. The comma category \((! \downarrow *) \cong \mathcal{I}\) (since there is only one object in \(\mathbf{1}\), every map in \(\mathbf{1}\) is \(\mathrm{id}_*\), and the objects of \((! \downarrow *)\) are pairs \((i \in \mathcal{I},\, !i = * \to *)\) i.e. just objects \(i \in \mathcal{I}\)). Therefore:

\[(\mathrm{Lan}_! F)(*) = \mathrm{colim}_{(! \downarrow *)} F \circ \Pi_* = \mathrm{colim}_{\mathcal{I}} F.\]

The right Kan extension case is dual. \(\square\)

Corollary. A category \(\mathcal{C}\) is cocomplete if and only if \(\mathrm{Lan}_!\) exists for every small diagram shape \(\mathcal{I}\) and every \(F: \mathcal{I} \to \mathcal{C}\).

5.2 Adjoints as Kan Extensions

This is the deepest unification result. Every adjoint functor is a Kan extension — of the identity functor.

Theorem (Adjoints as Kan Extensions). Let \(F: \mathcal{C} \to \mathcal{D}\) and \(G: \mathcal{D} \to \mathcal{C}\) with \(F \dashv G\). Then:

1. \(G \cong \mathrm{Ran}_F \mathrm{id}_{\mathcal{D}}\): the right adjoint \(G\) is the right Kan extension of the identity functor on \(\mathcal{D}\) along \(F\).
2. \(F \cong \mathrm{Lan}_G \mathrm{id}_{\mathcal{C}}\): the left adjoint \(F\) is the left Kan extension of the identity functor on \(\mathcal{C}\) along \(G\).

Proof sketch (for part 1). We show \(G\) satisfies the universal property of \(\mathrm{Ran}_F \mathrm{id}_{\mathcal{D}}\). The counit is \(\varepsilon: G \circ F \Rightarrow \mathrm{id}_{\mathcal{D}} \circ F = F\)… wait, let us be careful about variance. We need a natural transformation \(\varepsilon': (\mathrm{Ran}_F \mathrm{id}_{\mathcal{D}}) \circ F \Rightarrow \mathrm{id}_{\mathcal{D}}\). Take \(\varepsilon': GF \Rightarrow \mathrm{id}_{\mathcal{D}}\) to be the counit of the adjunction \(F \dashv G\). For any \(H: \mathcal{D} \to \mathcal{C}\) and \(\beta: HF \Rightarrow \mathrm{id}_{\mathcal{D}}\), the adjunction bijection gives a natural transformation \(\bar{\beta}: H \Rightarrow G\) as the composite:

\[H \xrightarrow{\eta \star H} GFH \xrightarrow{G \star \beta} G \cdot \mathrm{id}_{\mathcal{D}} = G.\]

One verifies \(\varepsilon' \circ (\bar{\beta} \star F) = \beta\) using the triangle identities for \(F \dashv G\). Uniqueness of \(\bar{\beta}\) follows from the bijectivity of the adjunction hom-set bijection. \(\square\)

Converse. The converse also holds: if \(\mathrm{Ran}_F \mathrm{id}_{\mathcal{D}}\) exists and is preserved by \(F\) (i.e. \(F \circ \mathrm{Ran}_F \mathrm{id}_{\mathcal{D}} \cong \mathrm{id}_{\mathcal{D}}\) in the appropriate sense), then \(\mathrm{Ran}_F \mathrm{id}_{\mathcal{D}}\) is a right adjoint to \(F\). This gives a Kan-extension-based criterion for the existence of adjoints, complementing the adjoint functor theorems of concepts/category-theory/04-adjoint-functor-theorems-monads|File 4.

5.3 The Yoneda Lemma and the Density Theorem

Let \(y: \mathcal{C} \to [\mathcal{C}^{op}, \mathbf{Set}]\) denote the Yoneda embedding \(c \mapsto \mathcal{C}(-, c)\) (as established in concepts/category-theory/02-adjoints-representables|File 2, §4).

Theorem (Density Theorem). For any small category \(\mathcal{C}\):

\[\mathrm{id}_{[\mathcal{C}^{op}, \mathbf{Set}]} \;\cong\; \mathrm{Lan}_y\, y.\]

That is, the identity functor on the presheaf category is (isomorphic to) the left Kan extension of the Yoneda embedding along itself.

Equivalently: every presheaf \(P: \mathcal{C}^{op} \to \mathbf{Set}\) is canonically a colimit of representable presheaves:

\[P \;\cong\; \mathrm{colim}\bigl(y \circ \pi: \textstyle\int_\mathcal{C} P \to [\mathcal{C}^{op}, \mathbf{Set}]\bigr),\]

where \(\int_\mathcal{C} P\) is the category of elements of \(P\) (objects are pairs \((c, x)\) with \(x \in P(c)\); morphisms \((c,x) \to (c', x')\) are \(f: c \to c'\) with \(P(f)(x') = x\)) and \(\pi: \int_\mathcal{C} P \to \mathcal{C}\) is the projection.

Proof sketch. The pointwise formula gives:

\[(\mathrm{Lan}_y\, y)(P) = \mathrm{colim}_{(y \downarrow P)} y \circ \Pi_P.\]

By the Yoneda lemma, an object of the comma category \((y \downarrow P)\) is a pair \((c, \alpha: y(c) \Rightarrow P)\), which by Yoneda corresponds to an element \(x \in P(c)\). Thus \((y \downarrow P) \cong \int_\mathcal{C} P\) (category of elements). The colimit of \(y \circ \Pi_P: \int_\mathcal{C} P \to [\mathcal{C}^{op}, \mathbf{Set}]\) is the colimit of representables indexed by elements of \(P\), which is \(P\) by the density theorem. \(\square\)

The co-Yoneda lemma (§7.3 below) gives the coend form:

\[P(c) \;\cong\; \int^{c' \in \mathcal{C}} \mathcal{C}(c', c) \times P(c'),\]

which says: the presheaf \(P\) evaluated at \(c\) is recovered as the coend of all hom-sets \(\mathcal{C}(c', c)\) weighted by \(P(c')\).

6. Derived Functors as Kan Extensions 🧮

The framework of Kan extensions subsumes the classical theory of derived functors in homological algebra. This section sketches the connection; see Riehl §6.4 for a full treatment.

Setup. Let \(\mathcal{A}\) be an abelian category with enough projectives (e.g. \(\mathbf{Ab}\), \(R\text{-}\mathbf{Mod}\)). Let \(F: \mathcal{A} \to \mathcal{B}\) be a right-exact functor. The left derived functors \(L_n F: \mathcal{A} \to \mathcal{B}\) are defined classically by choosing a projective resolution \(P_\bullet \xrightarrow{\sim} A\) and setting \(L_n F(A) = H_n(F(P_\bullet))\).

Kan extension formulation. Let \(\gamma: \mathcal{A} \to \mathbf{D}^-(\mathcal{A})\) be the localization functor from \(\mathcal{A}\) to its bounded-above derived category (obtained by inverting quasi-isomorphisms). The total left derived functor \(\mathbf{L}F: \mathbf{D}^-(\mathcal{A}) \to \mathbf{D}^-(\mathcal{B})\) is, when it exists, the left Kan extension of the composite \(\mathcal{A} \xrightarrow{F} \mathcal{B} \xrightarrow{\gamma_\mathcal{B}} \mathbf{D}^-(\mathcal{B})\) along \(\gamma_\mathcal{A}\):

\[\mathbf{L}F \;=\; \mathrm{Lan}_{\gamma_\mathcal{A}}\,(\gamma_\mathcal{B} \circ F).\]

The individual classical derived functors are recovered as \(L_n F(A) = H_n(\mathbf{L}F(A))\).

Key property. The Kan extension \(\mathbf{L}F\) is characterized by the fact that it agrees with \(F\) on projective objects: if \(P \in \mathcal{A}\) is projective, then \(\gamma_\mathcal{A}(P)\) already has a projective resolution \(P \xrightarrow{\mathrm{id}} P\) (a one-term resolution), so \((\mathbf{L}F)(\gamma P) \cong \gamma_\mathcal{B}(FP)\). This is the abstract counterpart to the classical fact that “derived functors of \(F\) agree with \(F\) on projective objects.”

Quillen’s perspective. In a model category \(\mathcal{M}\) (Quillen 1967), the localization \(\gamma: \mathcal{M} \to \mathrm{Ho}(\mathcal{M})\) inverts all weak equivalences. For a left Quillen functor \(F: \mathcal{M} \to \mathcal{N}\), the total left derived functor \(\mathbf{L}F: \mathrm{Ho}(\mathcal{M}) \to \mathrm{Ho}(\mathcal{N})\) is the left Kan extension of \(\gamma_\mathcal{N} \circ F\) along \(\gamma_\mathcal{M}\), computed by cofibrant replacement: \(\mathbf{L}F(X) = \gamma_\mathcal{N}(F(QX))\) where \(QX \xrightarrow{\sim} X\) is a cofibrant replacement.

7. Ends, Coends, and the Ninja Yoneda Lemma 🔬

Ends and coends are the “weighted” generalisations of limits and colimits that handle functors of mixed variance, \(T: \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{E}\). They are the natural language for the Kan extension coend formula of §3.4.

7.1 Ends

Definition (Wedge). Let \(T: \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{E}\) be a functor. A wedge from an object \(e \in \mathcal{E}\) to \(T\) is a family of morphisms \(\{w_c: e \to T(c, c)\}_{c \in \mathcal{C}}\) such that for every morphism \(f: c \to c'\) in \(\mathcal{C}\), the following dinatural (diagonal naturality) condition holds:

\[T(\mathrm{id}_c, f) \circ w_c = T(f, \mathrm{id}_{c'}) \circ w_{c'}: e \to T(c, c').\]

That is, both composites \(e \to T(c, c) \to T(c, c')\) and \(e \to T(c', c') \to T(c, c')\) agree.

Definition (End). The end of \(T\), written \(\int_{c \in \mathcal{C}} T(c, c)\) or \(\int_c T(c, c)\), is the universal wedge: an object \(\int_c T(c,c) \in \mathcal{E}\) together with wedge maps \(\{\pi_c: \int_c T(c,c) \to T(c,c)\}\) such that any other wedge \(\{w_c: e \to T(c,c)\}\) factors uniquely through a map \(e \to \int_c T(c,c)\).

When \(\mathcal{C}\) is small and \(\mathcal{E}\) is complete, \(\int_c T(c,c)\) is computed as the equaliser of two maps between products:

\[\int_c T(c,c) = \mathrm{eq}\Biggl(\prod_{c \in \mathcal{C}} T(c,c) \rightrightarrows \prod_{f: c \to c' \in \mathcal{C}} T(c, c')\Biggr),\]

where the two maps send the component indexed by \(c\) to the two morphisms \(T(\mathrm{id},f)\) and \(T(f, \mathrm{id})\).

7.2 Coends

Definition (Coend). The coend of \(T: \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{E}\), written \(\int^{c \in \mathcal{C}} T(c,c)\) or \(\int^c T(c,c)\), is the initial cowedge: an object \(\int^c T(c,c)\) with morphisms \(\{i_c: T(c,c) \to \int^c T(c,c)\}\) such that for every \(f: c \to c'\):

\[i_{c'} \circ T(f, \mathrm{id}_{c'}) = i_c \circ T(\mathrm{id}_c, f): T(c, c') \to \int^c T(c,c),\]

and any other such cowedge factors uniquely through \(\int^c T(c,c)\).

When \(\mathcal{C}\) is small and \(\mathcal{E}\) is cocomplete, \(\int^c T(c,c)\) is computed as the coequaliser:

\[\mathrm{coeq}\Biggl(\coprod_{f: c \to c'} T(c, c') \rightrightarrows \coprod_{c} T(c,c)\Biggr) = \int^c T(c,c),\]

where the two maps are \(T(\mathrm{id},f)\) and \(T(f,\mathrm{id})\).

7.3 The Ninja Yoneda Lemma

The co-Yoneda lemma (sometimes called the ninja Yoneda lemma in the coend-calculus literature) is the coend version of the Yoneda lemma.

Theorem (Co-Yoneda / Ninja Yoneda Lemma). For any functor \(F: \mathcal{C} \to \mathcal{E}\) (with \(\mathcal{C}\) small and \(\mathcal{E}\) cocomplete) and any \(c \in \mathcal{C}\):

\[\int^{c' \in \mathcal{C}} \mathcal{C}(c', c) \otimes Fc' \;\cong\; Fc,\]

naturally in \(c\). Here \(S \otimes e = \coprod_{s \in S} e\) is the copower of \(e\) with a set \(S\).

Proof sketch. This is exactly the left Kan extension formula with \(K = \mathrm{id}_\mathcal{C}\), \(d = c\): \((\mathrm{Lan}_{\mathrm{id}} F)(c) = \int^{c'} \mathcal{C}(c', c) \otimes Fc'\). But \(\mathrm{Lan}_{\mathrm{id}} F \cong F\) (extending along the identity recovers the functor), so the coend equals \(Fc\). \(\square\)

Alternatively, this follows directly from the ordinary Yoneda lemma: \([\mathcal{C}, \mathbf{Set}](y(c), F) \cong Fc\) where \(y(c) = \mathcal{C}(-, c)\)… but via coend calculus, the bijection is expressed as an isomorphism of coends.

7.4 Natural Transformations as Ends

Proposition. For functors \(F, G: \mathcal{C} \to \mathcal{D}\) (with \(\mathcal{C}\) small), the set of natural transformations is computed as an end:

\[[\mathcal{C}, \mathcal{D}](F, G) \;=\; \int_{c \in \mathcal{C}} \mathcal{D}(Fc, Gc).\]

Proof sketch. A natural transformation \(\alpha: F \Rightarrow G\) is a family \(\{\alpha_c: Fc \to Gc\}_{c \in \mathcal{C}}\) such that for every \(f: c \to c'\), the naturality square \(G(f) \circ \alpha_c = \alpha_{c'} \circ F(f)\) commutes. This is precisely the condition for \(\{\alpha_c\}\) to be a wedge into \(T(c, c') = \mathcal{D}(Fc, Gc')\) (where \(T(c,c) = \mathcal{D}(Fc, Gc)\)). The universal such wedge is the set of natural transformations. \(\square\)

References