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An end is a limit over the twisted arrow category, and a coend is a colimit over its opposite

Statement

Let $T : C^{o p} \times C \to D$ be a functor and let $π : Tw (C) \to C^{o p} \times C$ be the twisted arrow projection (The twisted arrow category and its projection to $C^{o p} \times C$ ).

Ends. The assignments $ω \mapsto ξ$ with $ξ_{f} : = T (1_{c}, f) \circ ω_{c}$ for $f : c \to c^{'}$ , and $ξ \mapsto ω$ with $ω_{c} : = ξ_{1_{c}}$ , are mutually inverse and give an isomorphism of categories $Wd (T) ≅ Cone (T π)$ that leaves the vertex unchanged (Wedges and cowedges, and the categories they form, Constant diagrams, cones, cocones, and their morphisms). Consequently an end is the limit over the twisted arrow category: a wedge $(e, ω)$ is an end of $T$ exactly when the corresponding cone is a limit of $T π$ (The end and the coend of a functor $C^{o p} \times C \to D$ , Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties), either exists exactly when the other does, and then

$\int_{c} T (c, c) ≅ \lim_{Tw (C)} T π$

by the unique isomorphism compatible with every component (Any two limits, or any two colimits, of one diagram are uniquely isomorphic compatibly with their structure maps).

Coends. Let $π^{s w} : Tw (C)^{o p} \to C^{o p} \times C$ (Opposite category $C^{o p}$ ) send an object $f : c \to c^{'}$ to the pair $(c^{'}, c)$ , with the domain and the codomain of $f$ swapped, and send the morphism $g \to f$ of $Tw (C)^{o p}$ determined by $(a, b) : f \to g$ to the morphism $(b, a) : (d^{'}, d) \to (c^{'}, c)$ . Then $π^{s w}$ is a functor, $Cwd (T) ≅ Cocone (T π^{s w})$ by mutually inverse assignments leaving the vertex unchanged, and a coend of $T$ is exactly a colimit of $T π^{s w}$ :

$\int^{c} T (c, c) ≅ {colim}_{Tw (C)^{o p}} T π^{s w} .$

Facts & Assumptions

Given: A functor $T : C^{o p} \times C \to D$ and the twisted arrow projection $π$ .

[F1]

A wedge from $d$ to $T$ is a dinatural transformation from a constant functor to $T$ : a family $ω_{c} : d \to T (c, c)$ with $T (1_{c}, f) \circ ω_{c} = T (f, 1_{c^{'}}) \circ ω_{c^{'}}$ for every $f : c \to c^{'}$ ; a cowedge from $T$ to $d$ is a family $ρ_{c} : T (c, c) \to d$ with $ρ_{c} \circ T (f, 1_{c}) = ρ_{c^{'}} \circ T (1_{c^{'}}, f)$ ; a morphism of wedges is a morphism of the vertices commuting with every component (Wedges and cowedges, and the categories they form).

[F2]

A cone over $D : J \to C$ with apex $c$ is a family $λ_{j} : c \to D (j)$ satisfying $D (u) λ_{j} = λ_{k}$ for $u : j \to k$ ; a cocone under $D$ is a family $ρ_{j} : D (j) \to c$ satisfying $ρ_{k} D (u) = ρ_{j}$ ; a morphism of cones $(c, λ) \to (c^{'}, λ^{'})$ is $h : c \to c^{'}$ with $λ_{j}^{'} h = λ_{j}$ (Constant diagrams, cones, cocones, and their morphisms).

[F3]

The objects of $Tw (C)$ are the morphisms of $C$ , a morphism $f \to g$ is a pair $(a, b)$ with $b f a = g$ for $a : d \to c$ and $b : c^{'} \to d^{'}$ , and the projection $π$ sends $f : c \to c^{'}$ to $(c, c^{'})$ and $(a, b)$ to $(a, b)$ (The twisted arrow category and its projection to $C^{o p} \times C$ ).

[F4]

A limit of $D$ is a terminal cone: explicitly, for every cone $(X, ξ)$ there exists a unique morphism $u : X \to L$ such that $λ_{j} u = ξ_{j}$ for every $j$ . A colimit is an initial cocone (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties).

[F5]

An end of $T$ is a terminal object of the category of wedges over $T$ and a coend an initial object of the category of cowedges under $T$ ; in short, an end is a terminal wedge and a coend an initial cowedge (The end and the coend of a functor $C^{o p} \times C \to D$ ).

[F6]

The opposite category has the same objects and reverses every morphism: $C^{o p} (A, B) = C (B, A)$ (Opposite category $C^{o p}$ ).

[L1]

If $(L, λ)$ and $(L^{'}, λ^{'})$ are limits of one diagram $D$ , there is a unique isomorphism $u : L \to L^{'}$ satisfying $λ_{j}^{'} u = λ_{j}$ for every $j$ ; dually for colimits (Any two limits, or any two colimits, of one diagram are uniquely isomorphic compatibly with their structure maps).

Proof

technique · direct

1.1

Let $ω$ be a wedge with vertex $X$ and put $ξ_{f} : = T (1_{c}, f) \circ ω_{c}$ for $f : c \to c^{'}$ , so that $ξ_{f} : X \to T π (f)$ . This family is a cone over $T π$ : for a morphism $(a, b) : f \to g$ of $Tw (C)$ , with $g : d \to d^{'}$ and $b f a = g$ , functoriality of $T$ gives $T (a, b) \circ T (1_{c}, f) = T (a, b f) = T (1_{d}, b f) \circ T (a, 1_{c})$ , the wedge equation at $a : d \to c$ gives $T (a, 1_{c}) \circ ω_{c} = T (1_{d}, a) \circ ω_{d}$ , and $T (1_{d}, b f) \circ T (1_{d}, a) = T (1_{d}, b f a) = T (1_{d}, g)$ ; hence $T π (a, b) \circ ξ_{f} = T (1_{d}, g) \circ ω_{d} = ξ_{g}$ .

F1F2F3

1.2

Let $ξ$ be a cone over $T π$ with apex $X$ and put $ω_{c} : = ξ_{1_{c}}$ . The pairs $(1_{c}, f) : 1_{c} \to f$ and $(f, 1_{c^{'}}) : 1_{c^{'}} \to f$ are morphisms of $Tw (C)$ , since $f \circ 1_{c} \circ 1_{c} = f$ and $1_{c^{'}} \circ 1_{c^{'}} \circ f = f$ , so the cone equation at each of them gives $T (1_{c}, f) \circ ω_{c} = ξ_{f}$ and $T (f, 1_{c^{'}}) \circ ω_{c^{'}} = ξ_{f}$ ; the two left-hand sides are therefore equal and $ω$ is a wedge.

F1F2F3

1.3

The assignment $π^{s w}$ is a functor: it sends the identity $(1_{c}, 1_{c^{'}})$ of $f$ to the identity of $(c^{'}, c)$ , and if $(a, b) : f \to g$ and $(a^{'}, b^{'}) : g \to h$ then their composite in $Tw (C)$ is $(a \circ a^{'}, b^{'} \circ b)$ , whose image $(b^{'} \circ b, a \circ a^{'})$ is exactly the composite in $C^{o p} \times C$ of the images $(b^{'}, a^{'})$ and $(b, a)$ , taken in the order these two morphisms compose in $Tw (C)^{o p}$ .

F3F6

2.1

The two assignments are mutually inverse: from a wedge, $ξ_{1_{c}} = T (1_{c}, 1_{c}) \circ ω_{c} = ω_{c}$ ; from a cone, the family rebuilt in step 1.1 has $T (1_{c}, f) \circ ξ_{1_{c}} = ξ_{f}$ by the first computation of step 1.2. A morphism $h$ of the vertices satisfies $ω_{c}^{'} h = ω_{c}$ for every $c$ exactly when it satisfies $ξ_{f}^{'} h = ξ_{f}$ for every $f$ , since each family determines the other by the displayed formulas. So $Wd (T)$ and $Cone (T π)$ are isomorphic categories over the identity on vertices, terminal objects correspond, and by [F5] and [F4] an end of $T$ is precisely a limit of $T π$ ; [L1] then supplies the unique component-compatible isomorphism between any two such limits.

F4F5L1step 1.1step 1.2

3.1

For cowedges, put $ρ_{f} : = σ_{c} \circ T (f, 1_{c})$ for a cowedge $σ$ and $f : c \to c^{'}$ , so $ρ_{f} : T π^{s w} (f) = T (c^{'}, c) \to X$ . Given $(a, b) : f \to g$ , so a morphism $g \to f$ of $Tw (C)^{o p}$ sent by $T π^{s w}$ to $T (b, a)$ , functoriality gives $T (f, 1_{c}) \circ T (b, a) = T (b f, a) = T (b f, 1_{c}) \circ T (1_{d^{'}}, a)$ , the cowedge equation at $b f : c \to d^{'}$ gives $σ_{c} \circ T (b f, 1_{c}) = σ_{d^{'}} \circ T (1_{d^{'}}, b f)$ , and $T (1_{d^{'}}, b f) \circ T (1_{d^{'}}, a) = T (1_{d^{'}}, b f a) = T (1_{d^{'}}, g)$ , so $ρ_{f} \circ T (b, a) = σ_{d^{'}} \circ T (1_{d^{'}}, g) = σ_{d} \circ T (g, 1_{d}) = ρ_{g}$ by the cowedge equation at $g$ ; conversely $σ_{c} : = ρ_{1_{c}}$ recovers a cowedge from a cocone through the two morphisms of step 1.2 read in $Tw (C)^{o p}$ , and the two assignments are mutually inverse exactly as in step 2.1. Hence $Cwd (T) ≅ Cocone (T π^{s w})$ , initial objects correspond, and a coend of $T$ is a colimit of $T π^{s w}$ over $Tw (C)^{o p}$ , the integrand being read with domain and codomain swapped.

F3F5F6step 1.3step 2.1∎

Remarks

The swap in the coend clause is not cosmetic and is not a consequence of formal duality applied carelessly. Dualising $D$ turns a cowedge under $T$ into a wedge over $T$ viewed in $D^{o p}$ , and the index category that then computes it is the opposite of the twisted arrow category, with the integrand evaluated at $(c^{'}, c)$ rather than $(c, c^{'})$ . Taking the colimit of $T π$ over $Tw (C)^{o p}$ instead — the same diagram whose limit is the end — gives a different object, and FALSE: under this page's convention a coend is the colimit of the same twisted-arrow diagram whose limit is the end computes both on a two-object category to show they differ.

No smallness hypothesis appears anywhere above: the two categories are isomorphic whatever the size of $C$ , and the statement is about which universal objects exist, not about whether the index category is a set. The size hypothesis enters only when existence is to be deduced from completeness, which is Ends exist over a small index category in a complete target, and coends in a cocomplete one.

Depends on

Used by

Dependency tree · two levels

16 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

F. Loregian, (Co)end Calculus (arXiv:1501.02503v7), Remark 1.2.3 (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory (author's draft), Proposition 4.5.3 (standard reference, not scraped)