Ends and Coends

1. Motivation of Ends
2. Different Definitions of Ends
3. Different Definitions of Coends
- 3.1. Elementary Definition of Coends

Ends and coends are the \( 1 \)-categorification of the following idea: let \( G \) be a group and \( X \) be an object (say a set) equipped with a left and a right \( G \)-action. Then we can ask for the subobject on which left and right action agree (end) or for the quotient that forces left and right action to agree (coend).

Indeed a set with a left \( G \)-action is just a functor \(\boldsymbol{B}G \rightarrow \Set \) and a set with with a right \(G \)-action is a functor \(\boldsymbol B G^{op} \rightarrow \Set \).

1. Motivation of Ends

We start with coends and motivate them as a \( 1 \)-categorification of tensor products.

2. Different Definitions of Ends

Let \( F \colon \mathcal C ^{op} \times \mathcal C \rightarrow \mathcal D \) be a functor (known as a bifunctor from \( \mathcal C \) to \( \mathcal D \)). Then \( F \) defines two actions an the object

\begin{equation*} \prod _{c \in \mathcal C} F(c,c) \end{equation*}

For every morphism \(f \colon a \rightarrow b \) in \(\mathcal C \) we get the morphisms

\begin{equation*} f^* \colon \prod _{c \in \mathcal C} F(c,c) \rightarrow \prod _{c \in \mathcal C} F(c,c), \; f_* \colon \prod _{c \in \mathcal C} F(c,c) \rightarrow \prod _{c \in \mathcal C} F(c,c) \end{equation*}

that are induced by

\begin{equation*} \prod _{c \in \mathcal C} F(c,c) \xrightarrow{\pi _b} F(b,b) \xrightarrow{F(f,1_b)} F(a,b) \end{equation*}

and

\begin{equation*} \prod _{c \in \mathcal C} F(c,c) \xrightarrow{\pi _a} F(a,a) \xrightarrow{F(f,1_b)} F(a,b) \end{equation*}

2.1. Elementary Definition

2.2. Definition of Ends using Limits

2.3. Definition of Ends using Kan Extensions

2.4. Definition of Ends using Adjunctions

3. Different Definitions of Coends

3.1. Elementary Definition of Coends

Let \(F \colon \mathcal C ^{op} \times \mathcal C \rightarrow \mathcal D \) be a functor. A cowedge (of \(F \) or wrt \(F \)) consists of

an object \( d \in \Ob \mathcal D \),
a morphism \(\iota_c \colon F(c,c) \rightarrow d \) for every \(c \in \mathcal C \),

such that

for every morphism \(f \colon c \rightarrow c' \) in \(\mathcal C \)
\begin{equation*} \xymatrix{ F(c',c) \ar[r]^{F(f,1_{c})} \ar[d]_{F(1_{c'},f)} & F(c,c) \ar[d]^{\iota _c}\\ F(c',c') \ar[r]^{\iota _{c'}} & d } \end{equation*}
commutes in \( \mathcal D\).

Alternatively we could define a cowedge to consist of an object \( d \) and a morphism \(\iota _f \colon F(c,c') \rightarrow d \) for every morphism \(f \colon c \rightarrow c' \) such that

\begin{equation*} \xymatrix{ d \ar[r]^{\iota _f} \ar[d]_{\iota _{g}} & F(c,c') \ar[d]^{F(1_c,g)}\\ F(c',c'') \ar[r]^{F(f,1_{c''})} & F(c,c'') } \end{equation*}

commutes for every morphisms \(c \xrightarrow f c' \xrightarrow g c'' \). If we add the diagonal map \(\iota _{gf} \colon d \rightarrow F(c,c'') \) to the diagram we get two commuting triangles.

Proof.

\begin{align*} \iota _{gf} = & F(gf,1 _{c''}) \circ \iota _{1 _{c''}} \\ = & F(f,1 _{c''}) \circ F(g,1 _{c''}) \circ \iota _{1 _{c''}} \\ = & F(f,1 _{c''}) \circ \iota _{g}. \end{align*}