Ends and coends with parameters

Let $\mathcal{P}$, $\mathcal{C}$ and $\mathcal{D}$ be categories and let

$T:\mathcal{P}\times {\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}⟶\mathcal{D}$

be a functor (Product category and its projection functors, Covariant functor, identity functor, composite functor, and contravariant functor). For an object $p$ of $\mathcal{P}$ write $T(p,-,-):{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$ for the functor obtained by holding the first variable at $p$ and the identity of $p$; that this is a functor is immediate from the functor laws for $T$ applied to morphisms whose first coordinate is $1_p$.

A parametrised end of $T$ is a choice, for every object $p$ of $\mathcal{P}$, of an end of $T(p,-,-)$ (The end and the coend of a functor ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$): that is, an end taken in the two dinatural variables with the remaining variables held fixed. Its vertex at $p$ is written ${\int }_{c}T(p,c,c)$ and its wedge components ${\omega }_c^p$ (Wedges and cowedges, and the categories they form). A parametrised coend is a choice of a coend of $T(p,-,-)$ for every $p$, with vertex ${\int }^{c}T(p,c,c)$ and cowedge components ${\rho }_c^p$.

The variable $p$ is the parameter and the variables in the second and third slots are the dinatural variables. Nothing here asserts that the vertices assemble into a functor of $p$: that is a further statement, and it is proved in A chosen family of ends is the object part of exactly one functor making the counit natural in the parameters from the choice given here.

The definition is stated as a choice rather than as an operation for a reason that is not bookkeeping. The parameter category may have a proper class of objects, and an end is only determined up to isomorphism, so "the" end at every parameter is not a well-defined assignment until one end has been selected at each parameter. Every statement below that treats a parametrised end functorially therefore carries that choice as a hypothesis.

Several parameters are covered by the same definition, since a product of parameter categories is again a parameter category. The case $\mathcal{P}={\mathcal{E}}^{\mathrm{o}\mathrm{p}}\times \mathcal{E}$ is the one that appears in the Fubini theorem, where the parameter itself is later made dinatural.