Wedges and cowedges, and the categories they form

Let $T:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$ be a functor (Product category and its projection functors, Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$) and let $d$ be an object of $\mathcal{D}$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection). Write $\mathrm{\Delta }d$ for the constant functor on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ at $d$ (Constant diagrams, cones, cocones, and their morphisms), which sends every object to $d$ and every morphism to $1_d$.

A wedge from $d$ to $T$ is a dinatural transformation from a constant functor to $T$ (Dinatural transformation between functors on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$), that is, a family

${\omega }_c:d⟶T(c,c)(c\in \mathrm{Ob}(\mathcal{C}))$

such that every $f:c\to c^{\mathrm{\prime }}$ satisfies the wedge equation

$T(1_c,f)\circ {\omega }_c=T(f,1_{c^{\mathrm{\prime }}})\circ {\omega }_{c^{\mathrm{\prime }}}$

between morphisms $d\to T(c,c^{\mathrm{\prime }})$. Dually, a cowedge from $T$ to $d$ is a dinatural transformation from $T$ to a constant functor, that is, a family

${\rho }_c:T(c,c)⟶d$

such that every $f:c\to c^{\mathrm{\prime }}$ satisfies the cowedge equation

${\rho }_c\circ T(f,1_c)={\rho }_{c^{\mathrm{\prime }}}\circ T(1_{c^{\mathrm{\prime }}},f)$

between morphisms $T(c^{\mathrm{\prime }},c)\to d$. The object $d$ is the vertex of the wedge or cowedge, and ${\omega }_c$, ${\rho }_c$ are its components.

A morphism of wedges $(d,\omega )\to (d^{\mathrm{\prime }},{\omega }^{\mathrm{\prime }})$ is a morphism $h:d\to d^{\mathrm{\prime }}$ of $\mathcal{D}$ satisfying ${\omega }_c^{\mathrm{\prime }}\circ h={\omega }_c$ for every object $c$; a morphism of cowedges $(d,\rho )\to (d^{\mathrm{\prime }},{\rho }^{\mathrm{\prime }})$ is a morphism $h:d\to d^{\mathrm{\prime }}$ satisfying $h\circ {\rho }_c={\rho }_c^{\mathrm{\prime }}$ for every $c$. Identities of $\mathcal{D}$ are morphisms of wedges and of cowedges, and a composite of two such morphisms again satisfies the displayed equation, so wedges over $T$ and their morphisms form a category $\mathrm{Wd}(T)$, and cowedges under $T$ and their morphisms form a category $\mathrm{Cwd}(T)$; associativity and the identity laws are inherited from $\mathcal{D}$.

The wedge and cowedge equations are the hexagon of Dinatural transformation between functors on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ with one side made constant. For a wedge the source is $\mathrm{\Delta }d$, so both $\mathrm{\Delta }d(f,1_c)$ and $\mathrm{\Delta }d(1_{c^{\mathrm{\prime }}},f)$ are $1_d$ and drop out; for a cowedge the target is $\mathrm{\Delta }d$ and the two outer morphisms on the target side drop out instead.

A wedge is not a cone over a diagram indexed by $\mathcal{C}$: its components sit at the diagonal values $T(c,c)$ and its equation involves the two off-diagonal values $T(c,c^{\mathrm{\prime }})$, whereas a cone (Constant diagrams, cones, cocones, and their morphisms) has one component per object of the index category and one equation per morphism, with no off-diagonal term. The precise comparison between the two shapes is An end is a limit over the twisted arrow category, and a coend is a colimit over its opposite, which replaces the index category by another one.