Limits and Colimits

The limit of a diagram is the universal object carrying a compatible cone of projections , terminal among all cones over . Dually, the colimit receives a compatible cocone and is initial among cocones, the “gluing” of the diagram. The opposite category exchanges the two, since . Products, pullbacks, and equalizers are the limits over discrete, cospan, and parallel-pair shapes, whereas coproducts, pushouts, and coequalizers are the dual colimits.

Example. Initial and terminal objects

Over the empty diagram a limit is a terminal object , admitting a unique map in from every object, and a colimit is an initial object , admitting a unique map out to every object. In , and , so the empty colimit is the empty set.

Example. Equalizer and inverse limit

The equalizer of is the universal with , the part of where the maps agree. A fixed set is the equalizer of and . Over , the inverse or tower limit of is the universal compatible family. The invariant of an endomorphism is this limit, taken over the tower with every map .

Definition. Coend

For , the coend is the colimit coequalizing the two actions of each morphism, a gluing with identifications built in that fuses an object appearing in several places into one. Along monomorphisms it agrees with the plain colimit.

Property. Over a poset

Over a partially-ordered-set, a limit is the greatest lower bound, or meet, of the diagram’s image, and a colimit is the least upper bound, or join. Gluing subobjects along overlaps is their union, identified where they agree.

Twisted-arrow indexing of ends and coends

The twisted-arrow category is the indexing shape that presents an end or a coend as an ordinary limit or colimit, so that a coend is an initial cowedge exactly as a colimit is an initial cocone. It is the category of elements of the hom-functor, , whose objects are the arrows of and whose morphisms conjugate into on the two sides. The canonical projection

sends an arrow to its source and target, so a functor restricts along it to a diagram on whose limit is the end and whose colimit is the coend .

Wedges and cowedges are the twisted analogues of cones and cocones, indexed through this projection, whereby each universal construction is again a terminal or initial object over :

construction	shape over	universal object
limit	cones	terminal cone
colimit	cocones	initial cocone
end	wedges	terminal wedge
coend	cowedges	initial cowedge

A natural transformation between such functors organises into a comma category whose objects are triples , since cowedges for are exactly its objects with constant and the coend is its initial object.

References

• Saunders Mac Lane, Categories for the Working Mathematician (1971), chs. III, V (limits, colimits, ends/coends)
• Fosco Loregian, (Co)end Calculus, Cambridge Univ. Press (2021)