An absolute colimit is a colimit which is preserved by any functor whatsoever. In general this happens because the colimit is a colimit for purely “diagrammatic” reasons. The notion is most important in enriched category theory.
Of course, there is a dual notion of absolute limit, but it is used less frequently.
The term “absolute colimit” is actually used for two closely related, but distinct, notions.
Note, however, that a conical colimit? in a -category may be preserved by all -functors without being preserved by all unenriched functors on the underlying ordinary category . Thus, for clarity we may speak of a colimit being -absolute.
For a given , a weight for colimits is an absolute weight, or a weight for absolute colimits, if -weighted colimits in all -categories are preserved by all -functors.
Absolute colimits of this sort are also called Cauchy colimits. A -category which admits all absolute colimits — that is, all -weighted colimits whose weights are absolute – is called Cauchy complete. By the characterization below, it is equivalent to admit limits weighted by all weights mature for absolute limits.
Both types of absolute colimits admit pleasant characterizations.
For a particular cocone under a functor (all in the Set-enriched world), the following are equivalent:
is an absolute colimiting cocone.
is a colimiting cocone which is is preserved by the Yoneda embedding .
is a colimiting cocone which is preserved by the representable functors (for all ) and .
There exists and such that
The equivalence of the first two is basically because the Yoneda embedding is the free cocompletion of . The third clearly follows from the second. The fourth follows from the third by inspecting exactly what preservation by those representables means in terms of colimits in Set (as is explained in more detail in the special case of absolute coequalizers). Finally, it is straightforward to check that the fourth implies that is colimiting, and it is clearly a property preserved by any functor.
It is also possible to prove directly that the third condition implies the first two, without extracting the fourth condition along the way. Namely, Let be the full subcategory of consisting of the objects and . Then defines a functor ; call it . Note that is also the colimit of in . Moreover, by the equivalence of the first two conditions, is an absolute colimit of , since by hypothesis it is preserved by all representable functors out of . Therefore, this colimit is in particular preserved by the inclusion , along with its composite with any functor out of ; so is an absolute colimit of .
is a weight for absolute colimits (i.e. -weighted limits in any -category are preserved by all -functors)
There is a weight such that -weighted colimits coincide naturally with -weighted limits.
Of course, every colimit weighted by a weight for absolute colimits is itself a particular absolute colimit. But it may also happen that a particuclar colimit may be absolute without all colimits of that shape being absolute. For example (in ordinary category theory, with ):
We can also say something about non-examples.
Initial objects (in -enriched categories) are never absolute. If is an initial object, then it is never preserved by the representable functor .
Similarly, coproducts in -enriched categories are never absolute.
In ordinary -enriched category theory there are not very many weights for absolute colimits, but we have
In fact, this example is “universal,” in that an ordinary category is Cauchy complete iff it has split idempotents, although not every absolute colimit “is” the splitting of an idempotent. More precisely, the class of absolute -limits is the saturation of idempotent-splittings.
In enriched category theory there can be more types of absolute colimits. For instance:
in Ab-enriched categories (or, more generally, categories enriched over commutative monoids), finite biproducts are absolute. Finite biproducts and splitting of idempotents together are “universal” absolute colimits for Ab-enrichment.
in SupLat-enriched categories, arbitrary small biproducts are absolute, and together with splitting of idempotents these generate all absolute colimits.
in Lawvere metric spaces, limits of Cauchy sequences are absolute. This is the origin of the name “Cauchy colimit.”
in categories enriched? over the bicategory (or double category) of relations in a site, gluings are absolute. In this case the enriched categories can roughly be identified with separated presheaves? and the Cauchy-complete ones with sheaves.
New kinds of absolute (co)limits also arise in higher category theory.
for (∞,1)-categories (enriched over -groupoids), splitting of idempotents is a universal absolute colimit.
Robert Pare, On absolute colimits, J. Alg. 19 (1971), 80-95.