nLab absolute colimit

Idea

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.

Definitions

The term “absolute colimit” is actually used for two closely related, but distinct, notions.

Particular absolute colimits

Definition

A particular colimit diagram in a particular category $C$ is an absolute colimit if it is preserved by every functor with domain $C$.

This definition makes sense also in enriched category theory: for any $V$, a weighted colimit in a particular $V$-enriched category $C$ is an absolute colimit if it is preserved by every $V$-functor with domain $C$.

Note, however, that a conical colimit in a $V$-category $C$ may be preserved by all $V$-functors without being preserved by all unenriched functors on the underlying ordinary category $C_o$.

Generalizing in a different direction, absolute colimits in $Set$-enriched categories can be regarded as the particular case of postulated colimit?s in sites where the site has the trivial topology.

Weights for absolute colimits

Definition

For a given $V$, a weight $\Phi\colon D^{op} \to V$ for colimits is an absolute weight, or a weight for absolute colimits, if $\Phi$-weighted colimits in all $V$-categories are preserved by all $V$-functors.

Absolute colimits of this sort are also called Cauchy colimits. A $V$-category which admits all absolute colimits — that is, all $V$-weighted colimits whose weights are absolute – is called Cauchy complete. By the characterization below, it is equivalent to admit limits weighted by all weights for absolute limits.

Characterizations

Both types of absolute colimits admit pleasant characterizations.

Particular absolute colimits

For a particular cocone $\mu \colon F \to \Delta A$ under a functor $F\colon I\to C$ (all in the Set-enriched world), the following are equivalent:

• $\mu$ is an absolute colimiting cocone.

• $\mu$ is a colimiting cocone which is is preserved by the Yoneda embedding $C \hookrightarrow [C^{op},Set]$.

• $\mu$ is a colimiting cocone which is preserved by the representable functors $C(F(i),-)\colon C\to Set$ (for all $i\in I$) and $C(A,-)\colon C\to Set$.

• There exists $i_0\in I$ and $d_0\colon A\to F(i_0)$ such that

  1. For every $i\in I$, $d_0 \circ \mu_i$ and $1_{F(i)}$ are in the same connected component of the comma category $(F(i) / F)$.
  2. $\mu_{i_0} \circ d_0 = 1_{A}$.

The equivalence of the first two is basically because the Yoneda embedding is the free cocompletion of $C$. 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 $\mu$ is colimiting, and it is clearly a property preserved by any functor.

Note that in particular, the fourth condition implies that $A$ is a retract of $F(i_0)$. Also, the first half of the fourth condition by itself characterizes absolute weak colimits.

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 $B$ be the full subcategory of $C$ consisting of the objects $F(i)$ and $A$. Then $F$ defines a functor $I\to B$; call it $F'$. Note that $A$ is also the colimit of $F'$ in $B$. Moreover, by the equivalence of the first two conditions, $A$ is an absolute colimit of $F'$, since by hypothesis it is preserved by all representable functors out of $B$. Therefore, this colimit is in particular preserved by the inclusion $B\hookrightarrow C$, along with its composite with any functor out of $C$; so $A$ is an absolute colimit of $F$.

Weights for absolute colimits

Let $V$ be a Bénabou cosmos and $J\colon K ⇸ A$ a $V$-profunctor. Then the following are equivalent:

• $J$ is a weight for absolute colimits (i.e. $J$-weighted colimits in any $V$-category are preserved by all $V$-functors)

• $J$ has a right adjoint $J^*$ in the bicategory $V$-Prof.

• There is a weight $J^*\colon A ⇸ K$ such that $J$-weighted colimits coincide naturally with $J^*$-weighted limits.

Examples

Particular absolute colimits

Of course, every colimit weighted by a weight for absolute colimits is itself a particular absolute colimit. But it may also happen that a particular colimit may be absolute without all colimits of that shape being absolute. For example (in ordinary category theory, with $V=Set$):

• split coequalizers are absolute, and figure in Beck’s monadicity theorem.
• More generally (of course), absolute coequalizers are absolute.
• absolute pushouts appear in elegant Reedy categories.

We can also say something about non-examples.

• Initial objects (in $Set$-enriched categories) are never absolute. If $0$ is an initial object, then it is never preserved by the representable functor $C(0,-)\colon C \to Set$.

• Similarly, coproducts in $Set$-enriched categories are never absolute.

Weights for absolute colimits

In ordinary $Set$-enriched category theory there are not very many weights for absolute colimits, but we have

• split idempotents.

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 $Set$-limits is the saturation of idempotent-splittings.

In enriched category theory there can be more types of absolute colimits. For instance:

• in categories with zero morphisms (that is, enriched over pointed sets), initial objects are absolute.

• 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 dg-categories (or more generally, categories enriched over graded sets), shifts/suspensions and mapping cones are absolute.

• in Lawvere metric spaces, limits of Cauchy sequences are absolute. This is the origin of the name “Cauchy colimit.”

• in posets, suprema of subsets with a greatest element are absolute.

• 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.

• in categories enriched over rational vector spaces, quotients by finite group actions are absolute.

New kinds of absolute (co)limits also arise in higher category theory.

• for (∞,1)-categories (enriched over $\infty$-groupoids), splitting of idempotents is a universal absolute colimit.

• in stable (∞,1)-categories (which are enriched, in the $(\infty,1)$-categorical sense, over the $(\infty,1)$-category of spectra), initial objects and pushouts are absolute, and therefore so are all finite colimits.

Generalisations

If $K : A \to C$ is a functor, then a colimit in $A$ is $K$-absolute if it is preserved by the nerve $N_K : C \to [A^{\mathrm{op}}, \mathscr{V}]$ of $K$. This generalises absoluteness, taking $K$ to be the identity functor.

References

• Ross Street, Absolute colimits in enriched categories, Cahiers 1983
• Robert Pare, On absolute colimits, J. Alg. 19 (1971), 80-95.
• Max Kelly, Basic Concepts of Enriched Category Theory, Cambridge University Press, Lecture Notes in Mathematics 64 (1982)
• Mike Shulman‘s answer to What are all of the exactness properties enjoyed by stable ∞-categories?