The n-Category Café

An absolute colimit is a colimit that’s preserved by any functor whatsoever. The term is used in two slightly different ways, which can be confusing if you don’t watch out for it.

On the one hand, a particular colimit in a particular category $C$ is absolute if it is preserved by any functor with domain $C$. On the other hand, if $V$ is a nice category to enrich over, and $W$ is a weight for colimits in $V$-categories, then $W$ is absolute if all $W$-weighted colimits in all $V$-categories are preserved by all $V$-functors.

There is obviously a close relationship between the two meanings; the subtlest part is probably that an ordinary colimit can be preserved by all $V$-functors without being preserved by all unenriched functors. For instance, initial objects are never absolute in the first, unenriched, sense — but the weight for initial objects is an absolute weight when $V=$ pointed sets or abelian groups.

In this post I’m interested in the first meaning of absolute colimit; no enrichments today.

I would guess that I’m fairly typical in that my first exposure to absolute colimits was through Beck’s monadicity theorem, in which one uses split coequalizers, and remarks that they are a special sort of absolute coequalizer. An obvious question to ask after you first meet these notions is “are there absolute coequalizers that aren’t split?” If you’ve never thought about this question, then I encourage you to go away and work on it for a little while. Go on, I’ll wait.

Back? So you might have realized that there is a very trivial sort of absolute coequalizer that is not split: if $f\colon A\to B$ is a morphism that is not a split epi, then $1_B$ is a coequalizer of $f$ and $f$, which is not split. But there can also be other more interesting absolute coequalizers that are not split.

If you’re a seasoned category theorist who automatically reaches for the Yoneda lemma, then you might also have realized that there’s a very clever way to characterize all absolute coequalizers, and which is a generalization of the notion of split coequalizer. I believe this is originally a result of Bob Paré from 1969. The key is this: suppose we have an absolute coequalizer diagram $$X\; \underoverset{f_1}{f_0}{\rightrightarrows}\; Y \overset{e}{\to} Z.$$ Then this coequalizer must, in particular, be preserved by the representable functor $hom(Z,-)$. Therefore, we have another coequalizer diagram $$hom(Z,X)\; \underoverset{f_1\circ -}{f_0\circ -}{\rightrightarrows}\; hom(Z,Y) \overset{e\circ -}{\to} hom(Z,Z)$$ But this is a coequalizer in $Set$, and we know what coequalizers in $Set$ look like. In particular, coequalizers in $Set$ are surjective, and so $(e\circ -)\colon hom(Z,Y) \to hom(Z,Z)$ must be surjective. Thus, in even more particular, there must be something in $hom(Z,Y)$ which maps onto $1_Z \in hom(Z,Z)$. That just says that $e$ is split epic (just as it must be in a split coequalizer).

Now our coequalizer diagram must also be preserved by $hom(Y,-)$, and from that and our knowledge of coequalizers in $Set$, we can extract a generalization of the rest of the split coequalizer condition. See absolute coequalizer for details.

It turns out that this technique works in arbitrary generality. In fact, by abstract nonsense, a colimit is absolute if and only if it is preserved by the Yoneda embedding $C\to [C^{op},Set]$. And by looking at presenvation by particular hom-functors, we can extract a characterization of general absolute colimits. Bob Paré did this in 1971.

Specifically, let $\mu \colon F \to \Delta A$ be a cocone under a functor $F\colon I\to C$. Then the following are equivalent:

• $\mu$ is an absolute colimiting cocone.

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

• 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. $\mu_{i_0} \circ d_0 = 1_{A}$.

In particular, there exists an $i_0$ such that $\mu_{i_0}$ is split epic, generalizing our above observation that absolute coequalizers are split epis. The rest of the characterization is likewise a generalization of the part of the characterization of absolute coequalizers that I didn’t mention.

So far, so good. Now what is a postulated colimit? Anders Kock introduces the notion as follows:

To say that a diagram

$$R \underoverset{b}{a}{\rightrightarrows} X \xrightarrow{q} Q$$

in the category of sets is a coequalizer may be expressed in elementary terms by saying that $q \circ a = q \circ b$, and that the following two assertions hold

(1.1) $q$ is surjective

(1.2) for any $x$ and $y$ in $X$ with $q(x) = q(y)$, there exists a finite chain $z_1, \dots, z_m$ of elements of $R$ with $x = a(z_1)$, $b(z_1) = a(z_2)$, … ,$b(z_m) = y$.

These assertions can be interpreted in any category where sheaf semantics is available; this means in any site…. If they hold for a given diagram in the site, we shall say that the diagram is a postulated coequalizer.

This already looks a bit familiar; those two characterizing facts about coequalizers in $Set$ are exactly the same properties that we used, after applying some representable functors, in order to characterize absolute coequalizers. But then Kock goes in a seemingly different direction: he takes these two statements and interprets them in the internal logic of a category.

Recall (if you knew it) that the internal logic of a category $C$ is a way of “interpreting” or “compiling” mathematical statements which look like they are talking about sets into statements which talk about objects of $C$ instead. For instance, given a function $q\colon X\to Q$ between two sets, the statement “for all $y\in Q$, there exists an $x\in X$ with $q(x)=y$” expresses the surjectivity of $q$. In the internal logic of a topos, however, our function would be replaced by a morphism $q\colon X\to Q$ between objects, and the same statement “for all $y\in Q$, there exists an $x\in X$ with $q(x)=y$” would get compiled into one which turns out to express that $q$ is an epimorphism.

So Kock is saying that in a topos, we can take his conditions (1.1) and (1.2) and interpret them “internally”. As I said above, condition (1.1) will just become the assertion that $q$ is an epimorphism. Condition (1.2) will become something somewhat more mysterious. Regardless, a fork with these properties is called a postulated coequalizer — “postulated” I guess because the internal logic “postulates” that it is a colimit.

More generally, we can do something analogous for a cocone under any diagram and obtain a notion of postulated colimit. In that case the analogue of condition (1.1) will assert that the coprojections in the cocone are jointly epic.

Finally, generalizing in another direction, we have an “internal logic” in any site, which is basically just the restriction of the internal logic of its topos of sheaves. So we can define postulated colimits in any site. In that case, the analogue of condition (1.1) will assert that the coprojections of the cocone form a covering family.

Now a priori, it may not be obvious that a postulated colimit even is a colimit! We have these odd conditions about epimorphisms, but why should that imply a universal mapping property? However, Kock proves that if the site is subcanonical, then a postulated colimit is indeed a colimit.

You could say that this works because in a subcanonical site, the covering families themselves are already colimits (namely, they are universally effective-epimorphic), so that that colimit-ness can be extended to the postulated colimits which are defined in terms of the covering families. Alternatively, you could say that it works because a subcanonical site embeds fully-faithfully in its topos of sheaves, and a topos is sufficiently set-like that the “internal” characterization of colimits works there for the same reason that it does in $Set$.

This leads us to another of Kock’s characterizations of postulated colimits: a cocone in a site $C$ is a postulated colimit if and only if it becomes a colimit in the topos of sheaves $Sh(C)$. In particular, a colimit in $C$ is a postulated colimit if and only if it is preserved by the sheafified Yoneda embedding $C\to Sh(C)$.

Now we can draw the loop closed. Recall that a colimit in $C$ is absolute if and only if it is preserved by the ordinary Yoneda embedding $C\to [C^{op},Set]$. But every category $C$ can be made into a site with a “trivial topology”, for which $Sh(C) \simeq [C^{op},Set]$. Therefore, a colimit is absolute if and only if it is postulated by the trivial topology. (Note that since the trivial topology is subcanonical, every postulated colimit in a trivial site is in fact a colimit.)

The especially nice thing is that when we take the definition of postulated colimit and “$\beta$-reduce it” in the case of the trivial topology, we recover (as we must) Paré’s characterization of absolute colimits. The simple half of this is easy to see: in the trivial topology, the covering families are precisely those which contain a split epimorphism, so Kock’s condition (1.1) for the trivial topology reduces exactly to the part of Paré’s condition which says that some coprojection is split epi. The correspondence of the other two conditions is more tedious, but basically the same.

Let me end with the following suggestive remark. Since the notion of postulated colimit is defined entirely in terms of the topology of a site, it’s immediate that postulated colimits are preserved by any morphism of sites. And since the sheafified Yoneda embedding $C\to Sh(C)$ is a morphism of sites, this property also characterizes postulated colimits.

However, this is reminiscent of the second type of absolute colimit I mentioned back at the beginning: the weights whose colimits are preserved by every enriched functor. Perhaps a topology on a category is something akin to an enrichment of it, and postulated colimits are the “absolute weights” for such an “enrichment”. Moreover, then the topos of sheaves, which is essentially a cocompletion under postulated colimits, would be the “Cauchy completion” relative to this “enrichment”. If you’ve read sections 6–8 of my exact completions paper, you may be able to guess what’s going on.