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Let $\mathcal V$ be a good enriching category. Recall that an enriched limit weight $\phi: D \to \mathcal V$ is called absolute if $\phi$-weighted limits are preserved by any $\mathcal V$-enriched functor whatsoever. If this is a familiar concept, please jump to my question at the end. But otherwise, I will provide some context. For instance,

  • Idempotent splitting is absolute for $Set$-enrichment and (hence) for any enriching category;

  • Finite products are absolute for enrichment in abelian groups, commutative monoids, etc;

  • Limits of Cauchy sequences are absolute for enrichment in $([0,\infty],\geq,+)$;

  • $\infty$-categorically, finite limits are absolute for enrichment in spectra;

  • Eilenberg-Moore objects for idempotent monads are absolute for enrichment in $Cat$.

There are a number of general things to say here:

  • (Street) A $\mathcal V$-weight $\phi: D \to \mathcal V$ is absolute if and only if $\phi$ has a left adjoint $\bar \phi: D^{op} \to \mathcal V$ in the bicategory of $\mathcal V$-categories and $\mathcal V$-profunctors.

  • Following Lawvere and the third example above, a $\mathcal V$-category with all absolute $\mathcal V$-limits is called Cauchy complete. The Cauchy completion of a $\mathcal V$-category $C$ is obtained by freely adjoining absolute $\mathcal V$-limits to $C$, and may be constructed as the $\mathcal V$-category of absolute weights on $C$. This construction exhibits the Cauchy complete $\mathcal V$-categories as a reflective subcategory of $\mathcal V\text{-}Cat$.

  • Thus a $\mathcal V$-weight $\phi: D \to \mathcal V$ is absolute if and only if it lies in the Cauchy completion of $D$.

These general facts are useful in working out explicit characterizations of the absolute weights for particular $\mathcal V$:

  • An ordinary ($Set$-enriched) category $C$ is Cauchy-complete iff $C$ has all split idempotents; a $Set$-enriched weight $\phi: D \to \mathcal V$ is absolute iff it is a retract of a representable (and in particular an ordinary conical limit is absolute iff the indexing diagram has a cofinal idempotent);

  • An $Ab$-enriched category $C$ is Cauchy-complete iff $C$ has split idempotents and finite products;

  • A generalized metric space is Cauchy-complete iff it is Cauchy-complete in the usual sense;

  • A spectrally-enriched $\infty$-category is Cauchy-complete iff it is stable with split idempotents;

But I don't know an analogous statement for $Cat$-enrichment.

Question: When does a 2-category $C$ have all absolute weighted 2-limits?

Clearly $C$ must have split idempotents and Eilenberg-Moore objects for idempotent monads -- do these suffice? Or are there more absolute 2-limit weights out there which can't be built from these?

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    $\begingroup$ I believe this is answered for general weak n-categories in arxiv.org/pdf/1905.09566.pdf (I'm not putting this as an answer because I havn't read the paper in enough detail to be really sure it answer your question, also it deals with weak, and not strict n-categories) $\endgroup$ Mar 25, 2020 at 19:04
  • $\begingroup$ @SimonHenry Thank you-- this paper is inspiring, and potentially fully answers my question! It's also frustrating -- they seem to be trying to work with weak $(n,n)$-categories in a "model-independent" way, but with the exception of one section at the end, they are not very clear about precisely what general properties they require their model of weak $n$-categories to satisfy. As a result, many of the definitions, statements, and proofs feel imprecise to me. In fact, I think I'd feel more comfortable if they just claimed that their results hold for strict $n$-categories rather than weak ones! $\endgroup$
    – Tim Campion
    Mar 26, 2020 at 14:10
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    $\begingroup$ I see your point, though I guess if you only care about the case of 2-categories that shouldn't really be a problem. For the record, if you plan to study this in more details, I would be interested if you wrote as an answer to your own question the details of what their results means for 2-categories. $\endgroup$ Mar 26, 2020 at 14:48
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    $\begingroup$ I just came across this question. Our paper arxiv.org/pdf/1905.09566.pdf mentioned by @SimonHenry claims an answer to Tim's question. I am very sympathetic, though, to Tim's comment about models and model independence. To respond to his comment about strict versus weak: I actually don't believe our results in too strict a world. In particular, if you input a strict 2-category, say, and then write down our Karoubi completion, it will not be strict. You could strictify for 2-categories, but I think for 3-categories you may find non-strictifiable things coming out of our construction. $\endgroup$ Jul 13, 2021 at 17:55

2 Answers 2

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In my paper Condensations in higher categories joint with Davide Gaiotto, we claim the following answer. It deserves someone to write a careful model-dependent paper confirming it. I believe Martin Szyld has been thinking about how to do this.

Define a 2-idempotent to be an endomorphism $e : X \to X$ together with a retract $\mu: e^2 \Leftrightarrow e : \mu^*$, by which I mean 2-morphisms $\mu$ and $\mu^*$ such that $\mu \cdot \mu^* = \mathrm{id}_e$, which is associative, coassiciative, and Frobenius. [In spite of the name "$\mu^*$", it is just another 2-morphism, and isn't required to be the adjoint or dual in any sense beyond the requirement $\mu \cdot \mu^* = \mathrm{id}_e$. In particular, $\mu$ constrains but does not determine $\mu^*$.]

By Frobenius I mean that the three natural maps $e^2 \Rightarrow e^2$, namely $\mu^* \cdot \mu$, $(\mu \circ \mathrm{id}_e) \cdot \mathrm{assoc} \cdot (\mathrm{id}_e \circ \mu^*)$, and $(\mathrm{id}_e \circ \mu) \cdot \mathrm{assoc}^{-1} \cdot (\mu^* \circ \mathrm{id}_e)$, are all equal. [Actually, the Frobenius axiom together with $\mu \cdot \mu^* = \mathrm{id}_e$ imply the associativity of $\mu$ and coassociativity of $\mu^*$.] Here my notation is that $\circ$ is the composition in the 1-morphism direction, and $\cdot$ for 2-morphisms, and $\mathrm{assoc} : e \circ e^2 \cong e^2 \circ e$ is the associator.

This same notion goes by many names. For instance, it is a nonunital separated monad, or a nonunital special Frobenius monad. It is almost but not quite the same as "separable monad" used by Douglas and Reutter. In our paper, we give it yet another name — "condensation monad". But Reutter and I have started saying "2-idempotent" when we talk to each other, and perhaps it is the best name.

A 2-idempotent $(X,e,\dots)$ splits when there is an object $Y$, 1-morphisms $f : X \leftrightarrows Y: f^*$ [again, in spite of the name, I don't require any duality/adjunction], and some 2-morphisms and equations which I will list. First, I require the data of a retract $\phi : f\circ f^* \Leftrightarrow \mathrm{id}_Y : \phi^*$, i.e. 2-morphisms such that $\phi \cdot \phi^* = \mathrm{id}_{\mathrm{id}_Y}$. Now, using the retract $(\phi,\phi^*)$, I claim that you can give the composition $f^*\circ f$ the data of a 2-idempotent. Specifically, you set $\mu : (f^* \circ f) \circ (f^* \circ f) \Rightarrow (f^* \circ f) \circ (f^* \circ f)$ to be what you get by using an assotiator $(f^* \circ f) \circ (f^* \circ f) \cong f^* \circ (f \circ f^*) \circ f$ and then applying $\mathrm{id}_{f^*} \circ \phi \circ \mathrm{id}_f$ and then applying some unitors; and $\mu^*$ is the reverse. The last datum needed to say that $(X,e,\dots)$ splits is an isomorphism $e \cong f^* \circ f$ of 2-idempotents on $X$.

To connect with things you know: 2-idempotents are a version of monads, and splittings are a version of Eilenberg–Moore objects. The difference is that mine are not unital and are separable, in fact separated.

Then our claim is that a weak 2-category is Cauchy complete when (and only when) it is locally idempotent complete [i.e. all hom-categories are idempotent complete] and also every 2-idempotent splits. If your 2-category is locally idempotent, then a splitting of a 2-idempotent is unique up to unique isomorphism. I forget if this is true without local idempotent completion.


We also claim an explicit construction of the 2- (indeed, $n$-)categorical Karoubi completion. Given any 2-category, the first step is to locally Karoubi complete it. [Easy exercise: write down the 1-morphism composition in the local Karoubi-completion of a 2-category.] Now given a locally Karoubi complete 2-category $\mathcal{C}$, I build a new 2-category whose objects are the 2-idempotents in $\mathcal{C}$. A morphism $(X,e,\dots) \to (Y,f,\dots)$ is a morphism $m : X \to Y$ together with retracts $m\circ e \Leftrightarrow m$ and $f \circ m \Leftrightarrow m$ such that a bunch of equations hold making $m$ into a bimodule and bicomodule and cetera. A "bi-bimodule", perhaps? The 2-morphisms are natural: they are the homomorphisms of these bi-bimodules.

The interesting thing is the composition. Given $(m, \dots) : (X,e,\dots) \to (Y,f,\dots)$ and $(n, \dots) : (Y,f,\dots) \to (Z,g,\dots)$, I can look at $n \circ m : X \to Z$. Now I claim you can write down an idempotent $n \circ m \Rightarrow n\circ m$. The trick is to map $n \circ m \Rightarrow n \circ f \circ m \Rightarrow n \circ m$ where you use the $f$-coaction on $m$, and then the $f$-action on $n$. Well, you could have used the $f$-coaction on $n$ and then the $f$-action on $m$, but it turns out you will get the same thing. Since your 2-category is by assumption locally idempotent complete, you can split this idempotent. Actually, $n \circ m$ was already a bi-bimodule, and the idempotent you write down on it is a moprhism of bi-bimodules. So the splitting is a bi-bimodule. This is the composition in the 2-Karoubi completion.

You can see that to write this down as a bicategory, say, would require making some arbitrary choices: I just told you "split the thing", and not which splitting to take, so even if you started life as a strict 2-category, you won't end up strict, and you should not expect our construction to lead to any statement of the form "this is the Cauchy completion" in the world of strict 2-categories and their functors.

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A 2-category is Cauchy complete (in the sense you describe) if and only if idempotents splits, which is, if and only if its underlying ordinary category is Cauchy complete.

This holds more generally in this context: the base of enrichement $\mathcal V=(\mathcal V_0,\otimes,I)$ is symmetryc monoidal closed and locally presentable, and the functor $\mathcal V_0(I,-):\mathcal V_0\to\mathbf{Set}$ is (weakly) cocontinuous and (weakly) strong monoidal. (weakly here means that the induced comparison maps need not be bijections, but just surjections.) In this case a $\mathcal V$-category is Cauchy complete if and only if it has splittings of idempotents. This is proven for example in Corollary 3.16 here.

The idea is that, given an absolute $\mathcal V$-weight $\phi\colon\mathcal D\to\mathcal V$, then the category of elements $\mathcal E$ of $\mathcal V_0(I,\phi_0-)\colon \mathcal D_0\to\mathbf{Set}$ is absolute (in the ordinary sense, i.e. $\mathcal E$-colimits are preserved by any functor) and that $\phi$ can be written as a conical colimit $$ \phi\cong \textit{colim}(\mathcal{E}_{\mathcal V}\to\mathcal D^{op}\to[\mathcal D,\mathcal V] )$$ indexed on $\mathcal E$.

Examples of such bases are of course $\mathbf{Cat}$ and $\mathbf{Set}$, but also $\mathbf{SSet}$, $\mathbf{Pos}$, and $\mathcal V\textit{-}\mathbf{Cat}$ whenever $\mathcal V$ is locally presentable.

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  • $\begingroup$ Thanks, this is really neat -- I will have to dig into your preprint! I'm a little surprised, in your last sentence, by the example of $Ab-Cat$ -- I would have thought $Ab-Cat$-enriched categories should have absolute finite coproducts. But thinking about it now, I suppose it's clear that $Ab-Cat-Cat$ doesn't have a zero object, dashing my expectations! $\endgroup$
    – Tim Campion
    Jul 20, 2021 at 13:03

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