## October 26, 2017

### Categorification and the Cosmic Cube

#### Posted by David Corfield

I see that Tobias Dyckerhoff’s A categorified Dold-Kan correspondence has just appeared, looking, as its title suggests, to categorify the nLab: Dold-Kan correspondence. As it says there, the latter

interpolates between homological algebra and general simplicial homotopy theory.

So with Dyckerhoff’s paper we seem to be dipping down to the lower layer of the flamboyantly named ‘cosmic cube’, see slide 10 of these notes by John, and discussed at nLab: cosmic cube. Via chain complexes of stable $(\infty, 1)$-categories Dyckerhoff speaks of a ‘categorified homological algebra’, and also through a categorified Eilenberg-Mac Lane spectrum, of a ‘categorified cohomology’.

For old time’s sake, let’s see if anyone is up for the kind of grand vision thing we used to talk about. For one thing we might wonder what plays the role of a categorified homotopy theory, the kind of world where Mike’s suggestions on directed homotopy type theory might find a home.

I see I was raising stratified spaces as relevant back there. In the meantime we now have useful models from A stratified homotopy hypothesis. It turns out that $(\infty, 1)$-categories are equivalent to ‘striation sheaves’, a certain kind of sheaf on ‘conically smooth’ stratified spaces. The relevant fundamental $(\infty, 1)$-category is the exit-path $(\infty, 1)$-category, rather than the entry and exit paths of our older discussions which brought in duals.

In line with Urs’s claim that cohomology concerns mapping spaces in $(\infty, 1)$-toposes (nLab: cohomology), perhaps for categorified cohomology we should be looking for parallels in $(\infty, 2)$-toposes, an important one of which will be that containing all $(\infty, 1)$-categories, or equivalently, all striation sheaves.

Posted at October 26, 2017 10:15 AM UTC

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### Re: Categorification and the Cosmic Cube

That particular suggestion on directed homotopy type theory, which has now become this paper, is, I would say, more about embedding it in ordinary homotopy theory. For a “natively” directed version, even the notion of (2,2)-topos is, I think, still poorly understood.

Posted by: Mike Shulman on October 26, 2017 4:52 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

Seeing that Voevodsky gained much by thinking directly about univalent bundles, defined in terms of what pullbacks it affords (nLab: univalence axiom), could one not use the opportunity of the nice geometric model of Ayala, Francis, and Rozenblyum to support intuitions?

They have an important striation sheaf called $\mathcal{B}un$, which plays a universal role. For instance, it classifies constructible bundles of stratified spaces. What would it need to be to act as a “directed” universe?

Anyway, the general question is whether some help on the geometric front could assist with a “natively” directed type theory, similarly to how it may be helpful to think in terms of points and paths in HoTT.

Posted by: David Corfield on October 27, 2017 8:40 AM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

Well, to play devil’s advocate a bit, I often feel these days that the longstanding implicit identification of $\infty$-groupoids with topological spaces was a serious inhibitor of progress in higher category theory, and one of the more important recent steps has been to disentangle the notions. So it seems somewhat backwards to suggest that a similar identification of $(\infty,1)$-categories with certain topological spaces would be helpful in understanding the former.

There are already plenty of candidates for directed universes: the $(\infty,1)$-category of $\infty$-groupoids; the $(\infty,1)$-category of $(\infty,1)$-categories and functors; its opposite; the $(\infty,1)$-category of $(\infty,1)$-categories and profunctors; etc. Each of them classifies something different, and satisfies some appropriate kind of “univalence”.

Posted by: Mike Shulman on October 27, 2017 9:09 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

I certainly see some philosophical commentators on HoTT being led astray by the ‘path’ language, and point out, to whoever listens, passages such as your

Classically, $\infty$-groupoids arose to prominence gradually, as repositories for the homotopy-theoretic information contained in a topological space… As we will see, however, the synthetic viewpoint emphasizes that this structure of a “homotopy space” is essentially orthogonal to other kinds of space structure, so that an object can be both “homotopical” and (for example) “topological” or “smooth” in unrelated ways. (Homotopy type theory: the logic of space)

Perhaps we need a ‘rational reconstruction’ of how one might have arrived at the univalence axiom through a line of reasoning which did not rely on topological understanding.

How might such a rational narrative look? Could the type theorists have done it by themselves? Would there have to have been some crucial category theoretic insight?

Posted by: David Corfield on October 28, 2017 9:54 AM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

Ah, I see you’re already a master of the rational reconstruction genre

Here’s a way they [higher homotopy groups] could have been invented…

Posted by: David Corfield on October 28, 2017 10:14 AM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

More later, but I believe Voevodsky was actually led to univalence through the simplicial set model of type theory, not topology at all.

Posted by: Mike Shulman on October 28, 2017 3:59 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

You actually don’t need very much homotopical/categorical input in order to arrive at univalence. Consider the analogy with function extensionality. Equalities in function types are not determined in plain MLTT: the most we can say is that for functions $f,g:A\to B$ there is a natural map $happly : (f=g) \to \prod_{x:A} f(x)=g(x)$. This naturally suggests the function extensionality axiom, which says that there is also a map in the other direction. This axiom, even stated so simply, implies that in fact the types $(f=g)$ and $\prod_{x:A} f(x)=g(x)$ are isomorphic/equivalent; this fact is also due to Voevodsky. (Of course, if you assume Uniqueness of Identity Proofs, then this is automatic since both are propositions and there are maps in both directions, but it’s true even in the absence of UIP.)

Now suppose you want to apply the same principle to equalities in the universe, which are also not determined in plain MLTT. The most natural analogue of the map $happly$ is a map $coe:(X=Y) \to (X\cong Y)$, where $X\cong Y$ is the type of isomorphisms/equivalences between $X$ and $Y$. Modern HoTT theorists are used to the need to define “equivalence” carefully, but in fact for what I’m about to say this is unnecessary: you can use even the most naive notion of isomorphism of types, $(X\cong Y) \coloneqq \sum_{f:X\to Y} \sum_{g:Y\to X} (\prod_{x:X} g(f(x))=x) \times (\prod_{y:Y} f(g(y))=y).$ Thus, a natural analogue of the function extensionality axiom would be asserting that there is a map in the other direction here, say $ua : (X\cong Y)\to (X=Y)$.

This isn’t quite univalence yet, but all we need is one more bit. Even to a set-theoretic mathematician, it is clear that two types can be isomorphic in more than one way (whereas functions – at least, functions between sets – cannot). Thus, $X\cong Y$ can contain a lot of information, unlike $\prod_{x:A} f(x)=g(x)$. And once you entertain the idea that an equality type might not be a proposition, and that we could take advantage of this fact rather than regarding it as a pathology, I think it would be natural to hope that our axiomatically asserted map $ua : (X\cong Y) \to (X=Y)$ doesn’t destroy this information. In other words, for an isomorphism $f:X\cong Y$, the equality $ua(f):X=Y$ should remember which isomorphism it came from; and the obvious way to extract this information is by the function $coe$ in the other direction. Thus, we could naturally assert in addition that $coe(ua(f))$ acts like $f$, i.e. that $coe(ua(f))(x) = f(x)$ for any $x:X$. And this is enough to imply the full univalence axiom!

There is admittedly a slight subtlety. By function extensionality, $\prod_{x:X} coe(ua(f))(x) = f(x)$ is equivalent to $coe(ua(f)) = f$ in the function type $X\to Y$. If we instead asserted that $coe(ua(f)) = f$ in the type $X\cong Y$ of isomorphisms, the result would be inconsistent. Of course, it works if we assert that $coe(ua(f)) = f$ in a coherent type of equivalences; but the point is that you don’t need to know a coherent definition of equivalence, or even that there is anything wrong with the naive notion of isomorphism, before you can assert a reasonable-sounding (pair of) axioms that implies (and is implied by) univalence. All you need is (1) the structuralist idea that isomorphic types should be “indistinguishable”, i.e. equal, and (2) the idea that an inhabitant of an identity type can remember nontrivial information.

This is not Voevodsky’s original approach; his original univalence axiom asserted that $X=Y$ was equivalent to a type of coherent equivalences, so that first he had to come up with a way to define the latter. That the above version is sufficient (and, in fact, you can get away with much less) is due to Orton and Pitts.

Posted by: Mike Shulman on October 28, 2017 11:52 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

Thanks! This should certainly find its way to nLab: univalence axiom. The Idea section there seems to promote the path/bundle aspects at the moment.

As a matter of historic fact, I’m fairly sure at the Oxford Clay Institute launch event that Voevodsky did relate his motivation for univalence in topological terms, classifying space for univalent bundles. John Baez was right beside me then, perhaps he can confirm.

Posted by: David Corfield on October 29, 2017 3:09 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

I guess there’s room for some dispute about what counts as “topological”. I don’t think of general classifying spaces as very topological, rather $\infty$-groupoidal. It does happen that some classifying spaces have nice topological representatives, like $\mathbb{C}P^\infty$, but that’s the exception rather than the rule; most classifying spaces can only be represented by a topological space that arises from a formal construction like the geometric realization of the more natural simplicial version.

I guess the main point I was making is that whatever intuition led Voevodsky to univalence in the $(\infty,1)$-situation, I think that intuition is already present in the category-theoretic version of the $(\infty,2)$-situation, without needing to bring in directed/stratified spaces, striation sheaves, or exit-paths.

Posted by: Mike Shulman on October 30, 2017 6:42 AM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

But maybe one day we’ll get interested in ‘cohesive’ $(\infty, 1)$-categories, with something like the adjoint quadruple involved between cohesive $\infty$-groupoids and $\infty Grpd$ (nLab: cohesive (infinity, 1)-topos).

Posted by: David Corfield on October 30, 2017 10:37 AM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

Dyckerhoff’s picture is interesting – I wonder if it can be filled in a bit more.

He constructs an equivalence between 2-simplicial stable $\infty$-categories and connective chain complexes of stable $\infty$-categories, both localized at levelwise equivalences. Here a 2-simplicial object in a 2-category $C$ is a 2-functor from the simplex 2-category to $C$, and a connective chain complex is the obvious thing.

So in his picture, stable $\infty$-categories are the analog of abelian groups, and 2-simplicial $\infty$-categories seem to be the analog of spaces – or perhaps it’s more like 2-simplicial $\infty$-categories with finite colimits. I suppose for the analog of pointed spaces, you’d use 2-simplicial right exact categories with zero object.

Chain complexes admit a “shift” functor which means you can “loop” and “deloop” them. One thing I’d like to know: is there a general notion of “loops” of a [2-simplicial right exact $\infty$-category with zero object] to which this corresponds? I don’t think it can be the usual “loops” defined as a pullback, since this is trivial when we have zero objects.

Posted by: Tim Campion on October 27, 2017 6:21 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

Is he using arbitrary (possibly even small) stable $(\infty,1)$-categories, or locally presentable ones? A locally presentable stable $(\infty,1)$-category bears a specific formal resemblance to an abelian group, or more precisely a module. Locally presentable $(\infty,1)$-categories admit a tensor product that represents two-variable functors preserving colimits in each variable (“bilinear maps”). The monoids for this monoidal structure (“rings”) are closed monoidal locally presentable $(\infty,1)$-categories, and the “modules” over such a monoid are enriched $(\infty,1)$-categories. Stable locally presentable $(\infty,1)$-categories are then exactly the modules over the monoidal $(\infty,1)$-category of spectra.

Posted by: Mike Shulman on October 27, 2017 9:42 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

He’s using arbitrary stable $\infty$-categories – perhaps it’s best to think of them as small. Of course, by taking Ind-categories everywhere, this could easily be recast as a story in the presentable setting. And anyway, the tensor product of presentable $\infty$-categories restricts to a tensor product of compactly-generated $\infty$-categories, or equivalently a tensor product of idempotent-complete $\infty$-categories with finite colimits, right?

It seems that Dyckerhoff’s functor $N$ from chain complexes of stable categories to 2-simplicial stable categories is a sort of iterated $S_\bullet$-construction. In particular,

1. If $B_\bullet$ is a stable $\infty$-category $B$ regarded as a chain complex of stable $\infty$-categories concentrated in degree 0, then $N(B)$ is basically the nerve of $B$$N(B)_n$ is the category of $[n]$-chains of morphisms in $B$.

2. If $B[1]_\bullet$ is a stable $\infty$-category $B$ regarded as a chain complex of stable $\infty$-categories concentrated in degree 1, then $N(B)$ is the $S_\bullet$ construction applied to $B$.

The first bullet tells us the standard way to regard an $\infty$-category $B$ as a 2-simplicial $\infty$-category $B_\bullet$: let $B_n = B^{\Delta[n]}$. This is a familiar construction from ordinary 2-category theory: $B$ is an object in a finitely-complete 2-category, then we get a 2-simplicial object by taking the powers $B^{[n]}$. In $Cat$, for example, this means we have simplicial maps between between $B^{[1]}$ and $B$, given by $codomain: B^{[1]} \to B$, $identityarrow: B \to B^{[1]}$, and $domain: B^{[1]} \to B$, and the 2-cells of a 2-simplicial object say that we have adjunctions $codomain \dashv identityarrow \dashv domain$ (which are fun to check yourself!). It’s an interesting fact that if $B$ is stable, then this adjoint string continues infinitely in both directions – I don’t know if this is relevant to Dyckerhoff’s construction.

The second bullet tells us that “delooping” is given by the $S_\bullet$ construction. And it seems roughly accurate to say that in general Dyckerhoff’s functor is a sort of “iterated $S_\bullet$” functor.

Posted by: Tim Campion on October 28, 2017 6:53 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

the tensor product of presentable ∞-categories restricts to a tensor product of compactly-generated ∞-categories, or equivalently a tensor product of idempotent-complete ∞-categories with finite colimits, right?

I didn’t know that. How does that work? And what do you mean by “restricts”? The one is not a subcategory of the other. (And what do you mean by “compactly generated”?)

Posted by: Mike Shulman on October 28, 2017 11:25 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

“Compactly-generated $\infty$-category” is Lurie’s terminology for “locally finitely-presentable $\infty$-category”.

Let me adopt the “implicit $\infty$” convention in the following. Let

• $Pres^L$ be the category of locally presentable categories and left adjoint functors,

• $CGen$ be the category of compactly-generated categories and functors with finitary right adjoints (equivalently, left adjoint functors preserving finite presentability of objects),

• $IRex$ be the category of small categories with finite colimits and split idempotents (and right exact functors).

Then Gabriel-Ulmer duality tells us that the functor $Ind: IRex \to CGen$ is an equivalence. We also have a non-full subcategory inclusion $CGen \to Pres^L$.

I’m pretty sure that $IRex$ admits a tensor product $\otimes^{irex}$ classifying “bilinear” functors, i.e. there’s a map $A \times B \to A \otimes^{irex} B$ such that a right exact functor $A \otimes^{irex} B \to C$ is the same as a functor $A \times B \to C$ which is right exact separately in each variable.

This induces a tensor product $\otimes^{cgen}$ on $CGen$, which classifies functors that separately in each variable have finitary right adjoints.

$Pres^L$ also has a tensor product $\otimes^L$, which classifies functors that separately in each variable have right adjoints (which need not be finitary). Now, my thinking was that the inclusion functor $CGen \to Pres^L$ really ought to be strong monoidal, but let’s see if I can justify that. The things to check are that:

1. If $C,D$ are compactly-generated, then $C \otimes^L D$ is compactly-generated, and

2. If $C,D,E$ are compactly-generated, and $\eta: C \times D \to C \otimes^L D$ is the universal functor, and $F: C \otimes^L D \to E$ is a left adjoint functor, then $F$ preserves finitely-presentable objects if and only if $F\eta(c,-)$ and $F\eta(-,d)$ preserve finitely-presentable objects for each $c\in C, d \in D$.

I think we can deduce (1) by writing $C = Ind(C_0)$, $D = Ind(D_0)$, and noting that $C \otimes^L D = Fun^R(Ind(C_0)^{op}, D)$ is a localization of the compactly-generated $Fun^R(P(C_0)^{op}, D) = Fun(C_0^{op},D) = Ind(C_1 \times D_0)$ (where $P$ is the presheaf category, and $C_1$ is the free completion of $C_0$ under finite colimits) at a set of morphisms between finitely-presentable objects.

This is moreover a localization of $P(C_0 \times D_0)$ at a set of morphisms between finitely-presentable objects, and thus a finitely-presentable object in $C\otimes^L D$ is a finite colimit of objects in the image of $\eta$, so to preserve them it suffices to preserve them separately in each variable, giving us (2).

Posted by: Tim Campion on October 29, 2017 11:39 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

Tim, this is the sort of material that the nLab was created to capture - it would be nice if this got written up and cross-linked. Wrinkles can be ironed out over time, so don’t worry about those.

Posted by: David Roberts on October 30, 2017 12:06 AM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

Hmm, sounds plausible. Does that mean that a small finitely-cocomplete idempotent-complete category is stable iff it is a module (under $\otimes^{\mathrm{irex}}$) over the category of finite spectra?

Posted by: Mike Shulman on October 30, 2017 6:41 AM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

That’s interesting! I suppose the argument you have in mind is the following. Let $St$ be the category of spectra, and $St_{fin}$ the category of finite spectra. Given an action $St \otimes^{irex} C \to C$ (which amounts to an action with respect to cartesian product which is right exact in each variable), we get an action $Ind(St_{fin}) \otimes^L Ind(C) \to Ind(C)$ by the above claim that $Ind$ is strong monoidal. But $Ind(St_{fin}) = St$, and it’s known that a module over $St$ with respect to $\otimes^L$ is the same thing as a stable locally presentable category. So $Ind(C)$ is stable, and since $C$ is a subcategory closed under finite colimits, $C$ is also stable.

(The split idempotents were superfluous – this could all be done just assuming $C$ has finite colimits (and not split idempotents) using a variant $\otimes^{rex}$ tensor. This isn’t so surprising, since a stable category is automatically idempotent-complete.)

There had better be a more direct argument – after all, a general finitely-cocomplete category $C$ need not have pullbacks or a terminal object. These limits do exist in $Ind(C)$; somehow this action of $St_{fin}$ forces them to lie in $C$ itself. Here is such an argument. Suppose that $C$ is finitely-cocomplete, and has an action $St_{fin} \times C \to C$ which is right exact in each variable.

First let’s show that $C$ has a zero object. The map $\mathbb{S} \to 0$ from the sphere spectrum (the monoidal unit) to 0 can be tensored with any object $c$ to yield a map from $c$ to the initial object 0 of $C$, which is natural with respect to all morphisms. By then composing with the map $0 \to d$, we obtain a family of maps $c \to d$, natural in $c,d \in C$. That is, $C$ is enriched in pointed spaces. So the initial object must in fact be a zero object.

A similar game shows that every object is canonically a comonoid with respect to coproduct, and all morphisms are comonoid homomorphisms – because this is true of $\mathbb{S}$. Thus $C$ is preadditive. Because $\mathbb{S}$ is in fact a cogroup object, $C$ is actually additive. Finally, because suspension is invertible in $St_{fin}$, it is also invertible in $C$. This is enough to see that $C$ is stable.

I suppose this says more generally that if $C$ has a separtely-right-exact action by a category $S$ which is pointed, (pre)additive, or pointed with invertible suspension, then $C$ also has that property. This reminds me of John Berman’s preprint, where lots of types of symmetric monoidal category are seen to be modules over certain semiring categories (commutative monoid objects with respect to the natural tensor on symmetric monoidal categories). I suppose module categories in $Rex$ or $IRex$ are probably also interesting.

Posted by: Tim Campion on October 30, 2017 1:44 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

Wow, thanks so much for pointing me to Berman’s preprint! I wasn’t aware of it before, but Moritz Groth and I have been independently developing a very similar theory. What’s available so far is this preprint, which uses this sort of module structure in the context of (left) derivators to characterize stability, pointedness, additivity, and so on in terms of limit-colimit commutativity. But the real project is to prove an abstract theorem such as:

For any class of (weighted) colimits $\Phi$, there is a universal solid semiring category (in Berman’s terminology) $V_\Phi$ such that $V_\Phi$-enrichment/module structure characterizes the absoluteness of $\Phi$-colimits.

Glancing over Berman’s paper, it seems he is proving this in the special cases of stability (absoluteness of finite colimits) and additivity (absoluteness of finite coproducts). Moritz and I have a plan for constructing $V_\Phi$ for any $\Phi$, but it’s taking a long time to write it out because we’re having trouble finding the right abstract context in which to do it. For a long time I hoped we could do it in “formal category theory” and thereby include the cases of 1-category theory and $(\infty,1)$-category theory simultaneously, but recently that hope seems to have been dashed, and working directly with $(\infty,1)$-categories is still kind of daunting for me.

Posted by: Mike Shulman on October 30, 2017 6:29 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

Here’s a speculative take on how to extend the analogy, which actually circumvents trying to understand what the analogs of “loops” or “spaces” are.

Dyckerhoff is pretty clear that:

• The analog of an abelian group is a stable $\infty$-category.

• The analog of a topological abelian group is a 2-simplicial stable $\infty$-category, localized at levelwise equivalences.

• The analog of a connective chain complex of abelian groups is a connective chain complex of stable $\infty$-categories, localized at levelwise equivalences.

And Dyckerhoff’s Dold-Kan theorem says that the latter two are equivalent, as expected. From what I’ve seen in the paper, it seems that

• The analog of delooping an abelian group is the $S_\bullet$ construction.

This jives with other contexts where $S_\bullet$ is seen as some kind of “suspension”. Let me get increasingly speculative:

• Note that for a 1-simplicial object to be 2-simplicial is basically a property rather than a structure: it says that the simplicial structure maps form adjoint strings. These strings of adjoints might be analogous to the trivialization of Postnikov invariants that happens for a topological abelian group. So perhaps the analog of a general connective spectrum is simply a 1-simplicial stable $\infty$-category. Then, for example, a double $\infty$-category (which is a certain type of 1-simplicial $\infty$-category which is typically not 2-simplicial) satisfying certain stability conditions would be an example of a “generalized connective spectrum”. This would accord with the fact that the $S_\bullet$ construction can be applied to double $\infty$-categories.

• In the ordinary context, we can think of an $\Omega$-spectrum as a sequence of infinite loop spaces $X_0, X_1, \dots$ with 1-coconnected infinite loop maps $BX_n \to X_{n+1}$. Analogously here, we would have a sequence of simplicial stable $\infty$-categories $X_0, X_1, \dots$ with maps $S_\bullet X_n \to X_{n+1}$. One sort of truly nonconnective example could be obtained from a suitable double $\infty$-category by taking $X_{n+1}$ to be like $S_\bullet X_n$, except without the condition that “diagonal” objects in the $S_\bullet$ construction be zero.

Posted by: Tim Campion on October 28, 2017 8:13 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

Note that for a 1-simplicial object to be 2-simplicial is basically a property rather than a structure: it says that the simplicial structure maps form adjoint strings.

This probably isn’t relevant for the rest of your comment, but I’m not sure what you mean by “basically” here, and taken literally the statement is certainly false: an adjunction between two given functors is a structure, not a property.

Posted by: Mike Shulman on October 29, 2017 3:08 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

Good catch – I thought I might be lying a bit here. But on closer inspection, I think the statement is accurate. In a 1-simplicial object, we have for example a canonical isomorphism $d_0 s_0 \cong 1$ which serves as the counit for the adjunction $d_0 \dashv s_0$ in a 2-simplicial object. So enhancing a 1-simplicial structure to a 2-simplicial structure only involves choosing a unit to match an already-chosen counit, which is a contractible-or-empty space of choices.

But to be honest, there’s more to check concerning the interaction between the various 2-cells (it’s not clear that just any adjunction data will work). So I might still be wrong here!

Posted by: Tim Campion on October 29, 2017 10:25 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

Ah, right. Now it seems at least plausible, since for instance a lax-idempotent 2-monad is one whose simplicial nerve extends to a 2-simplicial one, and lax-idempotence is a mere property (since it has the alternative characterization that every morphism between algebras has a unique structure of lax algebra-morphism.

Posted by: Mike Shulman on October 30, 2017 6:34 AM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

That’s a cool fact about lax-idempotence, I didn’t know that!

The complexity of the theory of $\infty$-monads is an oft-lamented stumbling block in $\infty$-fying categorical techniques, but I suspect that the theory of lax-idempotent $\infty$-monads is probably not so bad. It’s probably high time that somebody worked it out.

Posted by: Tim Campion on October 30, 2017 1:50 PM | Permalink | Reply to this

### Re: Categorification and the Cosmic Cube

It’s the definition of lax-idempotence in Kelly-Lack’s excellent paper On property-like structures.

I agree it is high time somebody worked out the theory of lax-idempotent $\infty$-monads.

Posted by: Mike Shulman on October 30, 2017 4:17 PM | Permalink | Reply to this

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