The n-Category Café

This leads to the following construction. Let $M$ be a 2-multicategory, equipped with a chosen object called $\bot$. (We don’t need to assume anything about $\bot$.) We define a 2-polycategory $Chu(M,\bot)$ as follows. Its objects are pairs $(A^+,A^-)$ of objects of $M$ equipped with a map $\hom_A : (A^+,A^-) \to \bot$. I won’t write out the general form of a poly-arrow, but here are a couple special cases to get the idea:

• A (1,1)-ary morphism $A\to B$ consists of morphisms $f:A^+ \to B^+$ and $g:B^-\to A^-$ in $M$, together with an isomorphism $\hom_B \circ_{B^+} f \cong \hom_A\circ_{A^-} g$ of morphisms $(A^+,B^-) \to \bot$.
• A (2,1)-ary morphism $(A,B) \to C$ consists of morphisms $f:(A^+,B^+)\to C^+$, $g:(A^+,C^-) \to B^-$, and $h:(C^-,B^+) \to A^-$ along with isomorphisms $\hom_C \circ_{C^+} f \cong \hom_B \circ_{B^-} g \cong \hom_A \circ_{A^-} h$ of morphisms $(A^+,B^+,C^-)\to \bot$.

In other words, we take the notion of $(n,m)$-variable adjunction, apply $(-)^{op}$ to all the left adjoints, then replace each occurrence of an opposite category $A^{op}$ with $A^+$ and each occurrence of a non-opposite category $A$ with $A^-$; then write out the representable functors like $B(f(a),b)$ and $A(a,g(b))$ as composites of multi-arrows in $Cat$ with target $Set$ and finally interpret them in $M$ but with $Set$ replaced by $\bot$. Composition is defined just like in $MVar$. This is the Chu construction $Chu(M,\bot)$ (or perhaps the “2-Chu construction”, since we’re doing it 2-categorically with isomorphisms instead of equalities).

It should now be clear, by construction, that $MVar$ embeds into $Chu(Cat,Set)$ by sending each category $A$ to its pair $(A^{op},A)$ with its “actual” opposite and its “actual” hom-functor $\hom_A : A^{op}\times A \to Set$. (Actually, we have to be a little careful with the directions of the 2-cells; the naive definition with this approach would lead to them pointing in the opposite direction from the usual. One option is to just redefine them by hand; another is to use $Chu(Cat,Set^{op})$ instead; a third is to define multivariable adjunctions to point in the direction of their right adjoints rather than their left ones.) This embedding is 2-polycategorically fully-faithful, i.e. induces an equivalence on categories of $(n,m)$-ary morphisms for all $n,m$.

Why is this interesting? Lots of reasons.

Firstly, it means that the notion of multivariable adjunction is not something we just wrote down because we saw it arising in examples. Not that there’s anything wrong with writing down a definition because we have examples of it, but it’s always reassuring if the definition also has a universal property. The Chu construction may seem ad hoc at first, but actually it is the “cofree” way to make a pointed symmetric multicategory into a polycategory with duals and counit (in the representable case, this is due to Dusko Pavlovic). And $MVar$ is (part of) what we get by applying this to $Cat$ pointed with $Set$; one might say that $Chu(Cat,Set)$ is what’s left of the construction of $MVar$ when we take away the mysterious fact that every category has an assigned opposite.

Secondly, if, like me, you never really understood the Chu construction (or maybe never even heard of it), but you are familiar with multivariable adjunctions, then now you have a new way to understand the Chu construction: it’s just an abstract generalization of $MVar$.

Thirdly, the objects of $Chu(Cat,Set)$ that aren’t in $MVar$ are not uninteresting: they are a kind of “polarized category”, with a “specified opposite” that may differ from their honest opposite. Related structures have been studied by Cockett-Seely and Mellies, among others, with semantics of “polarized linear logic” in mind. I haven’t seen anyone define before the exact sort of “polarized adjunction” that arise as the morphisms in $Chu(Cat,Set)$; and not really understanding polarized logic myself, I don’t know whether they are directly relevant to it. But they do seem quite suggestive to me. (This connection also explains my perhaps-odd-looking choice of notation $A^+ = A^{op}$ and $A^- = A$ rather than the other way around; the objects in $A^+$ are the “positive types” and those in $A^-$ are the “negative types”.)

Fourthly, Chu constructions are often more than just polycategories. If $M$ is not just a multicategory but a closed monoidal category, which moreover has pullbacks, then $Chu(M,\bot)$ is a representable polycategory — the analogue for polycategories of a multicategory that is representable by a tensor product (hence is a monoidal category). Surprisingly (at least, surprisingly if you’re used to thinking about PROPs instead of polycategories), a “representable polycategory” has two tensor products: poly-arrows $(A_1,\dots,A_n) \to (B_1,\dots,B_m)$ correspond to ordinary arrows $A_1\otimes \cdots\otimes A_n \to B_1 \parr \cdots \parr B_m$, where $\otimes$ and $\parr$ are two different monoidal structures. The two monoidal structures do have to be related, but the relationship takes the form of some somewhat odd-looking transformations $A\otimes (B\parr C) \to (A\otimes B)\parr C$ and $(A\parr B) \otimes C \to A \parr (B\otimes C)$; this is called a linearly distributive category.

A linearly distributive category that also has a dual for every object, in the polycategorical sense I mentioned last time, is called a star-autonomous category. Since $Chu(M,\bot)$ always has duals (the dual of $(A^+,A^-)$ is $(A^-,A^+)$), if it is linearly distributive then it is star-autonomous. In particular, this applies to $Chu(Cat,Set)$.

We can figure out what the tensor products in $Chu(M,\bot)$ look like by inspecting the universal property they would have to have. Recall that a (2,1)-ary morphism $(A,B) \to C$ in $Chu(M,\bot)$ consists of morphisms $f:(A^+,B^+)\to C^+$, $g:(A^+,C^-) \to B^-$, and $h:(C^-,B^+) \to A^-$ in $M$ along with isomorphisms between the three induced morphisms $(A^+,B^+,C^-)\to \bot$. A putative factorization $A\otimes B \to C$, on the other hand, would consist of morphisms $k:(A\otimes B)^+ \to C^+$ and $\ell : C^-\to (A\otimes B)^-$ and isomorphisms between two induced morphisms $((A\otimes B)^+,C^-) \to \bot$. It is easy to guess we should take $(A\otimes B)^+ = A^+ \otimes B^+$, so that $k$ is uniquely determined by $f$. For the others, note that when $M$ is closed, $g$ and $h$ correspond to maps $C^- \to [A^+,B^-]$ and $C^- \to [B^+,A^-]$, so $(A\otimes B)^-$ should involve $[A^+,B^-]$ and $[B^+,A^-]$ somehow. In fact it should be their pseudopullback over $[A^+\otimes B^+,\bot]$; the isomorphism involved in this pseudopullback encodes one of the two-variable adjunction isomorphisms, with what’s left being the induced ordinary adjunction isomorphism.

A similar, but simpler, argument shows that the unit object of $Chu(M,\bot)$ is $\mathbf{1}_\bot = (\mathbf{1},\bot,id_\bot)$, where $\mathbf{1}$ is the unit object of $M$ (if it exists). This has the universal property that it can be inserted anywhere in the domain list without changing the available morphisms, and in particular $(0,m)$-ary morphisms $() \to \Delta$ correspond to $(1,m)$-ary morphisms $\mathbf{1}_\bot \to \Delta$. Its dual $\mathbf{1}_\bot^\bullet = (\bot,\mathbf{1},id_\bot)$ has the dual property, and so in particular (0,0)-ary morphisms $()\to ()$ correspond to morphisms $\mathbf{1}_\bot \to \mathbf{1}_\bot^\bullet$ in $Chu(M,\bot)$, which simplify to just morphisms $\mathbf{1}\to \bot$ in $M$. This gives another solution to the puzzle at the end of my last post: since the embedding $MVar \to Chu(Cat,Set)$ is fully faithful on $(n,m)$-ary morphisms for $n+m\gt 0$, we should expect it to be fully faithful on $(0,0)$-ary morphisms too; but the $(0,0)$-ary morphisms in $Chu(Cat,Set)$ are morphisms $1\to Set$ in $Cat$.

Personally, I find this explanation of the tensor products in the Chu construction more transparent than any other I’ve heard, since it determines them by a universal property, rather than forcing us to guess what they should be. And it explains the odd condition in the definition of the star-autonomous $Chu(M,\bot)$ that $M$ should have pullbacks as well as being closed monoidal; it’s just to ensure that the polycategorical structure is representable. And it makes it much easier to construct the 2-categorical version of the Chu construction: the pseudopullback means that the tensor product in general will be only bicategorical, but by characterizing it by a polycategorical universal property (up to equivalence, rather than isomorphism) we can be guaranteed that it will be sufficiently coherent without having to delve into the definition of monoidal bicategory.

Finally, Chu constructions are interesting for other purposes. In particular, they often provide a “unified home for concrete dualities”. There is a nice explanation of this here for the case of $Chu(Set,2)$. You should read the whole thing, but the short version is that an “ambimorphic set” like $2$, which admits the structure of two different concrete categories $C$ and $D$, induces embeddings of $C$ and $D$ into $Chu(Set,2)$, and often the self-duality of $Chu(Set,2)$ restricts to a contravariant equivalence between $C$ and $D$. For instance, Stone duality can be represented in this way, while Pontryagin duality similarly arises from $Chu(Top,S^1)$.

The 2-Chu construction $Chu(Cat,Set)$ exhibits similar behavior, for instance involving Gabriel-Ulmer duality. Now the category $Set$ is an ambimorphic object of $Cat$: for instance, it carries the structure of both a category with finite limits and a locally finitely presentable category. This induces two embeddings of the 2-categories $Lex$ of finitely complete categories and $LFP$ of locally finitely presentable categories into $Chu(Cat,Set)$, and the self-duality of $Chu(Cat,Set)$ restricts to a contravariant equivalence $Lex^{op}\simeq LFP$.

I’m surprised that I’ve never seen the 2-Chu construction $Chu(Cat,Set)$ written down anywhere; it seems like a very natural categorification of $Chu(Set,2)$, and the latter has been studied a lot by many people. The only reference I’ve been able to find is a discussion on the categories mailing list from 2006 involving Vaughan Pratt, Michael Barr, and our own John Baez; you can comb through the archives to follow the thread here and here. Was this ever pursued further by anyone?