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I suspect there is a variant of the definition involving a transformation $R^Z_{X Y} \colon [X,Y] \to [[Y,Z],[X,Z]]$ rather than $L$. Is this correct? If so, how do these two definitions relate? Can one of them be expressed in terms of the other? Or is there a refined definition which comprises both $L$ and $R$?
Tobias, you’re right that there’s a variant of this type, just as there is a variant of closed monoidal category where it is the functors $b \otimes -$ that have right adjoints, not $- \otimes b$. (In fact, the underlying closed categories of such closed monoidal categories give instances of the phenomenon you’re asking about). As remarked at closed monoidal category, sometimes people use the terminology “left closed”, “right closed” to tell these things apart.
And just as we can manufacture a right closed monoidal structure from a left closed monoidal structure (by defining a new monoidal product obtained by switching the order of tensor factors), so we can manufacture right closed structures from left closed structures. One way of doing this might be to use the fact that any (left) closed functor can be embedded fully and “closedly” into a (left) closed monoidal category, switch the tensor factors, and then take the underlying (right) closed category of the right closed monoidal category that results. Off the top of my head, I couldn’t think of a way to do this more directly, but there may be a way.
Of course, a left closed symmetric monoidal category is also right closed. But there are also examples of left closed (monoidal) categories that aren’t right closed. An example that comes to my mind is the category of Schur functors with the plethystic monoidal product.
I expanded a bit in the section on “Embedding into closed monoidal categories”, with what I understand about the way LaPlaza’s embedding works, as well as an alternative embedding originally defined by Day.
While the definition of closed categories with $L^X_{YZ}$ matches the definition of monoidal (left) closed categories, the one with $R^Z_{XY}$ does not seem to.
Let’s denote the right adjoints of $X\otimes -$ and $-\otimes X$ by $-\Leftarrow X$ and $X\Rightarrow -$, respectively. On the one hand, the transformation $L^X_{YZ}\colon Z\Leftarrow Y\to(Z\Leftarrow X)\Leftarrow(Y\Leftarrow X)$ corresponds to $X\otimes((Y\Leftarrow X)\otimes(Z\Leftarrow Y)) \cong (X\otimes(Y\Leftarrow X))\otimes(Z\Leftarrow Y) \to Y\otimes(Z\Leftarrow Y) \to Z$. Similarly we have $R'^X_{YZ}\colon Y\Rightarrow Z\to(X\Rightarrow Y)\Rightarrow(X\Rightarrow Z)$. On the other hand, we only have $R^Z_{XY}\colon X\Rightarrow Y\to(X\Rightarrow Z)\Leftarrow(Y\Rightarrow Z)$. Note that in $R^Z_{XY}$ two closed structures $\Leftarrow, \Rightarrow$ are mixed. In a symmetric monoidal closed category, I suppose $R^Z_{XY}$ works fine.
That’s a fair point, Toshiki; thanks! There’s no explicit mention of the right-hand structure in the article; maybe you could add it? (Or I can, sometime later.) The query box could then be removed.
Have just fixed the link that Tobias created so it now points to this thread.
If anyone is still reading: can they clarify where $R^Z_{XY}$ “works” as a way to define closed categories? I ask because I was playing around with the category of dual Banach spaces (the morphisms are adjoints-in-the-TVS-sense of maps between the preduals, so maybe my category is isomorphic to $BSp^{op}$) and trying to get a closed structure. At the time I couldn’t remember whether $L^Z_{XY}$ or $R^Z_{XY}$ was the method used in the nLab page, so I tried both, but it seems that my method fails to give suitable $R^Z_{XY}$ while it might give a suitable $L^Z_{XY}$…
Suppose we have a biclosed monoidal category
$C(X\otimes Y,Z) \cong C(Y,X\rhd Z) \cong C(X,Z\lhd Y).$Then adjoint transposes give us evaluation maps
$\frac{(X\rhd Y) \to (X\rhd Y)}{X\otimes (X\rhd Y) \to Y}\qquad \frac{(Y\lhd X) \to (Y\lhd X)}{(Y\lhd X) \otimes X \to Y}$which we can put together to get double composition maps
$X\otimes (X\rhd Y) \otimes (Y\rhd Z) \to Y \otimes (Y\rhd Z) \to Z$ $(Z\lhd Y) \otimes (Y\lhd X) \otimes X \to (Z\lhd Y) \otimes Y \to Z.$Now applying adjoint transposes again to the first one, we get
$\array{\arrayopts{\rowlines{solid}} X\otimes (X\rhd Y) \otimes (Y\rhd Z) \to Z\\ (X\rhd Y) \otimes (Y\rhd Z) \to (X\rhd Z)\\ (Y\rhd Z)\to ((X\rhd Y) \rhd (X\rhd Z)) }$and to the second one we get
$\array{\arrayopts{\rowlines{solid}} (Z\lhd Y) \otimes (Y\lhd X) \otimes X\to Z\\ (Z\lhd Y) \otimes (Y\lhd X) \to (Z\lhd X)\\ (Z\lhd Y) \to ((Z\lhd X) \lhd (Y\lhd X)) }$If we write $X\rhd Y$ as $[X,Y]$, then the first one gives $L^X_{Y Z}$. But if we instead write $Y\lhd X$ as $[X,Y]$, then the second one gives… also $L^X_{Y Z}$! In other words, if you forget the presence of a monoidal structure, there is no difference between left closed and right closed.
I added to closed category a statement of the result about how a closed category becomes a monoidal category, including a warning about how it requires a stronger hypothesis than the monoidal-to-closed case.
(This came up recently in a discussion of how to prove in HoTT that the smash product is a 1-coherent monoidal structure.)
Hi Mike,
Thanks for all the additions and clarifications!
Re #12: yes, that is a good question about finding a notion of “non-unital closed category” equivalent to closed multicategories without unit, and the idea of asking for “left evaluation” operations seems natural. (By the way, it looks like there is a typo in the type of ev: that should be a “\to” rather than “\cdot”, right?) But I want to mention that there is also a natural notion of “non-unital closed category” where you simply drop the $i$’s and $j$’s: Bourke calls these “semi-closed” categories in this paper, and they are closely related to “skew semi-monoidal” categories (= categories equipped with a $\otimes$ which is only associative up to natural transformation, and which satisfies the pentagon axiom). Semi-closed categories are equivalent to closed “non-unitary”/“constantless” multicategories, i.e., closed multicategories with no multimaps of arity 0. To construct such a multicategory from a semi-closed category $C$, you define the multimaps of positive arity by $C(X_0,X_1,\dots X_n; Z) = C(X_0, [X_1,\dots,[X_n,Z]])$, without need for a unit.
Thanks for the comments Noam; I’ll have a look at Bourke’s paper. Since you’re interested, let me say a bit more about what’s going on in my head, in case you have anything to contribute to it.
My current suspicion is that my proposed kind of non-unital closed categories (which I am tempted to claim should really be the ones called simply “closed categories”; I don’t know why having a unit should be the default) can be defined as biased-lax monoids in $Prof$ whose binary multiplication profunctor is corepresentable in the first variable and satisfying one or two “normality” conditions. The “underlying-set” functor is then the unit profunctor $1 ⇸ A$, and the left evaluation is one of the biased-lax unit transformations (the other one, which I think must be required to be an isomorphism as one of the normality conditions, is the isomorphism $U([X,Y]) \cong C(X,Y)$). Since multicategories can be defined as normal unbiased-lax monoids in $Prof$, if this were true, then a positive answer to that MO question would almost automatically provide an abstract proof that non-unital closed categories are equivalent to closed multicategories. And by identifying monoidal categories (pseudo-monoids in $Cat$) with representable pseudo-monoids in $Prof$, it would also give an abstract viewpoint on the closed category / monoidal category correspondence.
Your comment about the semi-monoidal/constantless case also fits into this picture quite naturally: if we omit the underlying-set functor (and hence also the left evaluation), we would naturally get a “biased-lax semi-monoid” in $Prof$, which would of course be closely related to biased-lax semi-monoids in $Cat$ (skew semi-monoidal categories) and to unbiased-lax semi-monoids (constantless multicategories).
First, have you seen Street’s recent-ish paper on skew-closed categories? A skew-closed category is almost a closed category, but where $i : [I,X] \to X$ goes in the opposite direction and does not have to be invertible, and where the map $\gamma : C(X,Y) \to C(I,[X,Y])$ also does not have to be invertible (these two invertibility conditions are called “right normality” and “left normality”, so a closed category is a skew-closed category which is both left and right normal). In any case, Street gives a similar description of skew-closed/skew-monoidal categories as “skew-promonoidal” categories + representability conditions: see Proposition 22 on p.22.
Second, for your MO question, although they deal with skew units ($I \otimes A \to A \to A \otimes I$) rather than lax ones, you might be interested in Bourke and Lack’s recent paper on skew monoidal categories and skew multicategories. A skew multicategory is a certain kind of generalized multicategory, with two classes of multimaps called “loose maps” and “tight maps”, and with the latter embedding into the former. Bourke and Lack exhibit a correspondence between skew monoidal categories and left representable skew multicategories, where “left” representability basically means that the universal property of the multimaps $A_1,\dots,A_n \to A_1 \otimes \dots\otimes A_n$ only holds with respect to partial composition on the left. (I actually think this is a really natural idea, and wrote a paper recently that started to explore some closely related ideas from the perspective of proof theory.)
Thanks very much for those references! I will certainly take a look.
Indeed, I think that what you are proposing is very close to Street’s profunctorial definition of skew closed category - unit representability + left normality. The idea of retaining the unit profunctor while dropping its representability makes a lot of sense to me from the point-of-view of lambda calculus (where $U : C \to Set$ is the presheaf of closed terms of a given type), which I imagine is also part of your motivation.
In case you’re interested, I’ve also been trying to better understand aspects of the connection between linear lambda calculus and BCI combinatory logic, and wrote a bit about it in some musings that perhaps will make sense to you, see in particular Section 4 and the “topological completeness” result (Theorem 4.5) which is a variation of the BCI completeness theorem. The formulation given in the paper is pretty idiosyncratic, but the idea I originally had in mind is that the “imploid moves” on p.14 can be interpreted as natural transformations (going in the opposite direction) between profunctors constructed using the structure of a (symmetric, left normal skew) closed category. Now that you’ve mentioned non-unital closed categories, I think it makes even more sense to formulate things in that way, because in fact one doesn’t need (the moves corresponding to) representability of the unit to prove the theorem.
Thanks, I’ll have a look at your note too. What I have in mind, once a non-unital closed category is settled, is to reformulate it without assuming from the outset that we have a category at all. That is, since $C(X,Y) = U([X,Y])$, the homsets of $C$ are redundant data: all we need are the objects of $C$, the binary operation $[-,-]$ on them, the operation $U$ from objects to sets, and additional structure and axioms. I made a start on this here using unital closed categories, but it got kind of messy and I realized I actually wanted the non-unital version. My expectations, once this is worked out and fully simplified, are that:
One of these things where $U$ is valued in subsingletons will be precisely a (“semantic”) Hilbert system for the appropriate logic.
Given one of these things, the set $\coprod_X U(X)$ inherits an “application” partial binary operation. What properties of such an operation on a set are necessary in order to reconstruct a closed category in a “universal” way? I expect there is a way to make this precise in such a way that the answer is precisely a PCA of the appropriate sort (BCI, BCK, SKI, etc.), with the resulting closed category being its category of PERs. This would provide a nice “explanation” of realizability from a purely categorical point of view, even for a category theorist who starts out not caring at all about computability.
I did seem to be hitting a snag in the non-symmetric case. Do you know if there is a workable notion of Hilbert system or combinatory logic structure for ordered logic? Does it require the “biclosed” situation where there are two hom-objects? (If so, I could perhaps not bother about the skew case — apart from the aesthetic desire to see everything fit together well — since as noted in the Day-Laplaza paper, in the biclosed case (hence also the symmetric case) we do get an honest promonoidal category rather than a lax one.)
What I have in mind, once a non-unital closed category is settled, is to reformulate it without assuming from the outset that we have a category at all.
That sounds good!
I did seem to be hitting a snag in the non-symmetric case. Do you know if there is a workable notion of Hilbert system or combinatory logic structure for ordered logic?
The naive answer is to just drop the C combinator (thus “BI logic”, no pun intended), but are you referring to the fact that the functoriality rule for $[-,-]$ is not of the kind usually admitted in Hilbert systems/combinatory logic (since it is actually an infinite collection of axioms), and it is not clear how to otherwise derive it in the non-symmetric case? That is indeed an issue, and a year ago I ran into an old paper by Zielonka where he proved that there is no finite axiomatization of “unidirectional product-free lambek calculus with the empty string” (= free non-symmetric closed multicategories) where the only rule of inference is the cut rule.
However, I believe that this issue is resolved if you consider your combinator basis as generating terms also under $\eta$-expansion, and not just under $\beta$-reduction. (In the profunctor interpretation, $\beta$-reduction and $\eta$-expansion correspond to the unit/counit of the adjunction induced by representability of $[-,-]$.) Though it is not expressed exactly as a Hilbert system/combinatory logic, the version of the completeness theorem proved in my paper does restrict to the planar/non-symmetric case, and relies on $\eta$-expansion.
However, I believe that this issue is resolved if you consider your combinator basis as generating terms also under $\eta$-expansion, and not just under $\beta$-reduction.
But I am not sure whether this statement really makes sense in terms of combinatory logic. The more straightforward answer is that you do have to admit the functoriality principle as an additional rule in your Hilbert system. At some point Jason Reed showed me such a combinatory logic for (the even more restricted case of) ordered, constantless lambda calculus (corresponding to semi-closed categories, and “unidirectional product-free lambek calculus without the empty string”). In this system “combinators” are terms $x.M$ with a single free variable, and there is a primitive rule taking a pair of combinators $x : A_2 \vdash M : A_1$ and $y : B_1 \vdash N : B_2$ and returning a new combinator $y : A_1 \to B_1 \vdash \lambda x.N[y M/y] : A_2 \to B_2$.
are you referring to the fact that
I think so, but maybe you could make me more confident by explicitly stating the rule schema you’re talking about?
$\frac{A_2 \le A_1 \quad B_1 \le B_2}{A_1 \to B_1 \le A_2 \to B_2}$I think so, but maybe you could make me more confident by explicitly stating the rule schema you’re talking about?
(Corresponding to the third bullet point after “Add the following structure” in your notes.)
What does $\le$ mean in a Hilbert system? Or is that your point in #24 that it doesn’t mean anything?
I think of a Hilbert system as a deductive system whose judgments are simply $P\;true$ (usually written just “$P$”) for some proposition $P$, with no hypotheticals, and in which most properties of the system are expressed by axioms and often the rule that has premises is modus ponens $\frac{A \to B \quad A}{B}$. But some Hilbert systems have other rules, and it seems like this one could be formulated as $\frac{A_2\to A_1 \quad B_1 \to B_2}{(A_1\to B_1)\to (A_2\to B_2)}$. Is that right?
What does $\le$ mean in a Hilbert system? Or is that your point in #24 that it doesn’t mean anything?
Well, I guess my point is more that assuming one admits such binary judgments then functoriality is naturally formulated this way, but it is still a rule schema that would be required in addition to modus ponens.
I think of a Hilbert system as a deductive system whose judgments are simply $P\;true$ (usually written just “$P$”) for some proposition $P$, with no hypotheticals, and in which most properties of the system are expressed by axioms and often the rule that has premises is modus ponens $\frac{A \to B \quad A}{B}$.
Yeah, I’m okay with that terminology.
But some Hilbert systems have other rules, and it seems like this one could be formulated as $\frac{A_2\to A_1 \quad B_1 \to B_2}{(A_1\to B_1)\to (A_2\to B_2)}$. Is that right?
Right. What I understood from Zielonka’s paper is that he proves that having such a rule schema is in a sense unavoidable. But let me look again at his paper and try to be a bit more careful. Looks. Right…
First of all, he is talking about the (non-)existence of a “finite axiomatization where the only rule of inference is the cut rule”. This is not exactly a “Hilbert system” in your sense, but pretty close, what we would probably understand as a freely generated multicategory. He starts by setting up an infinite hierarchy of axioms, including all sequents of the form
$x/y, y \to x$as axioms of rank 0 (so he is interpreting $x/y$ as right division), then all sequents
$\to x/x$ $\to ((x/z)/(y/z))/(x/y))$as axioms of rank 1, and finally a sequent of the form
$\to x/(x/y)$of rank $n+1$ whenever $\to y$ is an axiom of rank $n$. Call $C$ the system with all such axioms (closed under the cut rule), and $C_n$ the system with axioms restricted to rank $\le n$. What Zielonka shows is that $C$ is logically equivalent to (the $/$-fragment of) Lambek’s sequent calculus $L$, but there are sequents which are derivable in $C_{n+1}$ which are not derivable in $C_n$. Therefore $L$ is not finitely axiomatizable.
Okay, so that is starting to ring some bells, and I see that it’s more interesting than I suggested above. Rather than a direct transcription of functoriality, the engine of his axiom hierarchy can be written as the following rule (shifting notation a bit)
$\frac{\vdash A} {\vdash (A \to B) \to B}$which is actually quite similar to your “evaluation” operation! (In my paper there is also an idea which I think is related of a “dni-closed” deductively closed subset [of an “imploid” = thin skew closed category], which serves as an analogue of a normal subgroup. The idea is that closure under double-negation introduction can be seen as a commutation condition necessary for the construction of quotients, and which is satisfied in particular for the subset of elements above the unit.)
Following the link from Bourke-Lack to their companion paper, I see that my MO question is answered there in the case of skew-monoidal categories. The biased-lax case where both unit morphisms go in the same direction ought to be basically the same but simpler.
Re: #29, are you saying that with only the “evaluation” operation he is able to get the functoriality rule somehow? (I don’t quite follow the notation.)
Re: #29, are you saying that with only the “evaluation” operation he is able to get the functoriality rule somehow?
Yes. (And I proved something similar as Prop. 2.23 in the above link.) The key is to prove the following rule as a lemma (in the notation of #28):
$\frac{A \to B}{(B \to C) \to (A \to C)}$Proof: Assume $A \to B$. By the “evaluation”/”dni” rule, we have $((A \to B) \to (A \to C)) \to (A \to C)$ (taking $A \to C$ as the “answer type” of double-negation introduction). Composing this with the “B” axiom $(B\to C) \to ((A \to B) \to (A \to C))$, we obtain $(B \to C) \to (A \to C)$. (Here I am using the auxiliary lemma that it is possible to compose a deduction $f : X \to Y$ with a deduction $g : Y \to Z$ to obtain a deduction $h : X \to Z$: first we apply modus ponens on the axiom $(Y \to Z) \to ((X \to Y) \to (X \to Z))$ together with $g$ to obtain a deduction $g' : ((X \to Y) \to (X \to Z))$, then we apply modus ponens on $g'$ with $f$ to obtain $h : X \to Z$.)
Now we can derive the functoriality rule
$\frac{A_2 \to A_1 \quad B_1 \to B_2}{(A_1 \to B_1) \to (A_2 \to B_2)}$as follows. Suppose given $f : A_2 \to A_1$ and $g : B_1 \to B_2$. First we apply modus ponens on the B axiom and $g$ to obtain $g' : (A_2 \to B_1) \to (A_2 \to B_2)$. Then we apply the lemma to $f$ to obtain $f' : (A_1 \to B_1) \to (A_2 \to B_1)$. Finally we compose $f'$ and $g'$ to obtain the desired $h : (A_1 \to B_1) \to (A_2 \to B_2)$.
Ah, nice! I thought there might be some argument like that, but I didn’t see it.
Pondering a bit more what you are saying and trying to connect it to things that I know, I think that the definition of “non-unital closed category” you are aiming for should begin with a skew monoid in $Prof$ (like I was suggesting in #21) rather than a biased lax monoid. Keeping track of the directions can be tricky (nice table at lax monoidal category, by the way!), so let me spell out my calculations.
Start with a right-skew monoid $(A, M : A \otimes A ⇸ A, U : 1 ⇸ A, \alpha, \lambda, \rho)$ in $Prof$, where we write $A = C^{op}$ for the opposite of some (non-unital closed) category $C$, so that $M$ and $U$ correspond to functors $M : C \times C \times C^{op} \to Set$ and $U : C \to Set$. Now we ask that $M$ is representable by a functor $[-,-] : C^{op} \times C \to C$ in the sense that
$M(c,a;b) = C(c,[a,b])$By profunctor composition the two halves of associativity $L = M(M(-,-),-)$ and $R = M(-,M(-,-))$ are calculated as follows
$L(d,e,a;c) = \int^b C(d, [e,b]) \times C(b, [a,c]) \cong C(d,[e,[a,c]]) \qquad R(d,e,a;c) = \int^b C(d,[b,c]) \times C(e,[a,b])$so that the right-to-left associator $\alpha : R \Rightarrow L$ yields the family of “compositor” maps
$L_{b,c}^a : [b,c] \to [[a,b],[a,c]]$in $C$. Now, consider the profunctors $U_L = M(U,-)$ and $U_R = M(-,U)$. I calculate them as follows:
$U_L(a;b) = \int^c U(c) \times C(c,[a,b]) \cong U([a,b]) \qquad U_R(c;b) = \int^a C(c,[a,b]) \times U(a)$Therefore the left unitor $\lambda : A \Rightarrow U_L$ corresponds to a transformation
$C(a,b) \to U([a,b])$while the right unitor $\rho : U_R \Rightarrow A$ corresponds to a transformation
$U(a) \to C([a,b],b)$So this is what one could call a “non-unital skew closed category”. If you additionally impose that $\lambda$ is an isomorphism (= left normality) then I think what comes out is what you are calling a “non-unital closed category”.
Those are the same as my calculations: it’s just that since $\lambda$ is an isomorphism, it could just as well point the other way. The reason I think it should point the other way is because the criterion of correctness for the definition should be that it coincides with closed multicategories, and we know that multicategories can be identified with normal unbiased lax monoids in $Prof$.
By the way, I think you meant $M : C^{op}\times C^{op}\times C\to Set$.
I think I see the point now that viewing them as left-normal skew monoids, whose corresponding LBC-algebras are also left-normal and hence equivalently left-normal lax monoids, would make the identification of non-unital closed categories with closed multicategories literally a consequence of Bourke-Lack rather than requiring the argument to be redone in the biased-lax case. Morally I still think the latter ought to be done, though… but I might be too lazy to do it.
Those are the same as my calculations: it’s just that since $\lambda$ is an isomorphism, it could just as well point the other way. The reason I think it should point the other way is because the criterion of correctness for the definition should be that it coincides with closed multicategories, and we know that multicategories can be identified with normal unbiased lax monoids in $Prof$.
Hmm. It’s subtle, but I have a different intuition. Or I guess it’s rather that I don’t want to start from an assumption of left normality. I think the notion of “skew” multicategory discussed by Bourke and Lack is quite natural, and there, what corresponds to the left unitor is the mapping from “tight” morphisms $A_0;A_1,\dots,A_n \to B$ to “loose” morphisms $A_0,A_1,\dots,A_n \to B$. My intuition coming from programming languages is that this corresponds to an inclusion of (“pure”) values into (“effectful”) computations, and so I think it will be interesting to consider closed skew multicategories which are not left normal (corresponding to extensions of the lambda calculus with effects). That is quite informal intuition, though.
In any case, I wanted to elaborate a bit more formally what I was alluding to at the end of #29, because I think it might be related to your ideas in #22, and possibly to this question of left normality. Given a (unital or non-unital, not necessarily left normal) skew closed category $C$, we can consider its category of covariant presheaves $[C,Set]$ as a right-skew monoidal category (or alternatively $[C,Set]^{op}$ as a left-skew monoidal category of course). Then it is interesting to consider _monoids_in $[C,Set]$, which correspond to “deductively closed presheaves” $R : C \to Set$, in the sense that the multiplication map gives an extranatural transformation
$m : R(a \to b) \times R(a) \to R(b)$corresponding to “modus ponens”, while the unit gives a natural transformation
$u : U(a) \to R(a)$corresponding to the inclusion of all “tautologies” into $R$.
Let me restrict now to the unital, 2-enriched case, because that’s what I considered in the aforementioned paper (but I think it can be extended to the non-unital, Set-enriched case). In this case, a monoid $R$ in $[C,2]$ (where $C$ is a unital skew closed preorder) corresponds to an upwards closed subset of $C$ such that the following pair of conditions hold:
$\frac{a \to b \in R \quad a \in R}{b \in R} \qquad \frac{}{I \in R}$Given such a monoid, we can construct a new preordered category $C/R = (C,\le_R)$ with the same objects as $C$ but where $a \le_R b$ iff $a\to b \in R$. The fact that this relation is transitive relies on both the multiplication $m$ of $R$ and the original associator $\alpha$/compositor $L$ of $C$, while the fact that it is reflexive relies on the unit $u$ of $R$ together with the left unitor $\lambda$ of $C$. There is also an evident embedding $i : C \to C/R$ (again relying on the left unitor $\lambda$), in the sense that the preorder $\le_R$ is an extension of $\le$.
But now we might wonder whether $C/R$ is (skew) closed, and whether we can legitimately see it as a quotient. And here is where things get more interesting: although we can show that the original operation $a\to b$ is monotone in $b$ relative to the coarser order $\le_R$, it need not be antitone in $a$. So this is where we need to impose an additional condition of closure under “double-negation introduction”:
$\frac{a \in R}{(a \to b) \to b \in R}$Note that this dni-condition is equivalent (at least in the 2-enriched case) to the existence of a natural transformation $-\otimes R \Rightarrow R\otimes-$, where $\otimes$ is the multiplication operation in $[C,2]$. Given this additional closure condition, we can prove that $a \to b$ is antitone in $a$ relative to $\le_R$ and thus defines a functor $(C/R)^{op} \times C/R \to C/R$, which together with the original unit $I$ turns $C/R$ into a left normal skew-closed category. Moreover we can show that $(C/R, i : C \to C/R)$ has the appropriate universal property of the quotient.
I did not try to work this out, but my intuition was that in the $Set$-enriched case, $R$ could be thought of as (generated by) a family of typed combinators, which is then used to construct a closed category $C_R$ with additional structure. Perhaps that aligns with some of your ideas?
By the way, I think you meant $M : C^{op}\times C^{op}\times C\to Set$.
Oops…I mean, that was just to prove my point that keeping track of all the directions/signs can be tricky! :-)
I don’t really understand all this skew business, and I’m not really motivated to put in the effort right now, since I don’t know or care about any of the relevant examples yet, and from a pure category-theoretic point of view it seems bizarre to me to have the unit morphisms pointing in different directions. (Even more bizarre than a noninvertible biased associator.) I accept that they do arise naturally in some contexts and are useful for some purposes, but the correspondence between non-unital closed categories and closed multicategories seems to me something much more basic, that should be simple to motivate, explain, and prove, without having to descend into this more exotic skew business.
I suppose a better name for “non-unital closed categories” would be “semi-closed categories”, by analogy to other uses of “semi-” for non-unitality (e.g. semigroup, semisimplicial set, semicategory…)
Re #41, I wouldn’t call these “semi-closed categories” since that name has already been used by Bourke (#16) for the weaker structure consisting of just a bifunctor $[-,-] : C^{op} \times C \to C$ equipped with the compositor transformations $[Y,Z] \to [[X,Y],[X,Z]]$ satisfying a pentagon axiom. And as you know (#17), that terminology is natural because such semi-closed categories indeed correspond to (biased-lax/skew) semigroups in $Prof$ with a representability requirement.
What’s tricky here is that a “non-unital” closed category is actually equipped with a unit (in $Prof$), it is just not representable. I’m not sure if there is a succinct way of conveying that…perhaps something like a “semi-closed category with a presheaf of global elements”?
Hrm, I suppose you’re right. There’s no name for a multicategory that has binary tensor products but not a unit, either.
“prounital”?
“prounital”?
Sounds good to me!
There’s no name for a multicategory that has binary tensor products but not a unit, either.
You mean a multicategory where there are universal binary maps $A,B \to A\otimes B$ but not a universal nullary map $\cdot \to I$? I don’t know a name for that, but I think the multicategory equivalent of a semigroup category is a multicategory with no nullary operations that is representable in the standard sense, so “representable constantless multicategory” (adopting the terminology you suggested before), or “representable non-unitary multicategory” (using Fresse’s terminology for the corresponding operads, which I don’t find particularly evocative).
You mean a multicategory where there are universal binary maps $A,B \to A\otimes B$ but not a universal nullary map $\cdot \to I$?
Yes, that’s what I meant. Since that’s the monoidal analogue of prounital closed categories, which (conjecturally) correspond to multicategories that have nullary maps, but have a “binary internal-hom” but not a “nullary one”.
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