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• CommentRowNumber1.
• CommentAuthorTim Campion
• CommentTimeSep 1st 2016

Todd has some interesting thoughts on the page non-unital operad, but I couldn’t find a discussion thread for the page, so I’ll start one here.

I was recently led to what seems to be a related perspective: there is a certain skew-monoidale $M$ in the monoidal bicategory Prof, with underlying object the groupoid $FB$ of finite sets and bijections. $M$ induces a lax monoidal structure on the category $Prof(1,FB)$, and a monoid therein is precisely a symmetric operad, defined in the $\circ_i$ style used for non-unital operads (I have the impression that the $\circ_i$ definition should be attributed to Markl, as opposed to the May-style definition which only works in the unital case – right?). I hadn’t made the connection to the Day convolution that Todd uses.

One thing I find intriguing about this approach is that you don’t need to construct a whole monad (using various infinite colimits in the process) in order to set up the definitions, nor do you need to introduce a substitution tensor product which might seem ad-hoc especially if you want to vary your groupoid of arities. So it’s a kind of “minimalist” approach to generalized operads. You might also be able to use a non-groupoidal category of arities and perhaps recover notions of Lawvere theory this way.

So I was wondering – Todd, is the material at the page non-unital operad based on the existing literature, or is this something original that you put up there? Because I want to find out as much as I can about this perspective!

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeSep 1st 2016
• (edited Sep 1st 2016)

Tim, I think of it just as a reconstructed version of the notion of operad that is implicit in Stasheff’s 1963 Homotopy Associativity of H-Spaces I (which did not have a unit; the spaces of operations $K_2, K_3, \ldots$ started at the 2-ary ones in $K_2$). So I’m not sure that the description in terms of the $sub_i$ or $\circ_i$ operations should really be attributed to Markl; I think I’ve seen them elsewhere too, but unfortunately don’t have a clear memory where. Maybe somewhere in the literature that is connected with Deligne’s conjecture.

You’re right that you can’t derive these operations from the May-style definition (whose data he originally wrote out in the form $e: 1 \to K_1$ and maps $m: K_m \times K_{n_1} \times \ldots \times K_{n_m} \to K_{n_1 + \ldots + n_m}$, and not in the slick way due to Kelly where an operad is a monoid in the monoidal category of species with plethystic monoidal product) without using the unit $e$.

I’ve not seen it packaged the way I propose, in terms of the graft product, except in my own scribblings. This stuff was in my mind mostly around 1997 or so. The particular advantage I saw to using the graft product is that the graft product is cocontinuous in both variables (probably this was mentioned somewhere on the non-unital operad page), whereas the plethystic monoidal product is cocontinuous only in its left argument. This makes possible the description of free operads in a way that is closer to that of free monoids in cosmoi, where you just take $\sum_{n \geq 0} X^{\otimes n}$. Well, it’s a little more complicated than that since a non-unital operad is not simply a semigroup with respect to graft product, but the advantage remains. I wanted to exploit it to give a more satisfactory account of bar constructions that I believe are in e.g. Ginzburg-Kapranov’s work on Koszul duality for quadratic operads, and I also wanted to use the graft product to give a more satisfactory account of cyclic operads (where the main datum is, in terms of the graft product, a pairing $Y' \ast Y' \to Y$, which when differentiated can be used to put an operad structure $Y'' \ast Y' \to Y'$ on the derivative $Y'$. The main guiding example for me is where $Y$ is the Lie operad, where $Y'$ becomes the associative algebra operad.

• CommentRowNumber3.
• CommentAuthorTim Campion
• CommentTimeSep 1st 2016

Thanks so much, Todd! The idea that the lax monoidal product is bicocontinuous (and hence biclosed, I suppose), and that this could have interesting applications, is very intriguing! I’ll have to read about cyclic operads – I think I was under the mistaken impression that they were just operads whose operations were equipped with cyclic group actions rather than symmetric group actions. From what you say, thinking about these things in terms of differentiation and Day convolution seems to tie in very nicely with important operads out there.

Let me see if I can make precise the perspective I was coming from and how it relates to yours. Let $V$ be a cosmos. There is a monoidal bicategory $V-Prof_g$ whose objects are groupoids, 1-cells $G \to H$ are $V$-bimodules between the free $V$-category $VG$ on $G$ and the free $V$-category $VH$ on $H$, and 2-cells are the usual morphisms of bimodules. The monoidal product is given by the product of groupoids, tensor product of bimodules, and tensor product of bimodule maps. (Lax) monoidal category objects (or “(skew) monoidales” as the Australians would say) in $V-Prof_g$ with underlying object $G$ can be identified via Day convolution with $V$-enriched (lax) monoidal structures on $Cat(G^{op},V) = VCat(VG^{op},V)$. I was still in the process of thinking about a natural way to derive the graft product from the $V-Prof_g$ / skew monoidale perspective when I found your work – I tried thinking about it in terms of an ambient category containing $G$ (analogous to $FinSet$ containing $FB$) either equipped with a factorization system or having certain pushouts – but probably your slick description in terms of the differentiation operator can be understood in a natural way from the skew monoidale perspective.

The skew monoidale perspective has the advantage (of perhaps little value!) that it still makes sense when $V$ isn’t sufficiently cocomplete to define the Day convolution, or (basically equivalently) composition of $V$-profunctors – so in some sense it’s more elementary; I suppose it basically makes sense in any “monoidal virtual equipment”. One could also imagine enlarging $V-Prof_g$ to get more notions of “generalized operad” – maybe the objects could be taken to be Hopf algebroids in $VCat$ or something, or even general $V$-categories (perhaps still with some kind of cocommutative coalgebra structure – I think this is probably necessary). I have a sneaking suspicion that from a suitably generalized perspective, various categories of spectra should be examples of operads where the base category $FB$ has been replaced with something a little more complex.

The subtlety that an operad is not just a lax monoid, but requires an additional axiom, is something I had completely missed. In hindsight the additional axiom is obviously necessary – a composite $(o \circ_i p) \circ_j q$ which is not of the of the form $o \circ_{i'} (p \circ_{j'} q)$ has $p$ and $q$ both directly plugged into $o$, and so should be re-expressible in the form $(o \circ_{i''} q) \circ_{j''} p$. This is something to ponder. There has been some serious study of skew monoidales in the last few years, for other reasons, and I wonder if similar conditions on lax monoids have come up in other contexts.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeSep 2nd 2016

Before getting back to you on your last comment (which will come later), I’d like to correct what I said about “cyclic operad”. Let me try that one again.

First, a cyclic operad (as presented by Getzler-Kapranov) involves an extension of $S_n$-actions to $S_{n+1}$-actions, so we are talking about a species $X$ that comes equipped with a given antiderivative $Y$: $Y' = X$. (Let’s assume $Y[0] = 0$. By the way, an antiderivative $Y$ such that $Y[0] = 0$ need not be unique: this is not property-like structure. But at least we’re not adding in extra “stuff” to $Y[0]$.) The $X$ here carries an operad structure which has to get along with the $S_{n+1}$-actions. I find it simplest to say it like this:

A cyclic (non-unital) operad is a species $Y$ equipped with a pairing $q: Y' \otimes Y' \to Y$ that is (1) symmetric, i.e., $q \circ \sigma = q$ where $\sigma: Y' \otimes Y' \to Y' \otimes Y'$ swaps the tensor factors, and (2) by taking the derivative of $q$ we get an operad structure on $Y'$. In other words, the composite

$Y' \ast Y' = Y'' \otimes Y' \hookrightarrow Y'' \otimes Y' + Y' \otimes Y'' = (Y' \otimes Y')' \stackrel{q'}{\to} Y'$

gives a structure $m: Y' \ast Y' \to Y'$ of non-unital operad. Of course, a cyclic operad is where the operad structure on $Y'$ is unital (a unit is property-like structure).

Let me remark that the data $(Y, m)$ uniquely determine the pairing map $q: Y' \otimes Y' \to Y$. This is because (1) $q': (Y' \otimes Y')' \to Y'$ uniquely determines $q: Y' \otimes Y'\to Y$, i.e., $q \mapsto q'$ is faithful under the initial value constraint $Y[0] = 0$, and (2) $q'$ equals $m$ on the summand $Y'' \otimes Y'$, and is thereby determined on the summand $Y' \otimes Y''$ by exploiting the symmetry axiom of $q$.

The classical example is this: the associative algebra operad $X$ is cyclic. Here the antiderivative is taken to be the species $Y$ of necklaces (a necklace on a finite set $S$ is a cyclic permutation $\alpha: S \to S$; a necklace structure on a pointed set $S + \ast$ is equivalent to a linear ordering of $S$, so $Y' = X$). The pairing $q: Y' \otimes Y' \to Y$ takes a pair of necklaces $(\alpha, \beta) \in Y[S + \ast] \times Y[T + \ast]$ to the necklace $\gamma$ on $S + T$ such that

1. For $s \in S$, $\gamma(s) = \beta(\ast)$ if $\alpha(s) = \ast$, else $\gamma(s) = \alpha(s)$;

2. For $t \in T$, $\gamma(t) = \alpha(\ast)$ if $\beta(t) = \ast$, else $\gamma(t) = \beta(t)$.

This pairing is manifestly symmetric, and you can check that the derivative reproduces the non-unital operad structure of the associative operad, which intercalates a list within a list to make a larger list.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeSep 3rd 2016

I think I was under the mistaken impression that they were just operads whose operations were equipped with cyclic group actions rather than symmetric group actions

No, you’re not mistaken; just saying it differently. The extension of a permutative group action $S_n \times X[n] \to X[n]$ to a suitable action $S_{n+1} \times X[n] \to X[n]$ is given by saying what the extension does to an $(n+1)$-cycle, i.e., how it should act on $X[n]$. Meanwhile, we have the species-theoretic meaning of the collective set of such extensions: it means we are giving an antiderivative in the sense of species.

Let me see if I can make precise the perspective I was coming from and how it relates to yours. Let $V$ be a cosmos. There is a monoidal bicategory $V-Prof_g$ whose objects are groupoids, 1-cells $G \to H$ are $V$-bimodules between the free $V$-category $VG$ on $G$ and the free $V$-category $VH$ on $H$, and 2-cells are the usual morphisms of bimodules. The monoidal product is given by the product of groupoids, tensor product of bimodules, and tensor product of bimodule maps.

Not to distract us here, but more a marginal note: for $V = Set$, this is one of my favorite examples of a cartesian bicategory: it’s a categorified example of a bicategory of relations in the sense of Carboni-Walters. Each object is $G$ is self-dual with a unit $1 \stackrel{!^\ast}{\to} G \stackrel{\delta}{\to} G \times G$ and a counit $G \times G \stackrel{\delta^\ast}{\to} G \stackrel{!}{\to} 1$; the 2-adjunction derives from the requisite Beck-Chevalley condition (or Frobenius condition): the canonical 2-cell $\delta \delta^\ast \to (1 \times \delta^\ast)(\delta \times 1)$ mated to the associativity isomorphism $(1 \times \delta)\delta \to (\delta \times 1)\delta$, is itself an isomorphism. That Frobenius holds is specific to groupoids; it fails for bimodules over general categories. In fact I have a memory that for any bicategory of relations (a cartesian bicategory satisfying this Frobenius condition), the locally full bicategory of objects, maps (left adjoints) and 2-cells is automatically locally groupoidal.

(Lax) monoidal category objects

What’s the official definition of lax monoidal category object? Is it consonant with the description of what lax monoidal category means in non-unital operad, where the associativity need not be invertible and at least one of the unit isomorphisms need not be invertible, and we impose just naturality and the usual pentagon and triangle conditions? (Now that I’m thinking about it, those two may no longer be sufficient for “all diagrams” commute, i.e., those famous lemmas of Kelly fail in the absence of some invertibility.)

Well, let me pretend for a while that the official sense and my sense meet up somehow. So we are contemplating possible niches for abstract graft products giving lax monoidal category objects.

Let me think out loud for just a moment. Please take this in the spirit of experimentation (or wild speculation); it might not be all that interesting in the end. (If it’s not meeting up with your own thinking, let me know.)

We have a lax associativity $F \ast (G \ast H) \to (F \ast G) \ast H$ which comes from $F' \otimes G' \otimes H \to (F' \otimes G)' \otimes H$, induced by a natural inclusion $F' \otimes G' \hookrightarrow F'' \otimes G + F' \otimes G' = (F' \otimes G)'$. Now in general, we may define the $S$-th derivative of a species by $F^{(S)}[T] = F[S + T]$, and attempt to define an “$S$-wise graft product” by the formula $F \ast_S G = F^{(S)} \otimes G$, and I’m wondering how far this generalizes from the case where our symmetric monoidal groupoid is $(FB, +)$ and $S = 1$. So we have in mind a (symmetric) monoidal groupoid/category $\mathbf{G}$, whose monoidal product I will continue to write as $+$ to maintain an analogical tie to $(FB, +)$.

First, it seems there is a suitable lax associativity $F \ast_S (G \ast_S H) \to (F \ast_S G) \ast_S H$. Writing this out, this means we have a canonical map $F^{(S)} \otimes G^{(S)} \otimes H \to (F^{(S)} \otimes G)^{(S)} \otimes H$ (here $\otimes$ is a Day convolution), induced by a map $F^{(S)} \otimes G^{(S)} \to (F^{(S)} \otimes G)^{(S)}$. This in turn would be a specialization of a “strength” which takes the form

$H \otimes G^{(S)} \to (H \otimes G)^{(S)}.$

Such a strength component corresponds to a natural family $H A \otimes G(S + B) \to (H \otimes G)(S + A + B)$ – which we have, using symmetric monoidal structure on $\mathbf{G}$. I have not bothered checking for the pentagon condition for the $S$-wise graft product so defined, but I imagine that holds (e.g., there is a pentagon condition on the strength).

A possible lax unit for this structure would be $\hom(-, S)$. This is where it gets intriguing according to my back-of-envelope calculations.

The right unit map $\rho_G: G^{(S)} \otimes \hom(-, S) \to G$ is given by a natural family with components $G(S + A) \otimes \hom(B, S) \to G(A + B)$. This is no problem at all: we have this for any symmetric monoidal $\mathbf{G}$.

The left unit map $\lambda_G: \hom(-, S)^{(S)} \otimes G \to G$ would be given by a natural family $\hom(S + A, S) \otimes G B \to G(A + B)$. Admittedly this looks a little weird at first (and you don’t see it in the $\mathbf{G} = FB$ case because $\hom(S + A, S)$ is empty unless $A = 0$ (cancel the $S$’s so to speak), so the Day convolution simplifies drastically). But look here: I think we have such a map if $\mathbf{G}$ is a traced monoidal category! (Thinking of tracing as a categorification of cancelling.)

(Comment too long; continued next comment.)

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeSep 3rd 2016
• (edited Sep 4th 2016)

In the simple case where $G$ is a representable $\hom(-, C)$, we do indeed have a map

$\hom(S + A, S) \otimes \hom(B, C) \stackrel{tensor}{\to} \hom(S + A + B, S + C) \stackrel{trace}{\to} hom(A + B, C).$

For more general $G$ we get a map $\hom(S + A, S) \otimes G B \to G(A + B)$ by writing $G$ as a colimit of representables ($G B = \int^C G C \otimes \hom(B, C)$) and playing the usual coend calculus games.

(Note: of course we don’t need the full force of a traced monoidal category; it’s enough to trace over just the given object $S$.)

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeSep 3rd 2016

In fact I have a memory that for any bicategory of relations (a cartesian bicategory satisfying this Frobenius condition), the locally full bicategory of objects, maps (left adjoints) and 2-cells is automatically locally groupoidal.

In the locally-posetal case, I think this is Carboni+Walters, Cartesian Bicategories I, Corollary 2.6(ii). I don’t recall seeing it in the non-posetal case.

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeSep 3rd 2016

Right, they even have locally discrete since groupoid+poset = discrete. The “memory” is of a personal calculation for the more general case, and that it was pretty straightforward.

• CommentRowNumber9.
• CommentAuthorTim Campion
• CommentTimeSep 4th 2016

Twice now I’ve started typing out replies and managed to lose them before posting! Here goes again.

a. These cyclic operads are interesting, and working through the associative / necklace example was fun! I read somewhere that in a cyclic operad you can permute the inputs with the output, but I don’t really see why that is.

b. The Leibniz rule is actually a bit delicate to prove – you take a map $A + B \to C + 1$ with $A$ mapping to the distinguished point, and look at it as a pair of maps $(A-1+B \to C, 1 \to A)$ with a funny $Aut(A)$-action, and show that when taking orbits you can reduce to the $Aut(A-1)$-action and throw out the map $1 \to A$. Is there a better way to understand this?

c. I think these cartesian bicategories might be useful. If I’m not mistaken, $V-Prof_g$ is a cartesian bicategory for any $V$ right? After all, the requisite structure is just the existence of certain cells, all of which flow functorially from having them in groupoids. I like this example because when $V$ is not cartesian, $V-Prof_g$ is a cartesian bicategory even though $\otimes$ is not the cartesian product. I think another example would be some sort of bicategory of Hopf algebroids in $V$, right?

d. There’s a way to construct the graft product using exactness properties of pushout squares. I’m not sure if you can get the associator etc. this way.

e. There’s also a way to construct the graft product by playing off (surjective, injective) factorizations versus minimal (injective, surjective) factorizations of constant maps between finite sets. I think this can be formulated in an arbitrary cartesian bicategory with duals if you use modules playing the role of injective and surjective maps as data. I don’t know if you can get the associator this way.

f. The “extra axiom” for a lax monoid $O$ to be an operad says that the action $O \ast O \to O$ coequalizes the two symmetry automorphisms of $O'' \otimes O \otimes O \to O$. This looks just like the equivariance condition on the binary operations of an algebra for an operad. Is there any chance that the iterates of the derivative form an operad themselves, in the category of endofunctors of the category of species? Certainly they form a symmetric sequence. (This sounds a lot like a fact I’ve heard in Goodwillie calculus: the derivatives of the identity form an operad. But I’m not sure it’s related.)

g. I suppose there’s a sort of 2-operad $C_1$ in the 2-category of categories with finite coproducts, which has a derivative operation, a binary $\otimes$ operation, and the Leibniz rule between them. There’s also a 2-operad $C_2$ encoding all these operations along with a nullary operation representing an actual operad, satisfying the axioms on the nlab page. So an operad is an extension of the $C_1$-algebra structure on the category of species to a $C_2$-algebra structure (along the natural map from $C_1$ to $C_2$). This gives one way to understand how all the pieces here fit together. But somehow it doesn’t feel terribly enlightening to me.

h. I really like this traced monoidal category construction of relative graft products! I want to work out what this looks like when $V = Vect$ and $\mathbf{G}$ is $FinVect$ and $S$ is 1-dimensional (I suppose the groupoid of finite vector spaces and automorphisms doesn’t support a trace operation…). Another interesting case should be when $V=Set$ and $\mathbf{G}$ is $FB$ but $S \neq 1$. In both cases, one should ask what an “operad” is, beyond being a lax monoid (since the Leibniz rule probably won’t hold verbatim)– and how to interpret it.

• CommentRowNumber10.
• CommentAuthorTodd_Trimble
• CommentTimeSep 5th 2016

Sorry you had trouble, Tim. That can be quite aggravating. If this means you lost work, then let me pass on something I learned from Toby Bartels: lazarus.com has software you can download where you never have to lose information you put in a web form like a comment box. This has saved me a lot of aggravation.

Let me respond to your comment in dribs and drabs, starting with (a).

These cyclic operads are interesting, and working through the associative / necklace example was fun! I read somewhere that in a cyclic operad you can permute the inputs with the output, but I don’t really see why that is.

Yes, that’s right. (By the way, I don’t know much about cyclic operads. In particular I’ve had trouble reading the Getzler-Kapranov stuff, for various reasons, and my take on it via species is a personal response to some of that.)

Here’s my intuition. A structure $\xi$ of species $Y$, as an element of let’s say $Y[n+1]$, can be diagrammatically depicted as a node labeled $\xi$ with $n+1$ spokes incident to it. When we consider the derivative of $Y$, given by $Y'[n] = Y[n + \{\ast\}]$, we are distinguishing one of those spokes, the one labeled $\ast$. This can be put to various uses, for example we may wish to consider that spoke as an output edge.

For example, let’s consider a cyclic operad structure $q: Y' \otimes Y' \to Y$. Diagrammatically, what this does is take an element $\nu \in Y[m + \ast]$ and another $\xi \in Y[n + \ast]$ and produces an element $q(\nu, \xi) \in Y[m + n]$. You can picture a structure of species $Y' \otimes Y'$ as the result of a “chemical bonding” of two spoked wheels where we merge the two edges marked $\ast$ into one, which becomes a bound edge (and there are now $m + n$ external edges after bonding). (Using this pictorial device, you can easily imagine what the free cyclic operad on a species looks like: the elements are molecular configurations of atomic spoked wheels whose nodes are labeled by elements of the species.) Then one contracts this bound edge with endpoints labeled $\nu, \xi$ into a node labeled $q(\nu, \xi)$, depicted as the center of a wheel with $m + n$ spokes.

Now consider differentiating the map $q: Y' \otimes Y' \to Y$. A structure of species $(Y' \otimes Y')'[S]$ is a structure of type $Y' \otimes Y'$ except that the outer derivative again means we are marking off one of the “spokes” $s \in S$ as part of the structure. Let’s follow Joyal and write $(F \otimes G)[S]$ as the convolution $\sum_{S = T + U} F[T] \otimes G[U]$ indexed over all ways of partitioning $S$ into two subsets. (Yes, you are right that we are using some special features of $FB$ when we do species calculus.) Relative to the decomposition $\sum_{S = T + U} Y'[T] \otimes Y'[U]$, the marked spoke $s$ belongs to $T$ or $U$, let’s say $T$. So we are contemplating a structure in $Y''[T] \otimes Y'[U]$, the two derivatives corresponding to two previously marked edges (one the shared spoke between the wheels, the other the external edge labeled $s$). Now we are viewing the edge $s$ as output, and regarding the entire configuration as a formal plugging of the $G$-wheel into the $F$-wheel, and the derivative map $q'$ itself as an operadic operation which applied to the configuration gives back an element of $Y'[S]$.

Of course $s$ might have belonged to $U$ and then we would be contemplating plugging the $F$-wheel into the $G$-wheel. The beauty of cyclic operads is this sort of directionlessness where there is no particular bias of what is being plugged into what; it’s more just a plugging together. Very much like some forms of relational calculus (which could be derived from a bicategory of relations where every object is self-dual) where one de-emphasizes domain vs. codomain. Instead there is a sense of just dealing with terms of type $X_1, \ldots, X_n$ which, if you have two lists with a type in common, then terms of those list-types can be “chemically bonded” (by a kind of tracing operation). This by the way was exactly the metaphor that Peirce was after in his Existential Graphs calculus, and I believe Sean Carmody wrote his thesis (under Martin Hyland) on this categorical topic.

In fact I would be willing to suppose that a “cyclic prop” (which would be to a cyclic operad as a prop is to an operad) should be such a compact closed category where every object is self-dual. I need to think about this more. From Getzler and Kapranov I gather that an important ingredient for constructing cyclic operads is a “universal quadratic form” which I suppose corresponds to self-duality. Again I need to think harder about this.

• CommentRowNumber11.
• CommentAuthorTodd_Trimble
• CommentTimeSep 5th 2016
• (edited Sep 5th 2016)

Okay, now (c).

If I’m not mistaken, $V-Prof_g$ is a cartesian bicategory for any $V$ right?

Not quite. I think $V$ had better be cartesian monoidal here.

So as you know, the notion of cartesian bicategory is meant to capture the structure of bicategories like $Rel$, $Span$, and $Prof$ where the 1-cells behave something like relations or bimodules and left adjoint 1-cells behave something like functions or functors, and much of the subject considers the interaction between these two bicategories. But the cartesian part refers to the fact that when we restrict the tensor product to the bicategory of 0-cells, left adjoint 1-cells (what Carboni and Walters call “maps”), and all 2-cells between them, the tensor product becomes a (cartesian) 2-product, and that the local hom-categories have cartesian products, and that that cartesian data at both levels is enough in fact to recapture the tensor product on the total bicategory (that uses all 1-cells). With a little tightening up, that can actually be turned into a definition (and in fact Cartesian Bicategories II, written IIRC by Carboni, Kelly, Verity, and Wood but which I’ve never actually read, defines cartesian bicategories in such a manner).

You can easily figure out how it works by considering the example $\mathbf{B} = Rel$. If you have two relations $R: A \to B$ and $S: C \to D$, then their tensor product (which we ordinarily denote as $R \times S: A \times C \to B \times D$, even though $(Rel, \times)$ is not cartesian monoidal – the cartesian product is in fact the coproduct $+$) is an intersection of two relations which we form as composites

$A \times C \stackrel{\pi_A}{\to} A \stackrel{R}{\to} B \stackrel{\pi_B^\ast}{\to} B \times D$ $A \times C \stackrel{\pi_C}{\to} C \stackrel{S}{\to} D \stackrel{\pi_D^\ast}{\to} B \times D$

where $A \times C$, $\pi_A$, etc. make reference to the 2-product structure on $Map(\mathbf{B})$ (here sets, functions, and inclusions between functions which are equalities), and $\pi_A^\ast$ is right adjoint to the 1-cell $\pi_A$. So we have the formula $R \times S = (\pi_B^\ast \circ R \circ \pi_A) \wedge (\pi_D^\ast \circ S \circ \pi_C)$. It works the same way for general cartesian bicategories: the tensor product on $\mathbf{B}$ is recaptured from its restriction to $Map(\mathbf{B})$ (where it is cartesian) by the formula

$R \otimes S = (\pi_B^\ast \circ R \circ \pi_A) \times (\pi_D^\ast \circ S \circ \pi_C)$

which uses the 2-product structure (the projections $\pi_A$ etc.) as well as local cartesian structure $\times$ on the local hom-category $\mathbf{B}(A \times C, B \times D)$.

One nice thing about this is that a cartesian bicategory structure on a bicategory $\mathbf{B}$ is manifestly property-like (i.e., essentially unique if it exists).

The nLab article cartesian bicategory (whose first editor was me) proceeds slightly differently and emphasizes lax (or colax) 2-adjunctions which become strong when restricted to maps, but of course the definitions are equivalent.

Mike emphasizes that all this should really be cast in the language of framed bicategories. This hasn’t been done yet in the nLab article (or even elsewhere AFAIK).

• CommentRowNumber12.
• CommentAuthorTim Campion
• CommentTimeSep 6th 2016
• (edited Sep 6th 2016)

Mostly a note to self on the symmetry condition $q\sigma = q$ on a cyclic operad $q: Y' \otimes Y' \to Y$ – I missed at first a subtlety in defining $\sigma$. Writing $Y' \otimes Y'[U] = \int^{S,T} Y'[S] \times Y'[T] \times FB(S+T,U)$, the symmetry does two things – it interchanges the two $Y'$ factors and also precomposes the bijection $S+T \to U$ with the symmetry isomorphism $S+T \cong T+S$.

I guess the conclusion on the business of permuting inputs and outputs is that for example it’s not the associative operad whose inputs and outputs you can permute, but rather its antiderivative, the necklace species, which should be thought of as having a directionless composition.

I guess I see that $V-Prof_g$ is not cartesian when $V$ is noncartesian, but I’m still having trouble seeing which of your axioms it fails to satisfy. It seems to me that the maps $\delta,\pi,\epsilon$ are all present, and satisfy all the equations you could want, because they are satisfied in groupoids. Your axioms must somewhere say something about how this structure relates to all 1-cells, but I’m not sure where. Actually there’s a further subtlety that the 2-morphisms between maps are not the same as 2-morphisms of groupoids, so perhaps this is really where the issue lies. From the Carboni-Walters perspective, I can see that there’s no natural way to make the bimodules into oplax comonoid morphisms. I wonder if there is a natural weakening of the notion of cartesian bicategory which will encompass these examples, since the basic idea is still present – the $\otimes$ is the cartesian product on maps, and it is somehow “derived” from its action on maps.

1. Hello Todd and others,

So I made a few small edits at operad to talk about the difference between “non-unital operads” in your sense and “non-unitary operads” in the terminology of Fresse’s book, which are operads with no nullary operations. Note that in their book, Markl, Shnider, and Stasheff actually define “operad” to mean “non-unitary operad”, and they refer to what you call non-unital operads as “pseudo-operads” (with the comment, “Markl in [Mar96c] defined ’pseudo-operads’ or ’non-unitary’ operads as $\Sigma$-modules with $\circ_i$-operations which satisfy axioms equivalent to the associativity and equivariance axioms […] but without a unit axiom”).

I am not sure what exactly are Fresse’s motivations for referring to operads with no nullary operations as “non-unitary”, or whether there is another name for this. Also, I haven’t looked closely enough in his book to see whether he has another term for non-unital operads. No doubt having such similar-sounding names is a potential source of confusion (and note that Stasheff’s associahedral operad is both non-unital and non-unitary!), but I don’t have any suggestions for alternatives. (However, I was looking for a word that covers exactly the concept of “non-unitary operad”.)

• CommentRowNumber14.
• CommentAuthorTodd_Trimble
• CommentTimeJul 20th 2017

Thanks, Noam!

Along the same lines as calling a ring without an identity a “rng”, I might whimsically propose “oprad” for what I had been calling a non-unital operad, without an (unary) operad unit. It sounds a little silly maybe, but so does “rng”. :-)

• CommentRowNumber15.
• CommentAuthorMike Shulman
• CommentTimeJul 20th 2017

I might call an operad without nullary operations “constantless”.