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In the in-depth conceptual treatment on operads over at operad, I noticed that there is something very similar to the construction of Hovey-Shipley-Smith of their tensor product of pointed simplicial sequences.
Their construction is essentially as follows: Let $S_*$ be the category of pointed simplicial sets equipped with the smash product of pointed simplicial sets. Then look at the functor category $S_*^\mathbb{P}$ where $\mathbb{P}$ is the groupoid of finite cardinals. The monoidal coproduct on $\mathbb{P}$ lifts to a canonical symmetric monoidal closed tensor product on $S_*^\mathbb{P}$ by the day convolution. The symmetric sphere spectrum is a monoid in this category, and we get the smash product of spectra as the tensor product of sphere-spectrum modules.
I’ve heard that before there was a good symmetric monoidal product on (symmetric) spectra, there was an operadic method to get $E_\infty$-rings. Does this have anything to do with the observation that the construction above looks like the first half of the construction of the monoidal category where operads are monoids?
Also, if you have time, could you explain how to construct the enriched day convolution for presheaf categories using universal properties (like we did in the recent thread on the join)? The difference here is that the category of enriched presheaves doesn’t seem to have the same universal property (the free cocompletion) that allows us to extend the monoidal product on the base.
Harry, first a niggle: can you use ’Day’ rather than ’day’? It does refer to the surname of a person, not a period of time. :-)
If I understand your question correctly you need a version of the Day convolution in this enriched context. The Day convolution is often written as an end or coend (I forget which) so we come back to them. Have you looked at that sort of formulation here, possibly replacing the end in the formula by a weighted end of some type.
I consider non-capitalization of a person’s name as a badge of honor (kan complex, abelian group, serre fibration, etc.), but if it matters to you, I will try to remember =)!
Regarding the end/coend thing, I’m hoping to avoid them like we were able to in the other thread (by means of a kan extension).
I’ve heard that before there was a good symmetric monoidal product on (symmetric) spectra, there was an operadic method to get $E_\infty$-rings.
You might want to get a hold of the old books by May, starting with The Geometry of Iterated Loop Spaces, and proceeding to $E_\infty$ ring spaces and $E_\infty$ ring spectra (Springer LNM 577).
I’m not quite sure what you’re driving at in your question, since the operad article gives a detailed description of the general notion of operad, whereas $E_\infty$ spaces are described as algebras of certain specific operads (or rather, of specific homotopy type, that of the operad which arises by applying the classifying space functor to the club for symmetric monoidal categories, if my memory can be trusted).
The difference here is that the category of enriched presheaves doesn’t seem to have the same universal property (the free cocompletion) that allows us to extend the monoidal product on the base.
Enriched presheaves form the free $V$-cocompletion (where $V$ is the base of enrichment). The development is completely analogous to the $Set$-based case. In particular, Day convolution is defined by the exact same coend formula, except that that the coend is an enriched coend, for which see Kelly’s book for details.
Edit: You really ought to try to get over your allergy to coends, because they are very handy. Of course, Day convolution is given by a tensor or weighted colimit construction
$F \otimes_{Day} G = (F \otimes_V G) \otimes_{M \otimes M} (y \circ \mu)$($F$, $G$ enriched presheaves on a $V$-monoidal category $M$ with monoidal product $\mu$) which in turn is a left Kan extension of $\mu: M \otimes M \to M \to V^{M^{op}}$ along the Yoneda embedding of $M \otimes M$, applied to the obvious object $F \otimes_V G$ in $V^{(M \otimes M)^{op}}$.
Regarding ends/coends they are easier to manipulate than Kan extensions.
On the other matter perhaps one Day we will refer to Kan extension in certain situations as gindi convolutions. =)
When I write articles with French mathematicians we often have the capitulisation discussion as Galois is théorie galoisienne as adjectives do not take capitals. (This occured in another thread as ’response normande’ not ’Normande’. My argument is that it can avoid some instances of confusion. A’ street fest’ is a festival in the street,; a Street fest’ is in honour of Ross Street!. It is really unimportant but fun to discuss from time to time.
There is in fact a Ross Street in downtown Sydney, which has become a standard joke you’ll hear. (The band Null Set plays at Zed Axis on Ross Street.) Good times, good times.
@Todd:
Regarding the second point first: Aha, got it.
Regarding the first point,
Hovey-Shipley-Smith is all about symmetric simplicial spectra (these are the only ones I have any interest in, because I find topological spaces abhorrent). In particular, there is a monoidal quillen equivalence between their category of symmetric spectra and EKMM’s category of S-modules. The (appropriate version of) May’s loop spectra become the fibrant objects in HSS’s model category.
Suppose for a moment that we wanted some $E_\infty$-ring. Normally this would be some algebra in the category of spectra for the $E_\infty$-operad, right? However, this algebra is somehow equivalent to a monoid in HSS’s category of symmetric spectra. The thing that looks similar is that doing the whole Day convolution thing with presheaves on the category $\mathbb{P}$ is halfway to the construction of the category of pointed simplicial species, for which the monoids are symmetric pointed simplicial operads.
Regarding the second point first: Aha, got it.
You’re welcome.
As for the rest, are you referring to this? Unfortunately I’m not really up on this, but I am interested.
Yup! I’m wondering sorta how their construction is related to the construction of symmetric operads in pointed simplicial sets, and further, how this is related to the $E_\infty$-operad.
At least, since the $E_\infty$ operad is an operad in $Top_*$, it has a corresponding operad in $sS_*$ mutatis mutandis.
So here’s what I’m thinking, in my completely uninformed intuition: Take some monoid in pointed simplicial species representing the $E_\infty$-operad (so far this thing at least exists). I’m guessing that this operad also has an underlying monoid for the Day tensor product (This is the part that’s really not based on anything at all!) on symmetric sequences (this has the same underlying category as simplicial species, but a different monoidal product, so maybe this is true?). If this is the case, then I would guess that this object is going to be stably equivalent to HSS’s symmetric sphere spectrum.
The sort of interesting thing here would be some kind of interaction between the Day tensor product and the tensor product of species.
Another unfounded idea that I’ll throw out there: Sort-of by the reasoning in the article operad, $E_\infty$ is a monad on symmetric sequences, and it is exactly the monad given by the repeated application of the Day tensor product on symmetric sequences. In particular, the symmetric sphere spectrum is just an algebra for this monad.
The sort of interesting thing here would be some kind of interaction between the Day tensor product and the tensor product of species.
It’s the same thing! The species monoidal product (usually denoted $\otimes$ by Joyal) is a Day convolution.
The species monoidal product I thought comes from endofunctor composition, which makes use of the canonical equivalence (isomorphism?) $[Psh(\mathbb{P}),Psh(\mathbb{P})]\cong Psh(\mathbb{P})$ induced by the universal property.
Oh, that one. I call that the “substitution product”. You kept referring to the “first half of the construction of the monoidal category” , which of course is vague and confusing. “First half” – I thought you meant the construction of the symmetric monoidally cocomplete category.
So now, what’s the question again? Of course there is interaction between Day convolution and substitution product.
Let’s save some time here, and see if there is a real (i.e., precise) question. There’s way too much futzing around.
So we’ve got the $E_\infty$ operad, which is a monad on the category of symmetric sequences of pointed simplicial sets. This presentation of the $E_\infty$ operad as a monad, I think, is just the free monoid monad on the category of symmetric sequences of simplicial sets (with the Day tensor product). Does that sound right?
For $A_\infty$ operads, the monad $\Omega \Sigma$ on pointed simplicial sets is homotopy equivalent to the James construction (which gives a free topological monoid). See for example theorem 2.1 of Baues-Brown, On relative homotopy groups of the product filtration, the James construction, and a formula of Hopf (downloadable via google search).
The situation for $E_\infty$ operads (where you have all higher homotopy commutativities as well) is I think slightly trickier, but I think I’ll pass along the question to someone else, because I don’t have time at the moment, and also I’d need to read a bit to answer with any confidence. Maybe Mike is up on this stuff.
Hmm.. Hey Todd, by the way, how do you deduce the formula for the substitution product from the composition of endofunctors and the universal property?
I’ll do the $Set$-based case; the general $V$-based case is similar. I’m pretty sure this must be at operad if you read carefully, though.
Let $G$ be a presheaf, written as $G: 1 \to Set^{\mathbb{P}^{op}}$ where $Set^{\mathbb{P}^{op}}$ has been endowed with the Day convolution product. Since $\mathbb{P}$ is the free symmetric monoidal category on $1$, there exists a symmetric monoidal functor $\mathbb{P} \to Set^{\mathbb{P}^{op}}$, unique up to symmetric monoidal isomorphism, that extends $G$ along the inclusion $1 \to \mathbb{P}$. At the object level, it takes the object (finite cardinal) $n$ to the $n$-fold Day convolution $G^{\otimes n}$.
By the universal property of Day convolution, the symmetric monoidal functor $G^{\bullet}: \mathbb{P} \to Set^{\mathbb{P}^{op}}$ extends uniquely (again up to symmetric monoidal isomorphism) to a symmetric monoidal cocontinuous functor $Set^{\mathbb{P}^{op}} \to Set^{\mathbb{P}^{op}}$. This takes a presheaf $F$ to what might be written as either side of
$F \otimes_{\mathbb{P}} G^{\bullet} = \int^{n \in Ob(\mathbb{P})} F(n) \otimes G^{\otimes n}$This is the formula for the substitution product $F \circ G$. Now it is just a matter of decoding this formula. Thinking of $\mathbb{P}$ as equivalent to the category of finite sets and bijections, the $n$-fold convolution is given by a familiar species formula
$G^{\otimes n}(S) = \sum_{S = S_1 + \ldots + S_n} \bigotimes_{i \in [n]} G(S_i)$where the sum is over all decompositions of $S$ into $n$ disjoint parts (allowing parts to be empty). Then
$(F \circ G)[S] = \sum_{n \geq 0} \sum_{S = S_1 + \ldots + S_n} F([n]) \otimes \bigotimes_{i \in [n]} G(S_i)$which gives the usual formula. The first couple of pages of my article Notes on the Lie Operad (which you can get at John Baez’s website) goes through this in the first few pages.
Yeah, what I’m not seeing is how you got that coend out of thin air. It looks like some kind of (pardon the pun) $G$-ometric realization.
Edit:
Ohhhhhh…..
I see.
$G$ determines the analog of a “cosimplicial object”, and the composition product is the realization with respect to that “cosimplicial object”.
Stupid question: What is the analog of the substitution product if we replace $\mathbb{P}$ by $\Delta_a$, the augmented simplex category with its monoidal product being the ordinal sum?
Harry, perhaps I did not express myself well. I’d like to try to say it again, focusing on the universal property of the free cocompletion.
First, any presheaf $F: C^{op} \to Set$ ($C$ small) is a colimit of representables, by the (co)Yoneda lemma. This is often expressed by a coend formula
$F(-) \cong \int^{c \in C} F(c) \otimes \hom(-, c)$but if you don’t like coends, we can be more fashionable and write $F(-) \cong F(?) \otimes_C \hom(-, ?)$. (“The hom is the unit bimodule/profunctor.”) You can also express the colimit of representables by using categories of elements, but the drawback to that is that it doesn’t translate well to the enriched context.
Now, suppose given any functor $\Phi: C \to D$ to a cocomplete category $D$. Then the unique (up to iso) cocontinuous extension $\Psi: Set^{C^{op}} \to D$ takes the presheaf
$F \cong \int^c F(c) \otimes \hom(-, c)$to the corresponding colimit in $D$:
$\Psi(F) \cong \int^c F(c) \otimes \Psi(\hom(-, c)) \cong \int^c F(c) \otimes \Phi(c)$Or, more fashionably, it takes $F(?) \otimes_C \hom(-, ?)$ to
$F(?) \otimes_C \Psi(\hom(-, ?)) = F(?) \otimes_C \Phi(?) \qquad (more simply, F \otimes_C \Phi).$So the formula for the cocontinuous extension (which is $Lan_y \Phi$) takes $F$ to the tensor product $F \otimes_C \Phi$.
In particular, for $\Phi = G^{\bullet}$, this should explain that mysterious coend formula: the cocontinuous extension takes $F$ to
$F \otimes_C G^{\bullet} = \int^{n \in \mathbb{P}} F(n) \otimes G^{\otimes n}$Now, as for your second question (which you feared was stupid): I am not aware of any substitution product analogue with $\Delta$ replacing $\mathbb{P}$. What makes this tick in the case $\mathbb{P}$ is the fact that symmetric monoidal categories are the (pseudo)algebras for a club, and $\mathbb{P}$ is the free symmetric monoidal category on $1$ (there’s a little more to it than that). Can you describe $\Delta$ as the free “category of structure type or brand $X$” generated by $1$? I don’t know how to express the universal property of $\Delta$ in quite that way; I don’t know what “$X$” would be. I know $\Delta$ is the “walking monoid”, i.e., (2-)initial monoidal category with a monoid object, but that’s a different kind of universal property.
Thus, I don’t know how to define a $\Delta$-operad. On the other hand, I do know how to define nonpermutative operads, because $\mathbb{N}$ is (equivalent to) the free monoidal category generated by 1. I do know how to define braided operads, because the braid category $\mathbb{B}$ is the free braided monoidal category generated by 1. I do know how to define cartesian operads = finitary Lawvere theories, because the category opposite to finite sets is the free category with finite products generated by 1.
@Todd: Yeah, I got it in the edit to my previous post. I didn’t realize that the functor here was $G^\bullet:\mathbb{P}\to Psh(\mathbb{P})$. This is the “cosimplicial object” thing I was talking about. However, thank you for the other part of the explanation regarding the coend in question.
Regarding the $\Delta$ thing, I was basing it on having read that “universal walking monoid” thing, but as you’ve explained, this isn’t the same type of universal property, so I’ve got that too. Thanks for all of the help.
@13-14: What is “the” $E_\infty$ operad? I only know the notion of an $E_\infty$ operad (a symmetric operad where all operation-spaces are contractible).
@Mike: Isn’t “the” $E_\infty$-operad well-defined up to weak equivalence of pointed topological (pointed simplicial) operads?
Sure, but the whole point of using operads is that you’re looking for a point-set level model of things. If you say “the contractible (∞,1)-operad” then I’m okay, but on the point-set level, when we’re looking for strict actions rather than up to homotopy, differences between different operads can make a difference. In particular, I don’t know how to make sense of your question 13 without picking a particular $E_\infty$-operad (and even then I’m not sure what you mean).
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