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October 30, 2009

Generalized Operads in Classical Algebraic Topology

Posted by Mike Shulman

There are lots of ways of categorically presenting “algebraic theories;” three of the most well-known are operads, Lawvere theories, and monads. In fact, operads and monads lie near opposite ends of a continuum of such notions, ranging from “less expressive and more controlled” (operads) to “more expressive and less controlled” (monads). One uniform framework for such “notions of theory” and their corresponding “functorial semantics” is the theory of generalized operads and multicategories.

My goal in this post is to explain how a couple of fairly obscure-seeming kinds of generalized operad are actually implicit in some very classical algebraic topology. In particular, they provide a way to “make good categorical sense” out of two constructions on topological operads that have always confused me: the “category of operators” associated to an operad, and the two different monads on based and unbased spaces associated to a “reduced” operad.

I’m going to assume you know at least a little bit about operads and Lawvere (or “finite-product”) theories. The nLab articles linked to above are good introductions, and there are plenty of others.

Generalized operads

Let me first state the definitions with an eye to generalizations. Let VV be a cartesian monoidal category (symmetric monoidal is enough for operads, but not really for finite-product theories). Think of V=SetV=Set or V=TopV=Top.

  1. Symmetric operads: Let N bijN_{bij} denote the category of natural numbers (regarded as finite sets) and bijections. A symmetric-monoidal collection in VV (more commonly, just a “collection”) is a functor N bijVN_{bij}\to V. There is a specially constructed monoidal structure on the category of symmetric-monoidal collections (not the Day convolution), and an (symmetric) operad in VV is a monoid in this monoidal category. Edit: A description of this “substitution” monoidal structure can be found here.

  2. Non-symmetric operads: Let N idN_{id} denote the category of natural numbers and identities (that is, the discrete category \mathbb{N}. A monoidal collection in VV is a functor N idVN_{id}\to V. There is a special monoidal structure on the category of monoidal collections, and a non-symmetric operad in VV is a monoid in this monoidal category.

  3. Lawvere theories: Let N funcN_{func} denote the category of natural numbers and all functions (that is, a skeleton of FinSetFinSet). A finite-product collection in VV is a functor N funcVN_{func}\to V. There is a special monoidal structure on the category of finite-product collections, and a finite-product theory or Lawvere theory in VV is a monoid in this monoidal category. (This is not the usual definition of Lawvere theory; see below for its equivalence.)

Now what’s so special about the categories N bijN_{bij}, N idN_{id}, and N funcN_{func}, and if I wanted to define a new type of “theory,” how would I know what to replace them by? The answer is that

  1. N bij opN_{bij}^{op} is the free symmetric (strict) monoidal category on one object,
  2. N id opN_{id}^{op} is the free (strict) monoidal category on one object, and
  3. N func opN_{func}^{op} is the free cartesian (strict) monoidal category on one object.

Moreover,

  1. we can talk about algebras for a (symmetric) operad in any symmetric monoidal (VV-)category,
  2. we can talk about algebras for a non-symmetric operad in any monoidal (VV-)category, and
  3. we can talk about algebras for a Lawvere theory in any cartesian monoidal (VV-)category.

In other words, given a putative type of theory X, we first choose a monad T XT_X (usually on CatCat or VCatV Cat) whose algebras are the categories with sufficient structure in which to consider models (algebras) for theories of type X. Then we define an X-collection in VV to be a functor T X(1) opVT_X(1)^{op}\to V, construct a monoidal structure on the category of X-collections, and define an X-theory or X-operad to be a monoid in that monoidal category.

In fact, usually there’s a bicategory Prof XProf_X whose objects are (VV-)categories and whose 1-cells ABA\to B are profunctors AT XBA\to T_X B; it’s a sort of Kleisli bicategory of the monad T XT_X. The monoidal category of X-collections is the hom-category Prof X(1,1)Prof_X(1,1). A monoid (aka a monad) in the bicategory Prof XProf_X is an X-multicategory or a colored X-operad. Numerous people have made this precise in different ways; I will cite this one not just because I’m biased, but because it includes references to a lot of the other literature. (Some still remain to be added in the next version. If your favorite is currently omitted, let us know, to make sure it’s scheduled for inclusion.) For now, let’s proceed informally, keeping our familiar examples in mind.

Functorial semantics

In such a general context, how can we talk about algebras/models for X-operads? One classical way to define algebras for an operad is via the endomorphism operad, defined as End(a)=C(a n,a)End(a) = C(a^{\otimes n},a) for any object aa in a (symmetric) monoidal category CC. Then for any operad 𝒪\mathcal{O}, an 𝒪\mathcal{O}-algebra structure on aa is an operad map 𝒪End(a)\mathcal{O}\to End(a).

This can be compared with the definition of an action of a group (or monoid) GG on an object aCa\in C, as simply a monoid homomorphism GEnd(a)=C(a,a)G\to End(a) = C(a,a). However, a more categorical way to define an action of GG on aa is simply as a functor GCG \to C, where we regard GG as a one-object category, and the functor takes this unique object to aa. By analogy, it makes sense to define the underlying multicategory U(C)U(C) of a monoidal category CC, and then define an action of an operad 𝒪\mathcal{O} on an object aCa\in C as a functor 𝒪U(C)\mathcal{O}\to U(C) of multicategories, where 𝒪\mathcal{O} is regarded as a multicategory with one object and the functor takes that unique object to aa.

This approach works for X-operads for arbitrary X. In particular, any monoidal category CC has an underlying multicategory U mon(C)U_{mon}(C) (= colored non-symmetric operad), any symmetric monoidal category has an underlying symmetric multicategory U sym(C)U_{sym}(C) (= colored symmetric operad), and any cartesian monoidal category has an underlying cartesian multicategory U cart(C)U_{cart}(C) (= many-sorted Lawvere theory), and in each case algebras can be defined as functors of the appropriate sort of multicategory.

The next observation is that the functor U X:X-structured categoriesX-multicategories U_X\colon \text{X-structured categories} \to \text{X-multicategories} has, in good situations, a left adjoint F XF_X. For example, for any operad 𝒪\mathcal{O} there is a free symmetric monoidal category F sym(𝒪)F_{sym}(\mathcal{O}) on 𝒪\mathcal{O}. It follows by adjointness that an 𝒪\mathcal{O}-algebra in a symmetric monoidal category CC can equally well be described as a functor F sym(𝒪)CF_{sym}(\mathcal{O}) \to C of symmetric monoidal categories. Thus F sym(𝒪)F_{sym}(\mathcal{O}) is the free symmetric monoidal category containing an 𝒪\mathcal{O}-algebra.

In fact, the functor F XF_X is usually faithful, so that an X-multicategory can be defined to be an X-structured category that is “in the image of F XF_X”. Lawvere theories are most commonly defined in this way, since in this case the image of F cartF_{cart} is quite simple: it consists of the cartesian (strict) monoidal categories whose monoid of objects is free (on one generator, in the classical one-sorted case). Operads can also be characterized as certain monoidal categories, but one needs to add extra conditions.

We need one more bit of structure: the relations between theories for different X. Let me draw a small part of the continuum of X-theories: mon. cats sym. mon. cats cart. mon. cats multicats sym. multicats cart. multicats \array{ \text{mon. cats} & \rightleftarrows & \text{sym. mon. cats} & \rightleftarrows & \text{cart. mon. cats} \\ \uparrow\downarrow && \uparrow\downarrow && \uparrow\downarrow \\ \text{multicats} & \rightleftarrows & \text{sym. multicats} & \rightleftarrows & \text{cart. multicats} } We’ve already mentioned the vertical adjunctions F XU XF_X \dashv U_X. The left-pointing arrows, which I’ll write as RR, are forgetful functors. In good situations, each such functor has a left adjoint which I’ll write as LL. Thus we see, for instance, that any non-symmetric operad 𝒪\mathcal{O} freely generates a symmetric operad L sym(𝒪)L_{sym}(\mathcal{O}), with the property that if CC is a symmetric monoidal category, then (by adjointness) L sym(𝒪)L_{sym}(\mathcal{O})-algebras in CC are the same as 𝒪\mathcal{O}-algebras in R(C)R(C) (that is, CC considered as a mere monoidal category). Likewise, any operad 𝒪\mathcal{O} generates a Lawvere theory L cart(𝒪)L_{cart}(\mathcal{O}) having the same algebras in cartesian monoidal categories.

Amusingly, we can also recover the monad associated to an operad or Lawvere theory in this way. Let V=SetV=Set for simplicity, and consider the case where X = arbitrary small products. Then T X(1) op=SetT_X(1)^{op} = Set (i.e. SetSet is the free category with small coproducts on one object), an X-collection is just a functor SetSetSet\to Set, and the monoidal structure on X-collections turns out to be just composition. Thus, an “X-operad” is simply a monad on SetSet. This value of X lives at the far right of the continuum, with forgetful functors RR landing in operads, Lawvere theories, and so on, which have left adjoints LL in good cases. It follows by adjointness, as usual, that every operad or Lawvere theory 𝒪\mathcal{O} gives rise to an associated monad L sprod(𝒪)L_{sprod}(\mathcal{O}) with the same algebras. One can then trace through the definitions and recover the usual explicit definition of this monad: a n𝒪(n)×a n a \mapsto \int^n \mathcal{O}(n) \times a^n where the coend is over N symN_{sym}, N idN_{id}, or N funcN_{func}, as appropriate. (To play this game for VV other than SetSet, we need to use cotensors instead of products.)

Semicartesian operads

The first novel type of theory I want to consider is that corresponding to semicartesian symmetric monoidal categories: symmetric monoidal categories whose unit object is the terminal object. I want to consider these not because there are interesting examples of semicartesian monoidal categories that are not cartesian (though there are some), but because of the resulting notion of operad.

In this case the relevant category T semicart(1) opT_{semicart}(1)^{op} can be identified with N injN_{inj}, the category of finite sets and injections. Therefore, for any injection γ:nm\gamma\colon n\to m, a semicartesian operad 𝒪\mathcal{O} comes with an operation 𝒪(n)𝒪(m)\mathcal{O}(n)\to \mathcal{O}(m). We think of this as saying “given an operation that takes nn inputs, we can make an operation that takes mm inputs by discarding those that are not in the image of γ\gamma.” That is, in a semicartesian operad, we can discard inputs (which we cannot do in an ordinary operad), but we cannot duplicate them (which we can do in a Lawvere theory).

Now here’s the interesting thing about semicartesian categories: any functor from a semicartesian monoidal category to a cartesian monoidal category is automatically (and uniquely) colax monoidal. This is well-known for functors between cartesian monoidal categories, but it actually only requires the domain to be semicartesian. Explicitly, let CC be semicartesian and DD be cartesian, and let G:CDG\colon C\to D be any functor. Then for any a,bCa,b\in C we have maps aba1aa\otimes b \to a\otimes 1 \cong a and ab1bba\otimes b \to 1\otimes b \cong b in CC, inducing maps G(ab)G(a)G(a\otimes b) \to G(a) and G(ab)G(b)G(a\otimes b)\to G(b). But DD is cartesian monoidal, so these two maps induce a map G(ab)G(a)×G(b)G(a\otimes b)\to G(a) \times G(b). Of course, GG is strong monoidal iff these canonical maps are isomorphisms.

If DD has moreover a notion of “weak equivalence,” it makes sense to say that GG is “homotopy strong monoidal” if these canonical maps are weak equivalences, or perhaps better, if the iterated canonical maps G(a 1a n)G(a 1)××G(a n) G(a_1 \otimes \dots \otimes a_n) \to G(a_1)\times \dots\times G(a_n) are weak equivalences. But now recall that if 𝒪\mathcal{O} is a semicartesian operad, 𝒪\mathcal{O}-algebras in a cartesian monoidal category DD are (semicartesian) strong monoidal functors F semicart(𝒪)R(D)F_{semicart}(\mathcal{O}) \to R(D). Thus it makes sense to say that a homotopy 𝒪\mathcal{O}-algebra is a functor G:F semicart(𝒪)DG\colon F_{semicart}(\mathcal{O}) \to D such that the above canonical maps are weak equivalences. (Tom Leinster has taught us the importance of colax functors for defining homotopy algebras. The point is that in the semicartesian case, the colax structure comes for free.)

The first punchline is now that if we start with an ordinary (symmetric) operad 𝒪\mathcal{O} in topological spaces, generate a free semicartesian operad L(𝒪)L(\mathcal{O}), then generate a free semicartesian symmetric monoidal category F semicart(L(𝒪))F_{semicart}(L(\mathcal{O})) from this, the result is precisely the category of operators 𝒪^\hat{\mathcal{O}} associated to 𝒪\mathcal{O} by May and Thomason in “The uniqueness of infinite loop space machines”. Moreover, the “homotopy 𝒪\mathcal{O}-algebras” defined above are basically their 𝒪^\hat{\mathcal{O}}-spaces (minus a cofibration condition).

In particular, for the operad 𝒪\mathcal{O} whose algebras are commutative monoids, we have F semicart(L(𝒪))=FinSet F_{semicart}(L(\mathcal{O})) = FinSet_∗, the category of finite based sets, which is the opposite of Segal’s category Γ\Gamma. A (special) Γ\Gamma-space is, quite classically, a functor from this category to spaces such that the above comparison maps are weak equivalences, i.e. a homotopy commutative monoid in the above sense.

Got all that? Great! Let’s move on to the second example.

Semi-cocartesian operads

Of course, a semi-cocartesian symmetric monoidal category is a symmetric monoidal category whose unit object is the initial object. There are actually lots of examples of these: let CC be any symmetric monoidal category with unit object ii, and consider the co-slice category i/Ci/C. It has a monoidal structure given by taking iai\to a and ibi\to b to iiiab, i \cong i \otimes i \to a\otimes b, and the unit object iii\to i is of course initial. Note that this monoidal structure is not the “smash product” often considered on co-slice categories. For instance, when C=SetC=Set (or TopTop) this is the category of pointed sets (or spaces) with its cartesian product.

In fact, i/Ci/C is a coreflection of CC into semi-cocartesian monoidal categories. That is, if BB is semi-cocartesian and G:BCG\colon B\to C is a symmetric monoidal functor, it lifts uniquely along the forgetful functor i/CCi/C \to C. For since BB is semi-cocartesian, any object bBb\in B comes equipped with a canonical map ibi\to b, which under the functor GG is mapped to iG(i)G(b)i\cong G(i) \to G(b); hence G(b)G(b) is naturally an object in i/Ci/C.

Now the category T semi-cocart(1) opT_{\text{semi-cocart}}(1)^{op} can be identified with the category N inj opN_{inj}^{op} of finite sets and injections pointing the other way. Therefore, any semi-cocartesian operad comes equipped with, for any injection γ:nm\gamma\colon n\to m, an operation 𝒪(m)𝒪(n)\mathcal{O}(m)\to \mathcal{O}(n), which we think of as saying “given an operation taking mm inputs, we produce an operation taking nn inputs by plugging these nn inputs into the given operation in the places specified by γ\gamma, and putting in the basepoint for all the rest.”

What basepoint, you ask? Well, as with any operad, there is a specified identity operation id𝒪(1)id \in \mathcal{O}(1), whose image under the function 𝒪(1)𝒪(0)\mathcal{O}(1)\to \mathcal{O}(0) induced by the obvious inclusion 010\hookrightarrow 1 gives a canonically defined 0-ary operation in any semi-cocartesian operad. That means that any 𝒪\mathcal{O}-algebra will have a canonically specified basepoint. Moreover, it’s kind of stupid if 𝒪(0)\mathcal{O}(0) contains any more than one 0-ary operation, since 0-ary operations in an 𝒪\mathcal{O}-algebra aa will correspond to maps iai \to a from the unit object; but since the unit object in a semi-cocartesian monoidal category is initial, there can be only one such map. Let’s call a semi-cocartesian operad non-stupid if 𝒪(0)\mathcal{O}(0) is a terminal object.

Now, invoking the appropriate notion of “equivariance,” it’s easy to see that for any injection γ:nm\gamma\colon n\to m, the induced map 𝒪(m)𝒪(n)\mathcal{O}(m)\to \mathcal{O}(n) really is given by “plug nn inputs into it in the places specified by γ\gamma, and put in the basepoint for all the rest.” That means that a non-stupid semi-cocartesian operad is completely determined by an ordinary (symmetric) operad such that 𝒪(0)\mathcal{O}(0) is terminal. In fact, it’s not hard to check that any symmetric operad with 𝒪(0)\mathcal{O}(0) terminal can uniquely be given the structure of a semi-cocartesian operad in the above way. May’s original definition of “operad” in “The Geometry of Iterated Loop Spaces” actually required that 𝒪(0)\mathcal{O}(0) be terminal; more recently he has referred to operads with this property as reduced.

The next observation we need is that if 𝒪\mathcal{O} is a reduced (hence non-stupid semi-cocartesian) operad, then the free symmetric monoidal category F sym(𝒪)F_{sym}(\mathcal{O}) that it generates is already semi-cocartesian. To show this carefully would require too much detail about the construction of F symF_{sym}, but here’s the idea. Morphisms mnm\to n in F sym(𝒪)F_{sym}(\mathcal{O}) are constructed as a colimit of 𝒪(m 1)××𝒪(m n)\mathcal{O}(m_1)\times \dots \times \mathcal{O}(m_n) over all partitions m=m 1+m nm = m_1 + \dots m_n. But when m=0m=0, the only possible partition is m 1==m n=0m_1 = \dots = m_n = 0, in which case 𝒪(m j)\mathcal{O}(m_j) is terminal for each jj, and thus so is F sym(𝒪)(0,n)F_{sym}(\mathcal{O})(0,n). In other words, 00 is an initial object in F sym(𝒪)F_{sym}(\mathcal{O}). It follows by universality that F sym(𝒪)F_{sym}(\mathcal{O}) is actually equivalent to F semi-cocart(𝒪)F_{\text{semi-cocart}}(\mathcal{O}).

Now if DD is an arbitrary symmetric monoidal category (possibly cartesian, possibly VV itself, so think of SetSet or TopTop), we have a sequence of bijections between

  • 𝒪\mathcal{O}-algebras in DD (considering 𝒪\mathcal{O} as an ordinary symmetric operad),
  • symmetric monoidal functors F sym(𝒪)DF_{sym}(\mathcal{O}) \to D (by adjointness),
  • symmetric monoidal functors F sym(𝒪)i/DF_{sym}(\mathcal{O}) \to i/D (since F sym(𝒪)F_{sym}(\mathcal{O}) is semi-cocartesian and i/Di/D is a coreflection),
  • symmetric monoidal functors F semi-cocart(𝒪)i/DF_{\text{semi-cocart}}(\mathcal{O}) \to i/D (since F sym(𝒪)F semi-cocart(𝒪)F_{sym}(\mathcal{O})\simeq F_{\text{semi-cocart}}(\mathcal{O})), and
  • 𝒪\mathcal{O}-algebras in i/Di/D (considering 𝒪\mathcal{O} as a semi-cocartesian operad).

Finally, recall that for any X-operad 𝒪\mathcal{O} and X-structured category DD, there is (in good situations) a monad on DD whose algebras are the 𝒪\mathcal{O}-algebras. The above sequence of bijections thus implies that there are two monads, one on DD (the category of “unbased” objects) and one on i/Di/D (the category of “based” objects) whose categories of algebras are the same. The first monad is the one that any category theorist would write down, and which we wrote down above. The second monad includes additional identifications coming from the basepoints of objects in i/Di/D, which are exactly specified by saying that it’s a coend over N inj opN_{inj}^{op}—as is appropriate for the monad associated to a semi-cocartesian operad.

The second punchline I’ve been aiming for is that this second monad is the one that May originally wrote down in “The Geometry of Iterated Loop Spaces.” (Kelly gives a different construction of May’s monad on based spaces starting from the more obvious monad on unbased spaces. However, I find the above explanation more satisfying.) And, of course, there’s quite a good reason that May chose the second monad. When 𝒪\mathcal{O} is the “little nn-cubes operad”, it is this second monad, acting on based spaces, which is equivalent to the nn-fold loop-suspension monad Ω nΣ n\Omega^n \Sigma^n. This is the essential starting point for the operadic study of iterated loop spaces.

Posted at October 30, 2009 2:37 AM UTC

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18 Comments & 1 Trackback

Re: Generalized Operads in Classical Algebraic Topology

Excellent! I’m enjoying trying to understand it.

When you say “a putative type of theory XX” and “This approach works for XX-operads for arbitrary XX”, what does that mean about XX? What collection does it belong to?

Posted by: David Corfield on October 30, 2009 5:30 PM | Permalink | Reply to this

Re: Generalized Operads in Classical Algebraic Topology

what does that mean about X? What collection does it belong to?

Haha, I was intentionally being kind of vague about that. Usually it’s some sort of (2-)monad—i.e. we define X by defining T XT_X. The precise definition depends on which theory of generalized multicategories you prefer. In Geoff’s and my theory, which aims to encompass all others as special cases, T XT_X is a monad on a “virtual equipment”. (Maybe some time I’ll write a whole post about what that means.) Intuitively, I think the best way to think of T XT_X is as a 2-monad on CatCat or V-CatV\text{-}Cat which is “nice enough” to allow the construction of X-multicategories.

One thing worth noting is that generalized multicategories are often defined (following Leinster and Burroni) using a cartesian monad on a 1-category, but (as far as I can tell) that’s not sufficient for most of the examples I’m interested in here. In fact, of the examples I mentioned, I think that that approach works only for non-symmetric operads.

Posted by: Mike Shulman on October 30, 2009 6:10 PM | Permalink | PGP Sig | Reply to this

Re: Generalized Operads in Classical Algebraic Topology

One thing worth noting is that generalized multicategories are often defined (following Leinster and Burroni) using a cartesian monad on a 1-category, but (as far as I can tell) that’s not sufficient for most of the examples I’m interested in here.

Actually, that’s an unfair thing to say. You can consider all of these monads (free-symmetric-monoidal-category, free-cartesian-monoidal-category, free-semicartesian-monoidal-category, etc.) as cartesian monads on CatCat. It’s just that when you do, the notion of generalized multicategory you get is more general than what I usually want. In order to recover what I usually want, you need to require that the span C 0C 1TC 0C_0 \leftarrow C_1 \rightarrow T C_0 in CatCat is a two-sided discrete fibration, and additionally that C 0C_0 is a discrete category.

In addition, this doesn’t work so well in the enriched case as far as I know, whereas if you use profunctors you can just change to using enriched profunctors. On the other hand, Tom Leinster has a much more general notion of enrichment for his sort of generalized multicategories; maybe he can tell us whether it works to give enriched multicategories of all these sorts.

Posted by: Mike Shulman on November 2, 2009 3:04 AM | Permalink | PGP Sig | Reply to this

Re: Generalized Operads in Classical Algebraic Topology

Thanks, Mike, for teaching us this stuff. I am reading…

First question: above you write

I’m not going to say any more about what this monoidal structure is; it’s not too hard to write down once you know the definition of an operad.

Now just help me here: are we or are we not talking about precisely the definition at operad – definition as a monoid?

Posted by: Urs Schreiber on October 31, 2009 12:07 AM | Permalink | Reply to this

Re: Generalized Operads in Classical Algebraic Topology

Yes, it is, thanks. I wrote most of this post offline and didn’t remember that the substitution tensor product was described in that article. I’ve now edited the entry to say so.

Posted by: Mike Shulman on October 31, 2009 2:10 AM | Permalink | PGP Sig | Reply to this

Re: Generalized Operads in Classical Algebraic Topology

Second question:

can you help me see how to the pattern that you describe would relate to the definition by Ieke Moerdijk and, more recently, the definition by Jacob Lurie?

Lurie considers fibrations over the Segal category. That must have some close connection to the monoidal categories that you, discuss, it feels. But I don’t get the picture right now.

Posted by: Urs Schreiber on October 31, 2009 12:18 AM | Permalink | Reply to this

Re: Generalized Operads in Classical Algebraic Topology

can you help me see how to the pattern that you describe would relate to the definition by Ieke Moerdijk and, more recently, the definition by Jacob Lurie?

Moerdijk and Lurie are both talking about (,1)(\infty,1)-operads, which are a generalization in a different direction than the generalized operads in use here. They both start with the ordinary notion of symmetric operad and (,1)(\infty,1)-categorify it in different ways. I’m not as well up on the (,1)(\infty,1)-world as you are, but here are some thoughts about the relationships.

I would expect that the picture of functorial semantics works as well in the (,1)(\infty,1)-context, i.e. there is a forgetful functor UU from the (,1)(\infty,1)-category (or maybe (,2)(\infty,2)-category) of symmetric monoidal (∞,1)-categories to (,1)(\infty,1)-multicategories, which has a left adjoint FF, so that an algebra for an (,1)(\infty,1)-operad 𝒪\mathcal{O} in a symmetric monoidal (,1)(\infty,1)-category CC could be defined either as a map 𝒪U(C)\mathcal{O}\to U(C) of (,1)(\infty,1)-multicategories or a map F(𝒪)CF(\mathcal{O})\to C of symmetric monoidal (,1)(\infty,1)-categories. It seems like this ought to work with either of the two definitions, but I don’t know whether Moerdijk or Lurie has written it down.

I would also expect that there are also (,1)(\infty,1)-versions of generalized multicategories (such as Lawvere theories, semicartesian operads, semi-cocartesian operads, and so on). IIRC Moerdijk has thought at least briefly about “planar” dendroidal sets which give non-symmetric (,1)(\infty,1)-operads. I don’t know whether he or anyone has thought about whether there is a version of dendroidal sets for, say, Lawvere theories, or whether a version could be constructed for some general class of monads on CatCat. It seems that it might be possible. On the other hand, I don’t (at present) see the potential for a general theory of generalized (,1)(\infty,1)-operads in Lurie’s approach.

However, it is true that Lurie’s definition of an (,1)(\infty,1)-operad is closely related to what I was talking about in this post, and this is probably what you were feeling. (First of all, let me point out that Lurie’s definition of (,1)(\infty,1)-operad does not include the full opfibration condition; that condition is what makes an (,1)(\infty,1)-operad representable and therefore a symmetric monoidal (,1)(\infty,1)-category.) Lurie starts out by defining, for any ordinary (symmetric) operad 𝒪\mathcal{O} in SetSet, a category 𝒪 \mathcal{O}^\otimes equipped with a map to what he calls Γ\Gamma and what is, I believe, the opposite of the category that Segal called Γ\Gamma; I’ll call it FinSet FinSet_∗ to be unambiguous. In fact, this category 𝒪 \mathcal{O}^\otimes is none other than the category of operators associated to 𝒪\mathcal{O} that I described above, which I claimed was equal to F semicart(L semicart(𝒪))F_{semicart}(L_{semicart}(\mathcal{O})). The map 𝒪 FinSet \mathcal{O}^\otimes\to FinSet_∗ is then obvious since FinSet FinSet_∗ is F semicartL semicartF_{semicart} \circ L_{semicart} applied to the terminal operad. Lurie then writes down conditions on the functor 𝒪 FinSet \mathcal{O}^\otimes\to FinSet_∗ which ensure that 𝒪 \mathcal{O}^\otimes actually arises from a symmetric operad, and generalizes them to the (,1)(\infty,1)-categorical context. This is a very clever way to define (,1)(\infty,1)-operads using only (,1)(\infty,1)-categories without needing any machinery like dendroidal sets, but as I said I don’t see as much potential for generalizing it to other sorts of generalized operads.

Posted by: Mike Shulman on October 31, 2009 5:02 AM | Permalink | PGP Sig | Reply to this

Re: Generalized Operads in Classical Algebraic Topology

I realize that it is getting too late for me here for following you in detail, so just one trivial remark on this:

Let’s call a semi-cocartesian operad non-stupid if…

When you write this up, may I ask you that you find another term here?

I once sat in a talk that introduced some structure and then the first example got called the “stupid example”. Unfortunately, the stupid example played a prominent role in the entire talk and the speaker had to mention the term every minute or so. It certainly wasn’t funny anymore after the first few times. When I think back about that talk, there is one single word overwhelmingly coming to mind.

Posted by: Urs Schreiber on October 31, 2009 12:34 AM | Permalink | Reply to this

Re: Generalized Operads in Classical Algebraic Topology

Let’s call a semi-cocartesian operad non-stupid if…

When you write this up, may I ask you that you find another term here?

Absolutely! What about just re-using “reduced”?

Posted by: Mike Shulman on October 31, 2009 2:12 AM | Permalink | PGP Sig | Reply to this

Re: Generalized Operads in Classical Algebraic Topology

Hi, just a small suggestion: I think it would be great to illustrate the concepts introduced here (when applicable) in the cases of the operads describing associative algebras and Lie algebras.

Posted by: yael fregier on October 31, 2009 2:10 AM | Permalink | Reply to this

Re: Generalized Operads in Classical Algebraic Topology

I think it would be great to illustrate the concepts introduced here (when applicable) in the cases of the operads describing associative algebras and Lie algebras.

Unfortunately, I don’t think they are applicable to those cases. Those operads are enriched in AbAb or VectVect, neither of which is cartesian monoidal. As far as I know, that means that cartesian and semicartesian operads don’t work very well, if at all, enriched in them.

Posted by: Mike Shulman on October 31, 2009 5:10 AM | Permalink | PGP Sig | Reply to this

Re: pagination

I like to print these discussions and read off line

Is there some way to know how many pages will print
so in e.g. chronological mode I can print just the new stuff?

Posted by: jim stasheff on October 31, 2009 1:43 PM | Permalink | Reply to this

Re: pagination

Your browser may have a “Print Preview” option in the “File” menu. If so, it will probably tell you how many pages of printout there will be; and using it you’ll be able to see which pages you want to print. You may then have the option to print just the ones you want.

Posted by: Tom Leinster on October 31, 2009 2:22 PM | Permalink | Reply to this

Re: pagination

my browser does for pdf etc but it’s not active when I’m directly looking at the cafe feed. Guess I could save as a file and then do as you suggest.
thanks

jim

Posted by: jim stasheff on November 1, 2009 1:10 AM | Permalink | Reply to this

Re: pagination

Jim wrote:

my browser does for pdf etc but it’s not active when I’m directly looking at the cafe feed.

I don’t know what the ‘cafe feed’ is, nor what browswer you use — but if you use Firefox to read this webpage (the blog entry we’re reading now!), and then click on ‘File’ in the upper left, and then click on ‘Print Preview’, you should get something that’ll let you print just the pages you want.

Posted by: John Baez on November 2, 2009 6:50 AM | Permalink | Reply to this

Re: pagination

JB wrote:

but if you use Firefox to read this webpage (the blog entry we’re reading now!), and then click on `File’ in the upper left, and then click on `Print Preview’, you should get something that’ll let you print just the pages you want.

Almost, but if it works for others as it does for me, `Print Preview’ is not an option, but clicking on `Print’ opens a window with a Preview option

thanks

Posted by: jim stasheff on November 2, 2009 1:57 PM | Permalink | Reply to this

Re: Generalized Operads in Classical Algebraic Topology

I’m a bit puzzled about the non-symmetric version of the “category of operators” construction; perhaps some reader can help me out. If we start with the terminal non-symmetric operad \mathcal{M}, whose algebras are monoids, generate the free non-symmetric semicartesian operad on it with an LL functor, and then generate the free non-symmetric semicartesian monoidal category on that with an FF functor, then I’m pretty sure that what we get is the category of finite linear orders and order-preserving partial functions, say FinLin parFinLin_{par}. (Note that FinSet *FinSet_* is equivalent to the category FinSet parFinSet_{par} of finite sets and all partial functions.)

Now FinLin parFinLin_{par} has all the right properties, e.g. any functor from it to a cartesian monoidal category DD is automatically colax, and if it is strong then it defines a (non-commutative) monoid in DD, whereas if we demand the colax comparison maps only to be weak equivalences we get a notion of homotopy non-commutative monoid. I’m pretty sure I’ve even seen this category somewhere before, although at the moment I can’t think where. However, homotopy non-commutative monoids are more usually defined as functors out of Δ op\Delta^{op}. If we define Δ op\Delta^{op} as the category of finite linear orders with distinct distinguished endpoints (and endpoint- and order-preserving maps), then it looks a bit like FinLin parFinLin_{par}, but as far as I can tell neither is even a subcategory of the other. Can anyone tell how they are related? And is there any way to get Δ op\Delta^{op} out of this machinery? It seems a bit unlikely, since it isn’t even a monoidal category.

Posted by: Mike Shulman on November 2, 2009 3:13 AM | Permalink | PGP Sig | Reply to this

Re: Generalized Operads in Classical Algebraic Topology

…the result is precisely the category of operators 𝒪^\hat \mathcal{O} of 𝒪\mathcal{O}

A little bit on this now here: category of operators

Posted by: Urs Schreiber on December 11, 2009 12:12 AM | Permalink | Reply to this
Read the post Generalized Multicategories
Weblog: The n-Category Café
Excerpt: A brief introduction to generalized multicategories, in honor of a new draft of a paper about them.
Tracked: January 4, 2010 2:25 AM

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