The n-Category Café

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 $V$ be a cartesian monoidal category (symmetric monoidal is enough for operads, but not really for finite-product theories). Think of $V=Set$ or $V=Top$.

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

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

3. Lawvere theories: Let $N_{func}$ denote the category of natural numbers and all functions (that is, a skeleton of $FinSet$). A finite-product collection in $V$ is a functor $N_{func}\to V$. There is a special monoidal structure on the category of finite-product collections, and a finite-product theory or Lawvere theory in $V$ 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 $N_{bij}$, $N_{id}$, and $N_{func}$, 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. $N_{bij}^{op}$ is the free symmetric (strict) monoidal category on one object,
2. $N_{id}^{op}$ is the free (strict) monoidal category on one object, and
3. $N_{func}^{op}$ 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 ($V$-)category,
2. we can talk about algebras for a non-symmetric operad in any monoidal ($V$-)category, and
3. we can talk about algebras for a Lawvere theory in any cartesian monoidal ($V$-)category.

In other words, given a putative type of theory X, we first choose a monad $T_X$ (usually on $Cat$ or $V Cat$) 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 $V$ to be a functor $T_X(1)^{op}\to V$, 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 $Prof_X$ whose objects are ($V$-)categories and whose 1-cells $A\to B$ are profunctors $A\to T_X B$; it’s a sort of Kleisli bicategory of the monad $T_X$. The monoidal category of X-collections is the hom-category $Prof_X(1,1)$. A monoid (aka a monad) in the bicategory $Prof_X$ 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 $End(a) = C(a^{\otimes n},a)$ for any object $a$ in a (symmetric) monoidal category $C$. Then for any operad $\mathcal{O}$, an $\mathcal{O}$-algebra structure on $a$ is an operad map $\mathcal{O}\to End(a)$.

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

This approach works for X-operads for arbitrary X. In particular, any monoidal category $C$ has an underlying multicategory $U_{mon}(C)$ (= colored non-symmetric operad), any symmetric monoidal category has an underlying symmetric multicategory $U_{sym}(C)$ (= colored symmetric operad), and any cartesian monoidal category has an underlying cartesian multicategory $U_{cart}(C)$ (= 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 $$U_X\colon \text{X-structured categories} \to \text{X-multicategories}$$ has, in good situations, a left adjoint $F_X$. For example, for any operad $\mathcal{O}$ there is a free symmetric monoidal category $F_{sym}(\mathcal{O})$ on $\mathcal{O}$. It follows by adjointness that an $\mathcal{O}$-algebra in a symmetric monoidal category $C$ can equally well be described as a functor $F_{sym}(\mathcal{O}) \to C$ of symmetric monoidal categories. Thus $F_{sym}(\mathcal{O})$ is the free symmetric monoidal category containing an $\mathcal{O}$-algebra.

In fact, the functor $F_X$ is usually faithful, so that an X-multicategory can be defined to be an X-structured category that is “in the image of $F_X$”. Lawvere theories are most commonly defined in this way, since in this case the image of $F_{cart}$ 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: $$\array{ \text{mon. cats} & \rightleftarrows & \text{sym. mon. cats} & \rightleftarrows & \text{cart. mon. cats} \\ \uparrow\downarrow && \uparrow\downarrow && \uparrow\downarrow \\ \text{multicats} & \rightleftarrows & \text{sym. multicats} & \rightleftarrows & \text{cart. multicats} }$$ We’ve already mentioned the vertical adjunctions $F_X \dashv U_X$. The left-pointing arrows, which I’ll write as $R$, are forgetful functors. In good situations, each such functor has a left adjoint which I’ll write as $L$. Thus we see, for instance, that any non-symmetric operad $\mathcal{O}$ freely generates a symmetric operad $L_{sym}(\mathcal{O})$, with the property that if $C$ is a symmetric monoidal category, then (by adjointness) $L_{sym}(\mathcal{O})$-algebras in $C$ are the same as $\mathcal{O}$-algebras in $R(C)$ (that is, $C$ considered as a mere monoidal category). Likewise, any operad $\mathcal{O}$ generates a Lawvere theory $L_{cart}(\mathcal{O})$ 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 $V=Set$ for simplicity, and consider the case where X = arbitrary small products. Then $T_X(1)^{op} = Set$ (i.e. $Set$ is the free category with small coproducts on one object), an X-collection is just a functor $Set\to Set$, and the monoidal structure on X-collections turns out to be just composition. Thus, an “X-operad” is simply a monad on $Set$. This value of X lives at the far right of the continuum, with forgetful functors $R$ landing in operads, Lawvere theories, and so on, which have left adjoints $L$ in good cases. It follows by adjointness, as usual, that every operad or Lawvere theory $\mathcal{O}$ gives rise to an associated monad $L_{sprod}(\mathcal{O})$ with the same algebras. One can then trace through the definitions and recover the usual explicit definition of this monad: $$a \mapsto \int^n \mathcal{O}(n) \times a^n$$ where the coend is over $N_{sym}$, $N_{id}$, or $N_{func}$, as appropriate. (To play this game for $V$ other than $Set$, 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 $T_{semicart}(1)^{op}$ can be identified with $N_{inj}$, the category of finite sets and injections. Therefore, for any injection $\gamma\colon n\to m$, a semicartesian operad $\mathcal{O}$ comes with an operation $\mathcal{O}(n)\to \mathcal{O}(m)$. We think of this as saying “given an operation that takes $n$ inputs, we can make an operation that takes $m$ inputs by discarding those that are not in the image of $\gamma$.” 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 $C$ be semicartesian and $D$ be cartesian, and let $G\colon C\to D$ be any functor. Then for any $a,b\in C$ we have maps $a\otimes b \to a\otimes 1 \cong a$ and $a\otimes b \to 1\otimes b \cong b$ in $C$, inducing maps $G(a\otimes b) \to G(a)$ and $G(a\otimes b)\to G(b)$. But $D$ is cartesian monoidal, so these two maps induce a map $G(a\otimes b)\to G(a) \times G(b)$. Of course, $G$ is strong monoidal iff these canonical maps are isomorphisms.

If $D$ has moreover a notion of “weak equivalence,” it makes sense to say that $G$ is “homotopy strong monoidal” if these canonical maps are weak equivalences, or perhaps better, if the iterated canonical maps $$G(a_1 \otimes \dots \otimes a_n) \to G(a_1)\times \dots\times G(a_n)$$ are weak equivalences. But now recall that if $\mathcal{O}$ is a semicartesian operad, $\mathcal{O}$-algebras in a cartesian monoidal category $D$ are (semicartesian) strong monoidal functors $F_{semicart}(\mathcal{O}) \to R(D)$. Thus it makes sense to say that a homotopy $\mathcal{O}$-algebra is a functor $G\colon F_{semicart}(\mathcal{O}) \to D$ 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 $\mathcal{O}$ in topological spaces, generate a free semicartesian operad $L(\mathcal{O})$, then generate a free semicartesian symmetric monoidal category $F_{semicart}(L(\mathcal{O}))$ from this, the result is precisely the category of operators $\hat{\mathcal{O}}$ associated to $\mathcal{O}$ by May and Thomason in “The uniqueness of infinite loop space machines”. Moreover, the “homotopy $\mathcal{O}$-algebras” defined above are basically their $\hat{\mathcal{O}}$-spaces (minus a cofibration condition).

In particular, for the operad $\mathcal{O}$ whose algebras are commutative monoids, we have $F_{semicart}(L(\mathcal{O})) = FinSet_∗$, the category of finite based sets, which is the opposite of Segal’s category $\Gamma$. A (special) $\Gamma$-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 $C$ be any symmetric monoidal category with unit object $i$, and consider the co-slice category $i/C$. It has a monoidal structure given by taking $i\to a$ and $i\to b$ to $$i \cong i \otimes i \to a\otimes b,$$ and the unit object $i\to i$ is of course initial. Note that this monoidal structure is not the “smash product” often considered on co-slice categories. For instance, when $C=Set$ (or $Top$) this is the category of pointed sets (or spaces) with its cartesian product.

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

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

What basepoint, you ask? Well, as with any operad, there is a specified identity operation $id \in \mathcal{O}(1)$, whose image under the function $\mathcal{O}(1)\to \mathcal{O}(0)$ induced by the obvious inclusion $0\hookrightarrow 1$ gives a canonically defined 0-ary operation in any semi-cocartesian operad. That means that any $\mathcal{O}$-algebra will have a canonically specified basepoint. Moreover, it’s kind of stupid if $\mathcal{O}(0)$ contains any more than one 0-ary operation, since 0-ary operations in an $\mathcal{O}$-algebra $a$ will correspond to maps $i \to a$ 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 $\mathcal{O}(0)$ is a terminal object.

Now, invoking the appropriate notion of “equivariance,” it’s easy to see that for any injection $\gamma\colon n\to m$, the induced map $\mathcal{O}(m)\to \mathcal{O}(n)$ really is given by “plug $n$ inputs into it in the places specified by $\gamma$, 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 $\mathcal{O}(0)$ is terminal. In fact, it’s not hard to check that any symmetric operad with $\mathcal{O}(0)$ 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 $\mathcal{O}(0)$ be terminal; more recently he has referred to operads with this property as reduced.

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

Now if $D$ is an arbitrary symmetric monoidal category (possibly cartesian, possibly $V$ itself, so think of $Set$ or $Top$), we have a sequence of bijections between

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

Finally, recall that for any X-operad $\mathcal{O}$ and X-structured category $D$, there is (in good situations) a monad on $D$ whose algebras are the $\mathcal{O}$-algebras. The above sequence of bijections thus implies that there are two monads, one on $D$ (the category of “unbased” objects) and one on $i/D$ (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 $i/D$, which are exactly specified by saying that it’s a coend over $N_{inj}^{op}$—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 $\mathcal{O}$ is the “little $n$-cubes operad”, it is this second monad, acting on based spaces, which is equivalent to the $n$-fold loop-suspension monad $\Omega^n \Sigma^n$. This is the essential starting point for the operadic study of iterated loop spaces.