ScalingStacks

The classical Eckmann–Hilton argument (EHA), introduced in [EH62], states that given a set $X$ with two unital (ie having a two-sided unit) binary operations

$$\circ,*\colon X\times X\to X,$$

if the two operations satisfy the “interchange law”

$$\forall a,b,c,d\in X,\quad\quad\left(a\circ b\right)*\left(c\circ d\right)=\left(a*c\right)\circ\left(b*d\right),$$

then they coincide and, moreover, this unique operation is associative and commutative. Even though it is easy to prove, the EHA is very useful. The most familiar applications are the commutativity of the higher homotopy groups of a space and the commutativity of the fundamental group of an $H$-space.

A natural language for discussing different types of algebraic structures and the interactions between them is that of operads (by which, for now, we mean one-colored, symmetric operads in sets). For example, the data of a unital binary operation on a set $X$ can be encoded as a structure of an algebra on $X$ over a certain operad $\mathbf{Uni}$. Similarly, the data of a unital, associative, and commutative binary operation on a set $X$ (namely, the structure of a commutative monoid) can be encoded as an algebra structure on $X$ over the operad $\mathbf{Com}$. Furthermore, the category of operads is equipped with a tensor product operation, introduced by Boardman and Vogt [BV06], such that given two operads $\mathcal{P}$ and $\mathcal{Q}$, a $\left(\mathcal{P}\otimes\mathcal{Q}\right)$-algebra structure on a set $X$ is equivalent to a $\mathcal{P}$-algebra structure and a $\mathcal{Q}$-algebra structure on $X$, which satisfy a certain natural generalization of the interchange law defined above. Specializing to the case at hand, one can rephrase the EHA as

$$\mathbf{Uni}\otimes\mathbf{Uni}\simeq\mathbf{Com}.$$

Noting that $\mathbf{Com}$ is the terminal object in the category of operads (as all operation sets are singletons), this formulation looks perhaps a bit less surprising than the classical one. One can further observe that we can replace $\mathbf{Uni}$ by more general operads. We call an operad $\mathcal{P}$ reduced if both the set of nullary and the set of unary operations of $\mathcal{P}$ are singletons (ie there is a unique constant and it serves as a unit for all operations). The classical proof of the EHA can be easily modified¹¹ 1 Eg see Proposition 3.8 of [FV15]. to show that given two reduced operads $\mathcal{P}$ and $\mathcal{Q}$ whose $n$-ary operation sets are non-empty for all $n$, we have

$$\mathcal{P}\otimes\mathcal{Q}\simeq\mathbf{Com}.$$

We call this the “operadic formulation of the EHA”.

In many applications of the EHA, the two binary operations one starts with are actually known to be associative in advance. This version, which of course follows from the general EHA, can be stated as

$$\mathbf{Ass}\otimes\mathbf{Ass}\simeq\mathbf{Com},$$

where $\mathbf{Ass}$ is the operad that classifies the structure of a (unital, associative) monoid. For future reference, we call this “the associative EHA”.

The language of operads already helps in organizing and systematizing the study of ordinary algebraic structures, but it is really indispensable for studying (and even defining) enriched and homotopy coherent algebraic structures. To start with, by replacing the sets of $n$-ary operations of an operad with spaces and requiring the various composition and permutation maps to be continuous, one obtains the notion of a topological operad. By further introducing an appropriate notion of a weak equivalence, one can study homotopy coherent algebraic structures. A fundamental example of such an object is the little $n$-cubes topological operad $\mathbb{E}_{n}$ for $0\leq n\leq\infty$ (see, eg [May72]). Loosely speaking, the structure of an $\mathbb{E}_{n}$-algebra on a space $X$ can be thought of as a continuous unital multiplication map on $X$ for which associativity holds up to a specified coherent homotopy and commutativity also holds up to a specified coherent homotopy, but only up to “level $n$’’ ²² 2 The situation is slightly different for $n=0$ as an $\mathbb{E}_{0}$-algebra is just a pointed space.. On a more technical level, $\mathbb{E}_{1}$ and $\mathbb{E}_{\infty}$ can be interpreted as cofibrant models for $\mathbf{Ass}$ and $\mathbf{Com}$, respectively, in a suitable model structure on the category of topological operads (eg [Vog03]). The sequence $\mathbb{E}_{n}$ serves as a kind of interpolation between them.

There are many approaches to modeling “homotopy coherent operads” (both one-colored and multi-colored). Among them, the original approach of J.P. May via specific topological operads [May72], via model structures (or partial versions thereof) on simplicial operads [BM03, Vog03, CM13b, Rob11] or dendroidal sets/spaces [CM11, CM13a], via “operator categories” of C. Barwick [Bar18] or intrinsically to $\left(\infty,1\right)$-categories via analytic monads [GHK17] or Day convolution [Hau17]. We have chosen to work with the notion of $\infty$-operads introduced and developed in J. Lurie’s [Lur] based on the theory of $\infty$-categories introduced by A. Joyal [Joy02] and extensively developed in [Lur09]³³ 3 See [CHH18] for a discussion of the comparison of the different models.. In this theory of $\infty$-operads (as in some of the others), there is a notion analogous to the Boardman–Vogt tensor product and it is natural to ask whether there is also an analogue of the EHA. For the associative EHA, one has the celebrated “additivity theorem”, proved by G. Dunn in the classical context [Dun88] and by Lurie in the language of $\infty$-operads [Lur, Theorem 5.1.2.2], which states that for all integers $m,k\geq 0$, we have

$$\mathbb{E}_{m}\otimes\mathbb{E}_{k}\simeq\mathbb{E}_{m+k}.$$

The goal of this paper is to state and prove an $\infty$-categorical version of the classical (non-associative) EHA. The key observation about the operadic formulation of the classical EHA is that both the hypothesis regarding the non-emptiness of the operation sets of $\mathcal{P}$ and $\mathcal{Q}$ and the characterization of $\mathbf{Com}$ as having singleton operation sets can be phrased in terms of connectivity bounds. For an integer $d\geq-2$, we say that a reduced $\infty$-operad $\mathcal{P}$ is $d$-connected if all of its operation spaces are $d$-connected. We prove

Unlike in the classical case, our result does not imply the additivity theorem (or vise versa), but the additivity theorem does demonstrate the sharpness of our result, since $\mathbb{E}_{n}$ is $\left(n-2\right)$-connected for all $n\geq 0$.

We shall deduce our $\infty$-categorical version of the EHA from a “relative” version, which might be of independent interest. For a reduced $\infty$-operad $\mathcal{P}$ and an integer $n\geq 0$, we denote by $\mathcal{P}\left(n\right)$ the space of $n$-ary operations of $\mathcal{P}$. We say that a map of spaces is a $d$-equivalence, if it induces a homotopy equivalence on $d$-truncations, and that a map of reduced $\infty$-operads $\mathcal{P}\to\mathcal{Q}$ is a $d$-equivalence, if for every integer $n\geq 0,$ the map $\mathcal{P}\left(n\right)\to\mathcal{Q}\left(n\right)$ is a $d$-equivalence.

This behavior of the Boardman–Vogt tensor product on reduced $\infty$-operads is somewhat analogous to the behavior of the join operation on spaces. Given a map of spaces $X\to Y$ that is a $d$-equivalence and a $k$-connected space $Z$, the map $X\star Z\to Y\star Z$ is a $\left(d+k+2\right)$-equivalence. Incidentally, for the space of binary operations we have $\left(\mathcal{P}\otimes\mathcal{R}\right)\left(2\right)\simeq\mathcal{P}\left(2\right)\star\mathcal{R}\left(2\right)$ (see [FV15, Proposition 4.8]), which relates the two phenomena.

The proof of the classical EHA is straightforward. One simply uses repeatedly the unitality and interchange law to deduce the various equalities. For $\infty$-operads, the situation is considerably more complicated as all identities hold only up to a specified coherent homotopy and keeping track of this large amount of data is very difficult. Consequently, there is probably no hope of writing down an explicit formula for the operation spaces of $\mathcal{P}\otimes\mathcal{Q}$ in terms of those of $\mathcal{P}$ and $\mathcal{Q}$, except for low degrees. Therefore, as usual with $\infty$-categories, one has to adopt a less direct approach.

The proof of 1.0.2 proceeds by a sequence of reductions, which we now sketch in an informal way (we refer the reader to the end of this section for a list of notational conventions). An $\infty$-operad is called an essentially $d$-operad if all of its multi-mapping spaces are homotopically $\left(d-1\right)$-truncated. With every $\infty$-operad we can associate an essentially $d$-operad, called its $d$-homotopy operad, by $\left(d-1\right)$-truncating the multi-mapping spaces. This operation constitutes a left adjoint to the inclusion of the full subcategory on essentially $d$-operads into the $\infty$-category of $\infty$-operads. Using this adjunction and the Yoneda lemma, a map of $\infty$-operads $f\colon\mathcal{P}\to\mathcal{Q}$ is a $d$-equivalence if and only if the induced map

$$\operatorname{Map}\left(\mathcal{Q},\mathcal{R}\right)\to\operatorname{Map}\left(\mathcal{P},\mathcal{R}\right)$$

is a homotopy equivalence for every essentially $\left(d+1\right)$-operad $\mathcal{R}$. Further analysis of the monad associated with a reduced $\infty$-operad shows that when $\mathcal{P}$ and $\mathcal{Q}$ are reduced, it is enough to check the above equivalence only for $\mathcal{R}$-s that are $\left(d+1\right)$-topoi endowed with the Cartesian symmetric monoidal structure.

Now let $\mathcal{P}\to\mathcal{Q}$ be a $d$-equivalence of reduced $\infty$-operads and let $\mathcal{R}$ be a $k$-connected $\infty$-operad. We want to show that the map $\mathcal{P}\otimes\mathcal{R}\to\mathcal{Q}\otimes\mathcal{R}$ is a $\left(d+k+2\right)$-equivalence. By the above reductions, it is enough to show that for every $\left(d+k+3\right)$-topos $\mathcal{C}$ endowed with the Cartesian symmetric monoidal structure, the induced map

$$\operatorname{Map}\left(\mathcal{Q}\otimes\mathcal{R},\mathcal{C}\right)\to\operatorname{Map}\left(\mathcal{P}\otimes\mathcal{R},\mathcal{C}\right)$$

is a homotopy equivalence. A key property of the tensor product of $\infty$-operads is that it endows the $\infty$-category of $\infty$-operads with a symmetric monoidal structure that is closed. Namely, for every $\infty$-operad $\mathcal{O}$, there is an internal hom functor $\operatorname{Alg}_{\mathcal{O}}\left(-\right)$ that is right adjoint to the tensor product $-\otimes\mathcal{O}$ ⁴⁴ 4 It is necessary to work here with multi-colored operads as developed in [Lur], since even though the full subcategory of one-colored, or even reduced, $\infty$-operads is closed under the tensor product, the induced symmetric monoidal structure would not be closed.. It is therefore enough to show that the map

$$\operatorname{Map}\left(\mathcal{Q},\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)\right)\to\operatorname{Map}\left(\mathcal{P},\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)\right)$$

is a homotopy equivalence. Let $\mathbf{Triv}$ be the trivial operad. There are essentially unique maps $\mathbf{Triv}\to\mathcal{P}$ and $\mathbf{Triv}\to\mathcal{Q}$ that induce a commutative triangle

and it is enough to show that the top map induces an equivalence on the fibers over each object $X$ of $\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)$. Fixing such an $X$, the fiber of the left map consists of the space of ways to endow $X$ with the structure of a $\mathcal{Q}$-algebra. Since $\mathcal{Q}$ is reduced, one can show that this is the space of maps from $\mathcal{Q}$ to the so-called “reduced endomorphism operad of $X$”. This is a reduced $\infty$-operad $\operatorname{End}^{\operatorname{\scriptsize{red}}}\left(X\right)$ whose space of $n$-ary operations is roughly the space of maps $X^{n}\to X$ for which plugging the unique constant in all entries but one produces the identity map of $X$. More formally, we have a homotopy fiber sequence

$$\operatorname{End}^{\operatorname{\scriptsize{red}}}\left(X\right)\left(n\right)\to\operatorname{Map}\left(X^{n},X\right)\to\operatorname{Map}\left(X^{\sqcup n},X\right)$$

over the fold map $\nabla\colon X^{\sqcup n}\to X$. Consequently, by applying analogous reasoning to $\mathcal{P}$ and some naturality properties, we are reduced to showing that for all $X$ in $\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)$, the induced map

$$\operatorname{Map}\left(\mathcal{Q},\operatorname{End}^{\operatorname{\scriptsize{red}}}\left(X\right)\right)\to\operatorname{Map}\left(\mathcal{P},\operatorname{End}^{\operatorname{\scriptsize{red}}}\left(X\right)\right)$$

is a homotopy equivalence. Since $\mathcal{P}\to\mathcal{Q}$ is a $d$-equivalence, it will suffice to show that $\operatorname{End}^{\operatorname{\scriptsize{red}}}\left(X\right)$ is an essentially $\left(d+1\right)$-operad. Namely, we need only to show that the spaces $\operatorname{End}^{\operatorname{\scriptsize{red}}}\left(X\right)\left(n\right)$ are $d$-truncated. Using the homotopy fiber sequence above, we may present $\operatorname{End}^{\operatorname{\scriptsize{red}}}\left(X\right)\left(n\right)$ as the space of lifts in the commutative square

The underlying $\infty$-category of $\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)$ is an essentially $\left(d+k+3\right)$-category (since $\mathcal{C}$ is); hence the right vertical map is $\left(d+k+2\right)$-truncated. We show that in a general presentable $\infty$-category, the space of lifts of an $n$-connected map against an $m$-truncated map is $\left(m-n-2\right)$-truncated. It is therefore enough to show that the map $X^{\sqcup n}\to X^{n}$ is $k$-connected in $\operatorname{Alg}_{\mathcal{R}}\left(\mathcal{C}\right)$. Under suitable conditions, which are satisfied in our situation, the $k$-connectedness of a map of algebras over an $\infty$-operad can be detected on the level of the underlying objects. Using the fact that $\mathcal{C}$ is an $\infty$-topos we are reduced to proving that the map $X^{\sqcup n}\to X^{n}$ has a section and that it becomes an equivalence after $k$-truncation in $\mathcal{C}$. For the first assertion, we show that one can construct a section rather easily using any $n$-ary operation of $\mathcal{R}$ for $n\geq 2$. The second assertion follows from the fact that $\mathcal{R}$ itself is $k$-connected, and so, roughly speaking, after $k$-truncation we can replace $\mathcal{R}$ with $\mathbb{E}_{\infty}$ and the coproduct of $\mathbb{E}_{\infty}$-algebras coincides with the product. The $\infty$-categorical EHA now follows easily from this by taking $\mathcal{Q}=\mathbb{E}_{\infty}$.

The paper is organized as follows. In Section 2, we develop some general theory regarding reduced (and unital) $\infty$-operads. The first theme is the construction and analysis of the reduced endomorphism operad. The second is an explicit formula for the associated map of monads induced from a map of reduced $\infty$-operads.

In Section 3, we recall from [SY19] some basic definitions and properties of essentially $d$-categories (and operads), as well as the notion of a $d$-homotopy category (and operad). We then proceed to prove that a map of $\infty$-operads is a $d$-equivalence if and only if it induces an equivalence on the spaces of algebras in every $\left(d+1\right)$-topos endowed with the Cartesian symmetric monoidal structure.

In Section 4, we prove some general results regarding the notions of $d$-connected and $d$-truncated morphisms in presentable $\infty$-categories.

In Section 5 we prove the main results of the paper. In particular we prove 1.0.2 and the $\infty$-categorical Eckmann–Hilton argument as a corollary. We also include a couple of simple applications.

For a more detailed outline we refer the reader to the individual introduction of each section.

Much of the length of the paper is due to the careful and detailed verification of many lemmas in $\infty$-category theory, whose proofs are arguably straightforward, but nonetheless do not appear in the literature. This refers mainly to the material up to subsection 4.3, from which the main theorems are 2.2.9, 3.1.8, and 3.2.6. Having said that, we believe that the theory and language of $\infty$-categories in general and $\infty$-operads in particular is still in an early enough stage of development to justify full detailed proofs of every claim that has no reference (known to the authors) in the literature. Hopefully, the added value in terms of rigor and accessibility to non-experts compensates for the loss in brevity and elegance of exposition.

We would like to thank Julie Bergner and Jim Stasheff, as well as all the participants of the Seminarak group, for useful discussions about the subject of this paper. We also thank the referee for helpful comments and corrections. The first Author was supported by the Alon Fellowship and the ISF grant 1588/18 and the second author was supported by the ISF grant 1650/15.

We work in the setting of $\infty$-categories (a.k.a. quasi-categories) and $\infty$-operads, relying heavily on the results of [Lur09] and [Lur]. Since we have numerous references to these two foundational works, references to [Lur09] are abbreviated as T.? and those to [Lur] as A.? while other references are cited in the standard way. As a rule, we follow the notation of [Lur09] and [Lur] whenever possible. However, we supplement this notation and deviate from it in several cases in which we believe this enhances readability. In particular:

1. 1.

   We abuse notation by identifying an ordinary category $\mathcal{C}$ with its nerve $N\left(\mathcal{C}\right)$.

2. 2.

   We use the symbol $\text{pt}_{\mathcal{C}}$ to denote the terminal object of an $\infty$-category $\mathcal{C}$ (or just pt if $\mathcal{C}$ is clear from the context).

3. 3.

   We abbreviate the data of an $\infty$-operad $p\colon\mathcal{O}^{\otimes}\to\mathbf{Fin}_{*}$ by $\mathcal{O}$ and reserve the notation $\mathcal{O}^{\otimes}$ for the $\infty$-category that is the source of $p$. Similarly, given two $\infty$-operads $\mathcal{O}$ and $\mathcal{U}$, we write $f\colon\mathcal{O}\to\mathcal{U}$ for a map of $\infty$-operads from $\mathcal{O}$ to $\mathcal{U}$. The underlying $\infty$-category of $\mathcal{O}$, which in [Lur] is denoted by $\mathcal{O}_{\left\langle 1\right\rangle}^{\otimes}$, is here denoted by $\underline{\mathcal{O}}$.

4. 4.

   When the $\infty$-operad is a symmetric monoidal $\infty$-category, we usually denote it by $\mathcal{C}$ or $\mathcal{D}$. We will sometimes abuse notation and write $\mathcal{C}$ also for the underlying $\infty$-category $\underline{\mathcal{C}}$ when there is no chance of confusion.

5. 5.

   By a presentably symmetric monoidal $\infty$-category, we mean a symmetric monoidal $\infty$-category $\mathcal{C}$, such that the underlying $\infty$-category $\underline{\mathcal{C}}$ is a presentable $\infty$-category and the tensor product preserves colimits separately in each variable.

6. 6.

   Given two $\infty$-operads $\mathcal{O}$ and $\mathcal{U}$, we denote by $\operatorname{Alg}_{\mathcal{O}}\left(\mathcal{U}\right)$ the $\infty$-operad $\operatorname{Alg}_{\mathcal{O}}\left(\mathcal{U}\right)^{\otimes}\to\mathbf{Fin}_{*}$ from Example A.3.2.4.4. This is the internal mapping object induced from the closed symmetric monoidal structure on $\mathbf{Op}_{\infty}$ (see A.2.2.5.13). The underlying $\infty$-category $\underline{\operatorname{Alg}}_{\mathcal{O}}\left(\mathcal{U}\right)$ is the usual $\infty$-category of $\mathcal{O}$-algebras in $\mathcal{U}$ (which in [Lur] is denoted by $\operatorname{Alg}_{\mathcal{O}}\left(\mathcal{U}\right)$). Moreover, the maximal Kan sub-complex $\underline{\operatorname{Alg}}_{\mathcal{O}}\left(\mathcal{U}\right)^{\simeq}$ is the space of morphisms $\operatorname{Map}_{\mathbf{Op}_{\infty}}\left(\mathcal{O},\mathcal{U}\right)$ from $\mathcal{O}$ to $\mathcal{U}$ as objects of the $\infty$-category $\mathbf{Op}_{\infty}$. Recall from A.3.2.4.4 that for a symmetric monoidal $\infty$-category $\mathcal{C}$, the $\infty$-operad $\operatorname{Alg}_{\mathcal{O}}\left(\mathcal{C}\right)$ is again symmetric monoidal and for every object $X\in\underline{\mathcal{O}}$, the evaluation functors $ev_{X}\colon\operatorname{Alg}_{\mathcal{O}}\left(\mathcal{C}\right)\to\mathcal{C}$ are symmetric monoidal functors.

7. 7.

   Let $\mathcal{C}$ be an $\infty$-category. We denote the corresponding coCartesian $\infty$-operad $\mathcal{C}^{\sqcup}\to\mathbf{Fin}_{*}$ by $\mathcal{C}_{\sqcup}$ (see Definition A.2.4.3.7). If $\mathcal{C}$ has all finite products, we denote the Cartesian symmetric monoidal $\infty$-category $\mathcal{C}^{\times}\to\mathbf{Fin}_{*}$ by $\mathcal{C}_{\times}$ (see Construction A.2.4.1.4).