MOTIVATING THE DEFINITION OF MONOIDAL ∞-

DANIEL FUENTES-KEUTHAN

Abstract. What follows is my attempt to motivate the definition of monoidal ∞-categories given in [Lur09b, § 1], which I first learned from the set of notes [Gro15]. In these works, monoidal ∞-categories are defined as certain fibra- tions of simplicial sets called cocartesian fibrations which seem to hide the actual monoidal structure.

Contents 1. in Topology 1 1.1. Towards Homotopy Coherent Monoids 1 1.2. The Bar Construction 3 1.3. The Commutative Story 5 2. Monoidal Categories 6 2.1. Monoidal Categories via the Bar Construction 6 2.2. The Grothendieck Construction 8 2.3. Monoidal Categories via opFibrations 10 3. Monoidal ∞-categories 13 3.1. Monoidal ∞-Categories via coCartesian Fibrations 13 3.2. Towards Higher Algebra 14 References 16

1. Monoids in Topology Perhaps the most appropriate place to start a study of multiplicative structures defined up to homotopy is in the world of topological spaces. Here the large gap be- tween strictly commuting structures and those relaxed to homotopy commutation allows the full complexity of the situation to come to light, providing greater admi- ration for the book-keeping power of recasting monoids in Top as certain simplicial objects. In this section we review the basics of topological monoids, H-spaces, and A∞ spaces, showing how to define good notions of homotopy coherent structures as weakenings of reformulations of strict structures.

1.1. Towards Homotopy Coherent Monoids. Recall the definition of a topo- logical Definition 1.1. A topological monoid is a topological space M together with a continuous associative binary operation µ: M × M → M and a distinguished element e ∈ M which acts as a two sided unit for µ. 1 2 DANIEL FUENTES-KEUTHAN

Our goal is to find an appropriate notion of this definition in the world of homo- topy theory. To get a sense for what appropriate means here, we will first consider homotopy associative H-spaces1 as a sort of first approximation. Note that the def- inition above can be recast diagrammatically: the continuous map µ: M ×M → M fits into the below commutative diagrams with the map η : ? → M that picks out the distinguished identity element

M×µ M×η η×M M × M × M M × M M M × M M

µ×M µ µ µ M × M M M

We will naively replace the identity 2-cells which fill the faces in these diagrams by unspecified homotopies.

Definition 1.2. A homotopy associative H-space is a topological space M together with a binary operation µH : M × M → M and a distinguished element e ∈ M. The operation µ is homotopically associative and unital with respect to e in that the diagrams below commute up to homotopy.

M×µ M×η η ×M M × M × M H M × M M H M × M H M ' ' µH ×M ' µH µH M × M M µH M

In other words, in a homotopy associative H-space the two multiplications (xy)z and x(yz) may not coincide exactly, but there is a continuous family of paths I×M 3 → M which interpolates between the two. Likewise, while the multiplication µH (e, −) might not exactly equal the identity, there is a homotopy to the identity I×M → M. Succinctly, a homotopy associative H-space is an associative monoid in the homotopy category of spaces H. While this sounds like a good thing, structures defined within H are notoriously poorly behaved. To get a sense of why such a structure does not satisfy our goal, consider the difference in our two definitions when we move to higher arity multiplications induced by the binary operations µ and µH . In the strict world of topological spaces and continuous maps the associativity n constraint for µ allows us to define an unambiguous n-ary operation µn : M → M. In fact, we could have taken this “unbiased” definition of a topological monoid, a space with compatible higher arity operations, in place of our original “biased definition” of a space with a binary operation and nothing would change. On the other hand, the unspecified homotopies in the definition of a homotopy associative H-space introduce ambiguities even when we move one step higher and consider induced operations M 4 → M. Consider the pentagon of homotopies relating mul- tiplications below

1H for Hopf not Homotopy. MOTIVATING THE DEFINITION OF MONOIDAL ∞-CATEGORY 3

x((yz)w)

x(y(zw)) (x(yz))w

(xy)(zw) ((xy)z)w The existing information does not allow us to relate two paths from x(y(zw)) to (x(yz))w in the diagram above: they represent legitimately different homotopies re- lating associativity of multiplication with four multiplicands. To solve this problem of ambiguity, we could ask axiomatically for the additional data of a 2-homotopy relating the various 1-homotopies (the paths above). However, we will still run into the same problem when we consider multiplication with more than four multipli- cands. What we need then is to ask, in the definition of homotopy coherently associa- tive monoid for the data of: a space M, a binary opertation µ, a 1-homotopy α1 witnessing associativity of µ, a 2-homotopy α2 which witnesses higher associativ- ity of α1, a 3-homotopy α3 witnessing the even higher associativity associated to α2, and so on ad infinitum. We strees that we must ask for this information as data, as opposed to obtaining it through asking for additional structure, such as the commutation of certain diagrams. In the next section we will take a simplicial approach to neatly wrapping up this tremendous amount of data.

1.2. The Bar Construction. Returning for a moment to (strict) topological n monoids, the associativity relations between higher arity multiplications µn : M → M and lower arity multiplications are generated entirely by relations regarding re- peated application of the operation µ. In other words, they are equivalent to asking that any combination of “face maps”, those maps which are products of the identity on M, together with a single application of µ, M n → M n−1 → M n−2 → · · · → M result in the same map M n → M. The unit conditions ask for the “degeneracies”, those maps M n−1 → M n which are products of the identity on M with η at exactly one position, to play nicely with these face maps. The names face and degeneracy op are of course chosen on purpose. Consider the BarM : ∆ → T op defined n on objects by [n] 7→ M , inner face maps di, 0 < i < n by applying µ at position i, outer facemaps d0, dn by projecting away from the first and last factor of M respectively, and on degeneracies si by applying η at position i. The above construction is referred to as the bar construction2. It allows us to associate a simplicial object in Top to a topological monoid. This functor has the benefit of lining up with the notion of unbiased monoid in the sense that it captures the data of higher arity operations, without preference to the binary operation, as composites of inner facemaps M n → M given by [1] → [n], 0 7→ 0, 1 7→ n. The functor also recovers the biased point of view via a monoidal structure on BarM [1]. To see this, note that for each n, the collection of ιi : [1] → [n], 0 < i ≤ n, which map 1 7→ i, 0 7→ i−1, assemble into a homeomorphism when applying BarM

Πιi ×n BarM [n] −−→ BarM [1]

2Note to Tslil: lower case ‘b’, single ‘r’. 4 DANIEL FUENTES-KEUTHAN

In particular, using the inverse to the map BarM [2] → BarM [1] × BarM [1], one obtains a binary operation µBar and an element ηBar ∈ BarM [1]

d1 µBar : BarM [1] × BarM [1] → BarM [2] −→ BarM [1] ∼ s1 ηBar : ? = BarM [0] −→ BarM [1] The simplicial identities amongst [0], [1], [2], [3], and [4] recover associativity and unit constraints on BarM [1] making it into a topological monoid. For example, the identity d2 ◦d1 = d1 ◦d1 : [2] → [3] corresponds to the equality x(yz) = (xy)z under the definition of µH . The topological monoids (BarM [1], µBar, ηBar) and (M, µ, η) are isormophic as monoids in T op. In fact a stronger statement is true. It turns out that the home- omorphism condition above, the “strict Segal condition”, is the defining charac- teristic of the bar construction in the following sense. Let M : ∆op → T op be a simplicial object in T op and suppose also that M satisfies the strict Segal condition, then the same construction as above produces a topological monoid M[1]. Furthermore, at the categorical level, a monoid map between topological monoids, f : M → N gives rise to a Barf : BarM ⇒ BarN by setting ×n Barf [n] = f . Commutation with simplicial maps in this setting is equivalent to compatibility with the monoidal structures present. The following result allows us to freely equate a topological monoid with its bar construction and vice versa. ∆op Theorem 1.3. Let T opsSegal denote the category of simplicial spaces satisfying the strict Segal condition, and let MonT op denote the category of (strict) topolog- ∆op ical monoids and monoid maps. The functor Bar : MonT op → T opsSegal is an equivalence of categories. The equivalence above suggests an alternative approach to a good definition of homotopy coherent monoid, namely that such a structure should manifest as a simplicial object satisfying certain properties which are weakened from those simplicial objects arising from the bar construction. Indeed we readily obtain such a structure by weakening the strict Segal conditions satisfied by a topological monoid. Definition 1.4. A homotopy coherently associative topological monoid (also re- op ferred to as an A∞ space) is a simplicial space A: ∆ → T op satisfying the Segal condition: for each n the map M[n] −−→Πιi M[1]×n is a homotopy equivalence.

As a first remark, note that by 1.3, composing an A∞ space with the localiza- tion functor T op → H yields a homotopy associative H-space. However not every homotopy associative H-space gives an A∞ space (see [Sta70, Thm 6.7] for an ex- ample involving the seven sphere), so this new type of object carries strictly more structure. This structure, conveniently hidden in the functoriality of the definition, corresponds essentially uniquely to a space with a coherently associative and unital binary . To see this we carry out the same construction as in the strict case: choosing a homotopy inverse to the Segal map A[2] −→' A[1] × A[1] yields a product µ: A[1] × A[1] −→' A[2] −→d1 A[1], and the Segal condition for 0 implies that the space A[0] is contractible, so that the first degeneracy operator acts as an identity η : ? ' A[0] −→s1 A[1]. The simplicial identities now imply that, for each n, MOTIVATING THE DEFINITION OF MONOIDAL ∞-CATEGORY 5 any path taken from A[1]×n → A[1] using the just-defined multiplication is homo- topic. For example, the previously mentioned simplical identity d2 ◦ d1 = d1 ◦ d1 now implies the existence of a homotopy x(yz) ' (xy)z. In fact the simplicial identities give something stronger: they specify all higher homotopies relating the various induced higher artiy operations. For example, in the pentagon of the last section, a face is specified between the paths (vertices) and 1-homotopies which rep- resents a 2-homotopy. Similarly, in the corresponding diagram for multiplication of 5 elements, which the reader can check is a 3-dimensional polytope, faces between 1-homotopies are specified, representing 2-homotopies, as well as a 3-homotopy re- lating these 2-homotopies. We hope the reader appreciates the power of the above definition after carefully drawing out this last example. From this point forth we shall freely equate an A∞ space A with its corresponding value at 1, A[1]. No uncertainy can arise. Indeed the only choice to be made is in the homotopy inverse to the Segal map but, as one may check, different choices lead to suitably homotopy equivalent A∞ structures. 1.3. The Commutative Story. If we are to encode the structure of a commuta- tive topological monoid in a functor, we must encode the flip map M ×M −→τ M ×M along with all other permutations of M ×n and their interactions with multiplica- tive operations. In other words, we will need to consider all maps between finite ordinals, not just those which are order preserving. Hence we will study out of a skeletal category F in? of finite, pointed sets, with elements denoted such as hni. Without wasting time, we state our preferred definition of a commutative topological monoid

Definition 1.5. A commutative topological monoid is a functor M : F in? → T op satisfying the strict Segal condition: for each pair of pointed sets S, T , the map M(S ∨ T ) → M(S) × M(T ) defined by the projections S ∨ T → S and S ∨ T → T which collapse a set to the basepoint is a homeomorphism. Note in particular that Mh0i =∼ ?. To see that this gives rise to a commutative monoid structure on Mh1i, we denote by m: h2i → h1i the map which sends only the basepoint of h2i to the basepoint of h1i. Note that m ◦ τ = m so that if we ∼ define µ: Mh1i × Mh1i −→= Mh2i −→m Mh1i, we obtain µ = µ ◦ τ, as desired. In the opposite direction, beginning with a commutative topological monoid M, S we obtain a functor SymM : F in∗ → T op by setting SymM (S) = M , viewed as a discrete topological space, and by assigning each map S −→α T to the map M S → M T defined by L M(α)({ms}s∈S)t = s0∈α−1(t) ms0 Just as in the noncommutative case, we can immediately weaken our Segal condition to obtain an appropriately homotopy coherent notion of commutativity.

Definition 1.6. A homotopy coherently commutative and associative monoid (E∞- space) is a functor M : F in? → T op which satisfies the Segal condition: for each pair of pointed sets S, T , the map M(S ∨ T ) → M(S) × M(T ) defined by the projections S q T → S and S q T → T which collapse a set to the basepoint is a homotopy equivalence. 6 DANIEL FUENTES-KEUTHAN

Choosing a homotopy inverse to the map Mh2i → Mh1i×Mh1i gives a multipli- ∼ cation µ: Mh1i × Mh1i −→= Mh2i −→m Mh1i, and the identity m = m ◦ τ now gives a homotopy µ ◦ τ ' µ, along with coherent higher homotopies when considering higher arity operations, providing Mh1i with a coherently commutative multipli- cation. Coherent associativity, via the underlying A∞ structure, can be witnessed op via the following inclusion: Let φ: ∆ → F in? be the map which sends [n] 7→ hni with 0 as the basepoint, and sends a α:[k] → [n] to the morphism φ(α) defined by φ(α)j = {min(i) ≥ j}, then precomposing the E∞ structure map with 3 φ yields an A∞ structure.

2. Monoidal Categories The homotopy coherent world of turns out to be much stricter than that of topological spaces. This is a good thing, as strict monoidal categories turn out to have less use than strict topological monoids. Unfortunately, this sim- plifies a vast amount of the complexity that we will need to consider when studying monoidal ∞-categories, potentially leading one4 to initially overlook this complex- ity. The issue is that categories live in a 2-dimensional world, and this allows for us to consider just a 2-truncated version of coherence. Topological spaces on the other hand naturally live in an infinite dimensional world. In other words, in category theory we will need to specify data in dimensions 0,1, and only structure in dimen- sion 3. In topology and ∞-category theory, we must specify data in all dimensions since these objects are naturally infinite dimensional. There is no point where we can cut off the data and instead specify structure. Nevertheless, this is a good place to introduce the Grothendieck Construction, a classifying construction which will let us reformulate monoidal categories again from their associated bar constructions to certain types of fibrations. We will ultimately use ∞-categorical analogs of these fibrations to define monoidal ∞-categories. In this section we quickly review (strict) monoidal categories and draw parallels to the Segal conditions used to define topological monoids. 2.1. Monoidal Categories via the Bar Construction. A not-so-close look at the previous section reveals that the results within have little to do with topologi- cal spaces themselves, only that there was an available notion of weakly invertible equivalence. Another such situation where this is the case is when studying monoid objects in Cat. Here we replace the notion of homotopy equivalence with equiva- lence of categories – homotopy equivalence with respect to the interval object given by the walking isomorphism, I = {0 =∼ 1}. Definition 2.1. A strict is a monoid object in (Cat, ×, e): that is, it is a category C together with a binary operation ⊗: C × C → C and a unary operation η : e → C which satisfy the usual associativity and unit monoid axioms, depicted diagrammatically by the following commutative diagrams C×µ C×η η×C C × C × C C × C C C × C C

µ×C µ µ µ C × C C C

3 op Interestingly, the composition of inclusions ∆ ,→ F in∗ ,→ Set∗ is the pointed simplicial circle. 4Such as the author. MOTIVATING THE DEFINITION OF MONOIDAL ∞-CATEGORY 7

Just as before (in fact even more so), this definition is far too strict. It is even less frequent than in topology that one runs into a monoidal structure on a category which is strictly associative. Of course one could naively replace the identity 2-cells in the faces of the diagrams with natural isomorphisms, obtaining a notion of “H- category”, but our experience now tells us that we seek a more coherent solution. Remarkably, unlike in the case of topological spaces, categories are much more rigid objects, and the Coherence Theorem of Maclane [Mac98] asserts that we get a lot of higher coherence for free by assuming much simpler axioms. In fact it suffices to deal only with the case of associativity of four objects. Definition 2.2. A monoidal category C is a category together with a bifunctor ∼= ∼= ⊗: C×C → C, unit object I: e → C, left and right unitors λ: I⊗− ⇒ C, ρ: −⊗I ⇒ C, ∼ and associator α:(− ⊗ −) ⊗ − ⇒= − ⊗ (− ⊗ −) which satisfy the triangle diagram

αA, ,B (A ⊗ I) ⊗ B I A ⊗ (I ⊗ B)

ρA⊗B A ⊗ B A⊗λB and the Pentagon diagram, ∀A, B, C, D ∈ obC (A ⊗ (B ⊗ C)) ⊗ D

((A ⊗ B) ⊗ C) ⊗ D A ⊗ ((B ⊗ C) ⊗ D)

(A ⊗ B) ⊗ (C ⊗ D) A ⊗ (B ⊗ (C ⊗ D))

We stress that what was just defined really is the highly coherent object that we seek, with the extra structure coming for ‘free’ from the Coherence Theorem. The same bar construction from the previous section lets us rewrite the definition op of a strict monoidal structure on a category via a functor BarC : ∆ → Cat, satisfying a strict Segal condition: for each n, we have an isomorphism of categories ∼ ×n BarC[n] = BarC[1] given by the product of the maps ιi. Strict op Theorem 2.3. There is an equivalence of categories, MonCat ' [∆ , Cat]sSegal, between the category of strict monoidal categories and strict monoidal functors, and the category of simplicial objects in Cat satisfying the strict Segal condition. We must be careful when weakening the Segal condition, as we now have two directions to weaken in. We can replace the assumption that the Segal maps give an isomorphism to instead ask for equivalences, as in the case of A∞ spaces, or we could also replace the defining functor by a pseudofunctor. Although any of these combinations will give you an object which produces a monoidal category, it turns out that the correct answer is to do both of these simultaneously:

op Theorem 2.4. There is an equivalence of categories, MonCat ' [∆ , Cat]Segal, between the category of monoidal categories and strong monoidal functors, and the category of simplicial objects in Cat satisfying the Segal condition. The proof of the above theorem requires one additional idea which we have not mentioned. The simplicial object that we obtain from a monoidal category is a 8 DANIEL FUENTES-KEUTHAN priori only a pseudo functor, but a standard result allows us to “rectify” a Cat valued pseudofunctor into a strict functor5. Henceforth we identify a monoidal category with its associated bar construction, taking to heart the idea that defining these structures through functors is a pow- erful book-keeping tool. In the categorical case we get another tool which allows us to turn our new functor definitions of monoidal structures into certain “lower dimensional” fibrations. 2.2. The Grothendieck Construction. The Grothendieck Construction identi- fies a class of functor, the opfibrations, with strict 2-pullbacks of a universal bundle along pseudofunctors. Let Cat?,l be the lax 2-slice of the terminal category inclusion and the identity 2-functor, e ,→ Cat = Cat. This is a 2-category with: • Objects pointed categories e −→Aa • 1-morphisms the 2-cell filled triangles e a b ⇒ A B F φ That is, pairs (F : A → B,F (a) −→ b). • 2-morphisms are described as follows: Let (f, φ) and (f 0, φ0) be two one cells e a A e a A φ φ0 ⇐ f ⇐ f b b B B A 2-cell (f, φ) ⇒ (f 0, φ0) is a 2-cell in Cat, ξ : f ⇒ f 0, such that φ0(ξa) = φ. U The universal Cat bundle is then the forgetful 2-functor Cat?,l −→ Cat. Now given a pseudofunctor F : C → Cat, where C is an ordinary 1-category, the Grothendieck Construction applied to F is the left leg of the strict 2-pullback R F Cat?,l y UF U C Cat Concretely, the 2-category R F has as objects pairs of objects (c ∈ obC, a ∈ f obF (c)), and as 1-morphisms pairs of morphisms (c −→ c0,F (f)(a) → a0). Since C is a 1-category, we won’t be interested in any 2-categorical properties of R F . The left leg of the pullback square above is the . Since R is a 2-, it gives rise to a 2-functor R :[C, Cat] → Cat/C. We are interested in classifying the essential image of this 2-functor, where R turns out to be an equivalence of categories. Consider an object c ∈ obC, together with a choice of a ∈ obF (c). This combi- f 0 nation constitutes an element of the fiber of UF over c. Given a morphism c −→ c in C, we can this to a morphism in R F

5Morally Cat has enough room that we can replace the target categories of the pseudofunctor with “larger” categories to move the strictness problem into the image categories themselves. MOTIVATING THE DEFINITION OF MONOIDAL ∞-CATEGORY 9

f (c,a) ! (c0,F (f)(a))

c c0 f Furthermore, this lifted morphism satisfies the following : sup- g pose c0 −→ c” is another morphism in C, and let h:(c, a) → (c”, a”) sit over the composite g ◦ f, then there exists a unique morphismg ˆ:(c0,F (f)(a)) → (c”, a”) lying over g such thatg ˆ ◦ f! = h. This situation is depicted below (c, a) (c0,F (f)(a)) ∃!ˆg h

(c”,a”)

f c c0 g g◦f c” Note that we could have also chosen any object a0 =∼ F (f)(a) when defining our initial lift and obtained an isomorphic lift in the fiber over f which satisfies the same universal property. We axiomatize this property of R F Definition 2.5. Let p: D → C be a functor, given a morphism φ: c → c0 and an 0 object d in the fiber Dc of p over C, a p-cocartesian lift of φ is an object d ∈ Dc0 φ and a morphism d −→! d0 lying over φ which together satisfy the universal property stated above. When the situation is clear, we may refer to such a morphism as a cocartesian lift, or simply a cocartesian morphism. As mentioned above, p-cocartesian lifts if they exist are unique up to unique isomorphism in c/C. This can be seen also by the following restatement of the above universal property. Lemma 2.6. Let p: D → C be a functor. A morphism f : d → d0 in D is a p-cocartesian lift of p(f) if and only if the uniquely induced morphism

f/D → p(f)/C ×p(d)/C d/D is an isomorphism. The definition of cocartesian morphism a priori only implies that the above functor is an isomorphism on objects, but the lift to category theory ends up being given for free. We will be interested in situations such as R F where there is a sufficient quantity of cocartesian morphisms. Definition 2.7. A functor p: D → C is a (Grothendieck) opfibration if every morphism in C has a p-cocartesian lift. The description of R F above shows that the Grothendieck construction sends any pseudofunctor to an opfibration. Conversely, the defining property of an opfibration implies that the fibers of such a functor vary covariantly pseudofunctorially in C: Suppose p: D → C is an opfibration. We will define a pseduofunctor P : C → Cat as follows 10 DANIEL FUENTES-KEUTHAN

• On objects we send c 7→ Dc α α! • Given c1 −→ c2 in C, we obtain a functor Dc1 −→ Dc2 : α! – Let d1 ∈ Dc1 . Choose a p-cocartesian lift d1 −→ d2 of α, and make the assignment d1 7→ d2. f1 0 α! – Let d1 −→ d1 be a morphism in Dc1 . Choose p-cocartesian lifts d1 −→ 0 0 α1 0 0 d2 and d1 −→ d2 of α beginning at d1 and d1. Let f2 be the unique induced morphism in the diagram below α! d1 d2

f1 f2

0 0 α! 0 d1 d2

α c1 c2

α c1 c2 We make the assignment f1 7→ f2. 0 • Now given morphisms c −→α c0 −→α c”, using the fact that p-cocartesian 0 0 morphisms compose, we obtain functors (α ◦ α)! and α! ◦ α!, which are re- lated using the universal property of cocartesian lifts by a canonical natural isomorphism. Theorem 2.8. The Grothendieck Construction furnishes an equivalence of 2-categories R : ps[C, Cat] → opfib/C, with inverse equivalence given by the construction above. As can be seen, choices of lifts are defined only up to unique isomorphism, so this construction really does give a pseudofunctor in general rather than a strict 2-functor. The Grothendieck construction allows one to recast structures with a lowered categorical dimension. Rather than worry about a functor whose target is Cat, having to deal with a 2-categorical world, we can move to the level of 1-categories. On the other hand, the passage from functor to pseudofunctor can be useful as well as it allows one to hide a lot of information in the structural transformations of the pseduofunctor. 2.3. Monoidal Categories via opFibrations. For a “concrete” example of the previous section’s developments, we will use the Grothendieck Construction on the bar construction of a monoidal category to recast it as an opfibration. To a monoidal category M: ∆op → Cat, this construction associates as part of its data a new category, M⊗. By the Segal condition, the objects of this category are finite tuples of objects in M (really M[1]), and a morphism from (M1,...,Mn) to k (N1,...,Nk) is a collection (α:[k] → [n], {fi}i=1), where each fi is a morphism fi : Mα(i−1)+1 ⊗ ... ⊗ Mα(i) → Ni (the domain of such a map is understood to be the monoidal unit when α(i − 1) + 1 > α(i). The associated opfibration is the forgetful functor from M⊗ → ∆op, sending an k n-tuple to the ordinal [n] and a morphism (α:[k] → [n], {fi}i=1) to α. Given a morphism α:[k] → [n] and an n-tuple (M1,...,Mn), a cocartesian lift of α is given by setting Li to be any object in the isomorphism class of Mα(i−1)+1 ⊗ ... ⊗ Mα(i), and each map fi to be an isomorphism. In particular, the morphisms ιi :[n] → [1] give rise to projections (M1,...,Mn) 7→ Mi. MOTIVATING THE DEFINITION OF MONOIDAL ∞-CATEGORY 11

⊗ By construction, the fibers M[n] of the associated opfibration depend pseudo- ⊗ ⊗ ×n functorially on n and satisfy the Segal condition: the map M[n] → (M[1]) is an equivalence of categories. In fact, as one might expect by now, due to the equivalence furnished by the Grothendieck Construction, we obtain the following result Theorem 2.9. An opfibration p: C → ∆op satisfying the Segal condition gives rise to a monoidal structure on C[1]. This structure is unique up to equivalence. In order to completely redefine monoidal categories using opfibrations we will need a notion of (lax) . Let F : M → N be a lax monoidal functor, that is F is equipped with a family of natural transformations αX1,...,Xn : F (X1) ⊗ ... ⊗ F (Xn) → F (X1 ⊗ ... ⊗ Xn) and 1N → F (1M) which are compatible with all associators, unitors, and other αXm,...,Xm+k . We will associate to F a functor p q Fˆ : M⊗ → N ⊗ between the associated opfibrations M⊗ −→ ∆op and N ⊗ −→ ∆op over ∆op, and seek to capture the required compatibility through unique maps associated to cocartesian lifts. The requirement that Fˆ be a functor over ∆op implies in particular that Fˆ send an n-tuple of objects of M to an n-tuple of objects of N. This together with the Segal condition imply that Fˆ is of the form (X1,...,Xn) 7→ (F (X1),...,F (Xn)). Now notice that the q-cocartesian lifts in diagrams of the form (0,n) (F (X1),...,F (Xn)) F (X1)⊗,..., ⊗F (Xn)

F (0,n)

F (X1⊗,..., ⊗Xn) (where the map (0, n): [1] → [n] picks out the elements 0, n) assemble into candidate transformations for our lax monoidal functor. Howevere these natural transforma- tions need not be compatible, so we must put additional structure on Fˆ. By way of example, let us suppose that M and N are strict monoidal, so that the desired compatibility requirment for three objects is that the diagram below commute FX ⊗ FY ⊗ FZ

F (X ⊗ Y ) ⊗ Z FX ⊗ F (Y ⊗ Z)

F (X ⊗ Y ⊗ Z) One way to ensure such commutation is to make it such that all three arrow paths FX ⊗ FY ⊗ FZ → F (X ⊗ Y ⊗ Z) are induced by q-cocartesian morphisms, so that they must be the same up to isomorphism. Compatibility on the right hand side of the above diagram is obtained from the diagram π (X,Y ,Z) 2,3 (Y ,Z)

π2,3

(Y ,Z) 1 Y ⊗ Z d! by applying F and extending upon a q-cocartesian lift of d1 12 DANIEL FUENTES-KEUTHAN

π d1 (F X,F Y ,F Z) 2,3 (F Y ,F Z) ! FY ⊗ FZ

∃? π2,3

(F Y ,F Z) 1 F (Y ⊗ Z) d! However, in order to ensure the existence of a unique morphism extending from FY ⊗ FZ → F (Y ⊗ Z) in the diagram above the top composite arrow must be q-cocartesian. This is the case of the functor F preserves the q-cocartesian arrow π2,3. In general we will need to require this for lifts of all convex map: maps in ∆op which are injective and whose image is of the form {[m], [m + 1],..., [m + k]} for some m, k. Lifts of convex maps can be seen to correspond to projections (X1,...,Xn) → (Xm,...,Xm+k), hence requiring Fˆ to take these to q-cocartesian arrows will make it such that Fˆ corresponds to a lax monoidal functor as it will have the neccesary compatibility requirements. The case of monoidal functors is simpler: Fˆ is monoidal if it takes all lifts of p-cocartesian arrows to q-cocartesian arrows. This ensures both arrows in the diagram

(0,n)! (F (X1), . . . ,F (Xn)) F (X1) ⊗ ... ⊗ F (Xn)

F ((0,n)!) F (X1 ⊗ ... ⊗ Xn) are q-cocartesian lifts of the same morphism, so that we have induced compatible ∼ natural isomorphisms F (X1) ⊗ ... ⊗ F (Xn) = F (X1 ⊗ ... ⊗ Xn). We summarize this discussion on monoidal functors

p Lemma 2.10. A functor Fˆ : M⊗ → N ⊗ over ∆op between opfibrations M⊗ −→ q ∆op and N ⊗ −→ ∆op corresponds to a lax monoidal functor if and only if it takes p-cocartesian lifts of convex arrows in ∆op to q-cocartesian arrows. Fˆ is monoidal if it takes all p-cocartesian lifts of arrows in ∆op to q-cocartesian arrows

We close this section with a quick exposition on symmetric monoidal cate- gories. Recall we could associate to a commutative topological monoid a functor F in? → T op which captured the symmetric monoidal structure essentially uniquely. In exactly the same way we associate to a symmetric monoidal category M a functor F in∗ → Cat. Applying the Grothendieck Construction to this functor we obtain ⊗ ⊗ an opfibration M → F in∗. The category M has as objects n-tuples of ob- jects of M, and as morphisms between (M1,...,Mn) → (N1,...,Nk) collections k (α: hni → hki, {fi}i=1). The morphisms fi have domain and target determined by N α: fi : j∈α−1(i) Mj → Ni. We have maps pi : hni → h1i which give rise to projections (M1,...,Mn) 7→ ⊗ ' ⊗ ×n Mi, and in turn to Segal equivalences Mhni −→ (Mh1i) . These equivalences determine a symmetric monoidal structure on the fiber over h1i uniquely up to equivalence. Let m: h2i → h1i be the map that sends 1, 2 7→ 1, and t: h2i → h2i be the twist map. The equality m = m ◦ t gives a natural isomorphism on the ⊗ ⊗ induced functors Mh2i → Mh1i, providing the symmetric structure. The underlying monoidal structure is obtained by pulling back along the pointed simplicial circle MOTIVATING THE DEFINITION OF MONOIDAL ∞-CATEGORY 13

⊗ ⊗ Munderlying M p

op φ ∆ F in?

Theorem 2.11. An opfibration M: C → F in? satisfying the Segal condition gives rise to a unique up to equivalence monoidal structure on the fiber Ch1i over h1i.

3. Monoidal ∞-categories Finally we are ready to state a definition of monoidal ∞-categories which ex- tends our unbiased, functor-based definitions of topological monoids and monoidal categories, but hides much of the messy combinatorics of associahedra governing higher coherence conditions. At the level of ordinary category theory, the pentagon axiom was sufficient to supply our monoidal structure with A∞ coherent associ- atvity. At the much less strict level of , we must specify the weak commutative of all associahedra axiomatically. It was for this reason that we chose to explore functors associated to monoidal structures in the first place. Here we will take this approach from the onset, and expect at this point that the reader knows exactly how the next section will proceed, indeed that is the point. As a national convention we freely conflate a category with its nerve.

3.1. Monoidal ∞-Categories via coCartesian Fibrations. We begin with the ∞-categorical analogs of cocartesian morphisms and opfibrations. Definition 3.1. Given a functor p: C → D of ∞-categories, a morphism f : c → c0 is said to be a p-cocartesian lift of p(f), or a p-cocartesian morphism, if the induced map of ∞-categories

f/C  p(f)/D ×p(c)/D c/C is a trivial Kan fibration. Note that this is a much stronger universal property than in ordinary category theory, where there the universal property was really on the level of sets. The universal property of this definition is truly infinite dimensional. As in the case of opfibrations we are interested in the case where a sufficient number of p-cocartesian morphisms are present.

Definition 3.2. A functor p: C  D of ∞-categories is a cocartesian fibration if p is an inner fibration, and every morphisms in D has a p-cocartesian lift. The two additional model theoretic assumptions deserve a brief explanation. Asking for the functor in 3.1 to be a trivial Kan fibration has the effect of weakening the universal property of cocartesian morphism to no longer induce unique up to isomorphism maps, but to instead induce a space of maps which is contractible. This is the appropriate homotopy theoretic weakening of the notion of uniqueness up to unique isomorphism: there is at least one solution to the problem, and the choice of solution is homotopically irrelevant. Asking for cocartesian fibrations to be 0 inner fibrations guarantees that the fibers of p, defined as the pullback Cd = ∆ ×d C, are themselves ∞-categories. The ∞-categorical analog of the Grothendieck construction, the straightening and unstraightening constructions of Lurie [Lur09a, Sec. 3], which establishes an 14 DANIEL FUENTES-KEUTHAN equivalence between cocartesian fibrations and functors valued in Cat∞, establishes that the fibers of a cocartesian fibration vary essentially covariantly. Definition 3.3. A monoidal ∞-category 6 is a cocartesian fibration M⊗ → ∆op ⊗ ⊗ ×n which satisfies the Segal condition: for all n, the functor M[n] → (M[1]) is a categorical equivalence. ⊗ The Segal condition in particular implies that M[0] is a contractible Kan com- 1 ⊗ ⊗ plex, so that the first degeneracy map s : M[0] → M[1] essentially picks out an object that will serve as our unit. Choosing an inverse to the equivalence ⊗ ⊗ ⊗ M[2] → M[1] × M[1] gives a functor 1 ⊗ ⊗ ' ⊗ d ⊗ ⊗: M[1] × M[1] −→M[2] −→M [1] The definition of monoidal ∞-categories implies that the functor ⊗ is a associ- cative and unital (via s1) up to coherent homotopy, indeed a weaker statement is captured in the following result Theorem 3.4. Let p: C → ∆op be a monoidal ∞-categories, then the functor Ho(p): Ho(C) → ∆op endows Ho(C) with the structure of a monoidal category. In other words, a monoidal ∞-categories is a monoidal category “up to ho- motopy”, but in a highly coherent way where all of the relating homotopies are explicitly captured by the defining cocartesian fibration. Likewise, a symmetric monoidal ∞-categories is a cocartesian fibration p: M⊗ → ⊗ ⊗ ×n F in? satisfying the Segal condition: for all n, the functor Mhni → (Mh1i) is a ⊗ categorical equivalence. The structure on Mh1i is defined similarly to the non ∞- categorical case, and the Segal condition implies symmetry of the monoidal product up to E∞ operations. Taking motivation from 2.10, we make the following definition of lax monoidal and monoidal functors of ∞-categories. In this setting all compatibility require- ments hold up to coherent higher homotopy. Definition 3.5. A functor Fˆ : M⊗ → N ⊗ over ∆op between monoidal ∞-categories p q M⊗ −→ ∆op and N ⊗ −→ ∆op is a lax monoidal functor if a it takes p-cocartesian lifts of nerves of convex morphisms in ∆op to q-cocartesian arrows. Fˆ is monoidal if it takes all p-cocartesian lifts of morphisms in ∆op to q-cocartesian arrows 3.2. Towards Higher Algebra. In this section we prove Theorem 1.3 in a round- about way which will bring us to a definition of monoid object in a monoidal ∞-categories. As mentioned previously, the fact that we could represent monoid objects in Top as certain simplicial objects satisfying the strict Segal condition was in no way specific to Top, and indeed we took this same approach to define strict monoidal categories. We take the following standard result as our base Proposition 3.6. Let V be a monoidal category. There is an equivalence of cate- gories, Mon(V) ' Cat(?, V), between monoid objects in V and lax monoidal func- tors out of the terminal category. From the face that the terminal category is represented as a monoidal category by the opfibration ∆op = ∆op, we obtain the following upon recasting to our fibration definitions of monoidal category and lax monoidal functor

6Really a monoidal structure on an ∞-category. MOTIVATING THE DEFINITION OF MONOIDAL ∞-CATEGORY 15

Proposition 3.7. Let V : V⊗ → ∆op be a monoidal category. There is an equiva- lence of categories Mon(V) ' Cat/∆op(∆op, V)lax. Proof. For clarity we provide a few details towards a direct proof. In one direction, a lax monoidal functor ∆op → V⊗ is specified by a monoid A by sending [n] 7→ (A, . . . , A). Going the other way, note that the object A[n] is determined under 1 n the Segal equivalence by an n-tuple of objects, (A[n],...,A[n]). In particular the 1 2 1 object A[2] ' (A[2],A2]). A cocartesian lift of d : [2] → [1] is given by a morphism 1 2 1 2 1 (A[2],A[2]) → A[2] ⊗ A2]. Alternatively, evaluating A on d yields a morphism 1 2 1 1 2 µA 1 (A[2],A[2]) → A[1]. The cocartesian condition gives a morphism A[2] ⊗A2] −−→ A[1]. Send A is lax monoidal it sends the opposites of the convex morphisms ι1, ι2 : [1] → [2] to cocartesian morphisms. Hence the diagram below, along with its counterpart with ι2

1 2 ι1 1 (A[2],A2]) A[2] ∼= A(ι1) 1 A[1]

[2] ι1 [1]

ι1 [1]

1 ∼ 1 ∼ 2 imply that A[2] = A[1] = A[2]. Hence µA is really an arrow of the form µA : A[1] ⊗ 1 A[1] → A[1]. The unit is given by ηA = A(s ), and the simplicial identities together with the fact that A is lax monoidal imply that A[1] is a monoid object in V.  Now suppose that V has the cartesian monoidal structure. Due to the strict Segal isomorphisms defining this strict monoidal structure, any functor ∆op → V⊗ which appears in the fibration definition of a lax monoidal functor induces a functor ∆op → V which is strict monoidal, with structure maps exactly those as in the strict Segal condition. Conversely, any functor A: ∆op → V satisfying the Segal condition, hence representing a monoid object A, gives rise to a lax monoidal functor Aˆ∆op ⇒ V⊗ by sending [n] 7→ (A, . . . A). Hence we obtain Theorem 3.8. Let V be a cartesian monoidal category. There is an equivalence of categories Mon(V) ' Cat(∆op, V)sSegal. Which lets us settle our unproven claims from §1 and §2 Corollary 3.9 (proof of 1.3). Strict monoid objects in Top, and Cat are equivalent to their bar constructions. Note that as a side effect of 3.8 we have successfully recast monoid objects in classical category theory into our world of fibrations: they are certain sections to the opfibrations defining a monoidal category. In the world of ∞-categories we take this as a definition

Definition 3.10. An (A∞ or associative) algbera object in a monoidal ∞-categories M: M⊗ → N(∆op) is a lax monoidal functor N(∆op) → M. 16 DANIEL FUENTES-KEUTHAN

Associative algebras in a monoidal ∞-categories assemble into an ∞-categories ⊗ A∞(M ). Furthermore, an analog of 3.8 holds. There is a notion of carte- sian monoidal structure, which once established allows us to form a monoidal ∞- categories M × → N(∆op) on an ∞-categories with finite products. Theorem 3.11. There is an equivalence of ∞-categories × op Segal A∞(M ) ' Cat∞(N(∆ ), M) × Applying this to the cartesian monoidal category Cat∞ gives the following special case: by the ∞-categorical Grothendieck construction there is an equivalence of op Segal op Segal Mon categories Cat∞(N(∆ ), Cat∞) ' cocart/N(∆ ) = Cat∞ . Hence we Mon × obtain an equivalence Cat∞ ' A∞(Cat∞) of objects in the cartesian monoidal structure on Cat∞ and monoidal ∞-categories. As a final remark, we note that all appropriate analogs of results in this section hold for symmetric monoidal categories and E∞ algebra objects. Reaching further, a full theory of modules over algebras in monoidal ∞-categories exists, as in [Lur11] and [Lur09b], complete with analogs of many classical results.

References [Gro15] Moritz Groth, A Short Course on ∞-categories, https://arxiv.org/abs/1007.2925, 2015. [Lur09a] Jacob Lurie, Higher Theory, volume 170 of Annals of Mathematics Studies, Prince- ton University Press, Princeton NJ, 2009. [Lur09b] Jacob Lurie, Derived Algebraic Geometry II: Noncommutative Algebra, http://www- math.mit.edu/ lurie/papers/DAG-II.pdf, 2009. [Lur11] Jacob Lurie, Higher Algebra. http://www.math.harvard.edu/ lurie/papers/HA.pdf, 2011. [Mac98] , Categories for the Working Mathematician, 2nd edition, Springer GTM, 1998. [Maz15] Aaron Mazel-Gee, A user’s Guide to co/cartesian Fibrations, https://arxiv.org/abs/1510.02402, 2015. [Sta70] James Stasheff, H-Spaces from a Homotopy Point of View, volume 161 of Lecture Notes in Mathematics, Springer, 1970.