Homotopy-coherent algebra via Segal conditions

Many homotopy-coherent algebraic structures can be described by Segal-type limit conditions determined by an"algebraic pattern", by which we mean an $\infty$-category equipped with a factorization system and a collection of"elementary"objects; examples include $\infty$-categories, $(\infty,n)$-categories, $\infty$-operads, and algebras for an $\infty$-operad in spaces. We characterize the monads on presheaf $\infty$-categories that arise in this way as precisely the cartesian monads that are local right adjoints, and exhibit an $\infty$-category of these monads as a localization of an $\infty$-category of algebraic patterns.


Introduction
Homotopy-coherent algebraic structures, where identities between operations are replaced by an infinite hierarchy of compatible coherence equivalences, have played an important role in algebraic topology since they were first introduced in the 1970s in work of Boardman-Vogt [6], May [24], and others, and have since found many applications in other fields. From a modern perspective, they can be considered as the natural algebraic structures in the setting of ∞-categories (which are themselves the homotopy-coherent analogues of categories).
It turns out that many interesting homotopy-coherent algebraic structures can be described by "Segal conditions", i.e. they can be described as functors satisfying a specific type of limit condition. The canonical (and original) example is Segal's [28] description of commutative (or E ∞ -)monoids in spaces as "special Γ-spaces". In ∞-categorical language, these are functors F : F * → S where F * is a skeleton of the category of pointed finite sets, with objects n := ({0, 1, . . . , n}, 0), and S is the ∞-category of spaces (or ∞-groupoids), satisfying the following condition: For all n, the map induced by the morphisms ρ i : n → 1 given by is an equivalence.
Other key examples of structures described by Segal conditions include: • associative (or A ∞ -or E 1 -)monoids, using the simplex category ∆ op (in unpublished work of Segal), • ∞-categories, again using ∆ op , in the form of Rezk's Segal spaces [26], • (∞, n)-categories, using Joyal's categories Θ op n , also in work of Rezk [27], • ∞-operads, using the dendroidal category Ω op of Moerdijk-Weiss [25], in work of Cisinski and Moerdijk [8], • algebras for an ∞-operad O (in the sense of [23]) in S, using the "category of operators" O itself. Given these and other examples (many of which we will discuss below in §3), we might wonder why so many different algebraic structures can be described by Segal conditions. Our main results in this paper provide an explanation of this situation, by answering the following question:

Question 1.1. Which homotopy-coherent algebraic structures can be described (in a reasonable way) by Segal conditions, and how canonical is this description?
Before we describe our answer, we need to formulate a more precise version of this question, by defining the terms that appear. First of all, we will consider algebraic structures on (families of) spaces, which we take to mean algebras for monads on functor ∞-categories Fun(I, S) (where I is any small ∞-category). Next, let us specify what precisely we mean by "Segal conditions". Returning to the example of special Γ-spaces, the category F * has the following features that we wish to abstract: • A morphism φ : n → m is called inert if |φ −1 (j)| = 1 for j = 0, and active if φ −1 (0) = {0}. The inert and active morphisms form a factorization system on F * : every morphism factors as an inert morphism followed by an active morphism, and this decomposition is unique up to isomorphism. • The morphisms ρ i are precisely the inert morphisms n → 1 .
• If F int * denotes the subcategory of F * with only inert morphisms, then the special Γ-spaces are precisely the functors F : F * → S such that the restriction F | F int * is a right Kan extension of F | { 1 } . These features recur in our other examples, which suggests that the input data for a class of "Segal conditions" should consist of an ∞-category O equipped with a factorization system (whereby every morphism factors as an "active" morphism followed by an "inert" morphism) and a class of "elementary" objects (or generators). From this data, which we will refer to as an algebraic pattern 1 , we obtain the relevant Segal-type limit condition on a functor F : O → S by imposing the requirement that for every O ∈ O the object F (O) is the limit over all inert morphisms to elementary objects, we say that such a functor F is a Segal O-space in C. 2  algebraic patterns in our sense, and should not be confused with the notion of "pattern" considered by Getzler [11]. general, however, it is not possible to describe this monad explicitly (as the left adjoint involves an abstract localization), and we only want to consider a pattern to be "reasonable" if T O is given by a concrete formula, as where Act O (E) is the space of active morphisms to E in O. We call such patterns O extendable, and give explicit necessary and sufficient conditions for a pattern to be extendable in Proposition 8.8.
We can now state the precise version of the previous question that we will address: Question 1.2. Which monads on presheaf ∞-categories can be described as the free Segal O-space monad for an extendable algebraic pattern O, and how canonical is this description?
Our first main result precisely characterizes these monads as the polynomial monads, by which we mean the monads that are cartesian 3 and whose underlying endofunctors are local right adjoints: We prove part (i) in §10 and part (ii) in §11. Note that part (ii) amounts to an ∞-categorical version of Weber's nerve theorem [30].
Our second main result addresses the uniqueness of the canonical pattern: The canonical pattern W(T ) is thus the unique saturated pattern describing the polynomial monad T . We will prove this in §12.
We will also see that the constructions of both theorems are compatible with appropriate notions of morphisms of polynomial monads and of patterns. It follows that the ∞-category of polynomial monads is given by a localization of an ∞-category of slim extendable patterns, and is equivalent to the full subcategory of saturated patterns.
1.1. Overview. In the first part of the paper we set up a general categorical framework for algebraic patterns and Segal objects. In §2 we introduce these objects more formally and prove some of their basic properties, before we look at examples of algebraic patterns and their Segal objects in §3. We then introduce morphisms of algebraic patterns in §4 and construct an ∞-category of algebraic patterns in §5, where we also prove that this has limits and filtered colimits. Next, we provide conditions under which Segal objects are preserved by right and left Kan extensions in §6 and §7, respectively.
In §8 we apply our work on left Kan extensions to analyze free Segal objects; in particular, we obtain necessary and sufficient conditions for a pattern O to be extendable, meaning that free Segal O-spaces are described by a colimit formula. In §9 we study (weak) Segal fibrations, which generalize Lurie's definitions of symmetric monoidal ∞-categories and symmetric ∞-operads. We show that any weak Segal fibration over an extendable base is again extendable, and moreover left Kan extension along any morphism of weak Segal fibrations preserves Segal objects; this recovers, 3 I.e., the multiplication and unit transformations are cartesian natural transformations, meaning that their naturality squares are all cartesian.
4 This is a mild technical condition, satisfied in all examples. Moreover, any extendable pattern can be replaced by a full subcategory that is slim and determines the same monad.
for example, the formula of [23] for operadic left Kan extensions of ∞-operad algebras in cartesian monoidal ∞-categories.
In §10 we introduce polynomial monads, and prove that the free Segal O-space monad for any extendable pattern is polynomial. Then we show in §11 that, conversely, any polynomial monad is described by a canonical pattern. Finally, in §12 we study the relationship between an extendable pattern and the canonical pattern of its free Segal space monad; under a mild hypothesis there is a functor between these, and we give a simple criterion for this to be an equivalence.

Related Work.
There is an extensive literature on using (finite) limit conditions to describe algebraic structures in category theory, going back at least to Lawvere's thesis [21], where he introduced algebraic theories. Our work is in particular closely related to the "nerve theorem", one version of which says that a strongly cartesian monad on a presheaf category is described by Segal conditions; this version was first proved in unpublished work of Leinster (though his proof did not use the factorization system), and later extended by Weber [30] to a weaker description of certain weakly cartesian monads. 5 We were particularly inspired by the simpler proof given by Berger, Melliès, and Weber [5].

Algebraic Patterns and Segal Objects
In this section we introduce the basic structures we will study in this paper, namely algebraic patterns and their Segal objects.
, the morphisms in which we refer to as the inert and active morphisms in O, • a full subcategory O el ⊆ O int whose objects we call the elementary objects of O. Unless stated otherwise, we will assume by default that algebraic patterns are essentially small. Remark 2.2. Here a factorization system on an ∞-category C means a pair of subcategories (C L , C R ) such that both contain all objects of C, and for every morphism f : Remark 2.3. We will often abuse notation and conflate an algebraic pattern with its underlying ∞-category O, i.e. we will simply say that O is an algebraic pattern.
is an equivalence. We write Seg O (C) for the full subcategory of Fun(O, C) spanned by the Segal O-objects.
Notation 2.7. We will often refer to Segal O-objects in the ∞-category S of spaces as Segal Ospaces, and to Segal O-objects in the ∞-category Cat ∞ of ∞-categories as Segal O-∞-categories.
is an equivalence.
Proof. The object F is local with respect to C ⊗ y(O) Seg → C ⊗ y(O) precisely when the morphism of spaces is an equivalence. Here we have equivalences using the Yoneda Lemma, and similarly F (E) is an equivalence it suffices to consider C in C κ , which proves (ii).
It follows that if C is presentable, then Seg O (C) is the full subcategory of objects in Fun(O, C) that are local with respect to a set of morphisms. Parts (iii) and (iv) then follow from [22,Proposition 5.5.4.15].

Examples of Algebraic Patterns
In this section we will briefly describe some examples of algebraic patterns and their associated Segal objects.
Example 3.1. We write F ♭ * for the algebraic pattern structure on F * given by the inert-active factorization system we discussed above in the introduction, with F el * containing the single object 1 . Then a Segal F ♭ * -space is precisely a commutative monoid, or equivalently a special Γ-space in the sense of [28].
Example 3.2. We can also consider another pattern structure on F * : We define F ♮ * by the same factorization system, but now F el * contains the two objects 0 and 1 , with the unique inert morphism where the right-hand side denotes an iterated fibre product over F ( 0 ); this is equivalently a commutative monoid in the slice C /F ( 0 . Example 3.3. We write ∆ for the simplex category, i.e. the category of non-empty finite ordered sets [n] := {0, . . . , n} and order-preserving maps between them. A morphism f : and active if f preserves the endpoints, i.e. f (0) = 0 and f (n) = m. Every morphism in ∆ factors uniquely as an active morphism followed by an inert one, so this determines an inert-active factorization system on ∆ op . Using this factorization system we can define two interesting algebraic pattern structures on ∆ op : In particular, a Segal ∆ op,♮ -space is precisely a Segal space in the sense of Rezk [26], which describes the algebraic structure of an ∞-category. On the other hand, a Segal ∆ op,♭ -object F satisfies and describes an associative monoid.
Example 3.4. For any integer n the product ∆ n,op := (∆ op ) ×n has a coordinate-wise factorization system (i.e. a morphism is active or inert precisely when all of its components are). Using this we can define two algebraic pattern structures ∆ n,op,♮ and ∆ n,op,♭ , where These are both special cases of products of algebraic patterns (Corollary 5.5). Segal ∆ n,op,♮ -spaces are n-uple Segal spaces, which describe internal ∞-categories in . . . in ∞-categories. A special class of these was first introduced by Barwick as a model for (∞, n)-categories. On the other hand, the Dunn-Lurie additivity theorem [23, Theorem 5.1.2.2] implies that Segal ∆ n,op,♭ -objects are equivalent to E n -algebras, i.e. algebras for the little n-disc operad.
Example 3.5. Let Θ n be defined inductively by Θ 0 := * and Θ n := ∆ ≀ Θ n−1 , where for any category C the category ∆ ≀ C has objects [n](C 1 , . . . , C n ) with C i ∈ C, and morphisms [n](C 1 , . . . , C n ) → [m](C ′ 1 , . . . , C ′ m ) given by morphisms φ : [n] → [m] in ∆ together with maps ψ ij : C i → C j in C whenever φ(i − 1) < j ≤ φ(i). (This category was first considered in unpublished work of Joyal; the "wreath product" definition is due to Berger [4].) Then Θ n has an inductively defined factorization system (first defined in [3,Lemma 1.11]): the morphism above is inert (or active) if φ is inert (active) and each ψ ij is inert (active). We can again use this to define two algebraic patterns. To do so we need some notation: We inductively define objects C 0 , . . . , C n in Θ n by C 0 := [0]() and C n := [1](C n−1 ), starting with C 0 being the unique object of Θ 0 . Then • Θ op,♮ n is defined by taking Θ op,♮,el n to contain the objects C 0 , . . . , C n , • Θ op,♭ n is defined by taking Θ op,♭,el n to contain the single object C n .
Segal Θ op,♮ n -spaces are then precisely Rezk's Θ n -spaces [27], which describe the algebraic structure of (∞, n)-categories. On the other hand, Segal Θ op,♭ n -objects are again equivalent to E n -algebrasthis follows from [1,Theorem 4.12] together with the Dunn-Lurie additivity theorem. Example 3.6. All the examples considered so far are special cases of the following construction, due to Barwick: Suppose Φ is a perfect operator category in the sense of [1], and let Λ(Φ) be its Leinster category, which is the Kleisli category of a certain monad on Φ. This has an active-inert factorization system by [1,Lemma 8.3], where the active morphisms are the free morphisms on morphisms of Φ. Using this factorization system we can define two natural algebraic patterns: • Λ(Φ) ♭ is defined by taking Λ(Φ) ♭,el to consist only of the terminal object * ∈ Φ, • Λ(Φ) ♮ is defined by taking Λ(Φ) ♮,el to contain all objects E such that there is an inert map * → E in Λ(Φ). If O denotes the category of (possible empty) ordered finite sets then Λ(O) ≃ ∆ op , while if F denotes the category of finite sets then Λ(F) ≃ F * , and these pattern structures agree with those defined above. The same holds for Θ op n , which can be described as the Leinster category of a wreath product O ≀n of operator categories. Let Ω be the dendroidal category of Moerdijk and Weiss [25, §3]; this can be defined as the category of free operads on trees. This has a natural active-inert factorization system, described for example in [19] (where the inert maps are called "free" and the active ones "boundarypreserving"). Using this we can define an algebraic pattern Ω op,♮ where Ω op,♮,el consists of the corollas C n (i.e. trees with one vertex) and the plain edge η. Segal Ω op,♮ -spaces are the dendroidal Segal spaces introduced by Cisinski and Moerdijk [8], which describe the algebraic structure of ∞operads. The Segal condition says that the value of a Segal object at a tree decomposes as a limit over the corollas and edges of the tree. (We can also consider a pattern Ω op,♭ where the elementary objects are just the corollas; then Segal Ω op,♭ -spaces describe ∞-operads with a single object.)   14], the inertactive factorization system on Gr restricts to Υ, so we can give Υ op a pattern structure Υ op,♮ with the same elementary objects as before. According to [20, 2.4.14], the category Υ is equivalent to the category of graphs defined and studied by Hackney, Robertson and Yau in [14]. Segal Υ op,♮ -spaces are therefore equivalent to the model of ∞-properads as "graphical spaces" satisfying a Segal condition that is briefly discussed in [13]; this is presumably equivalent (after imposing a completeness condition) to the model of ∞-properads as certain presheaves of sets on Υ constructed in [14]. Moreover, by work in progress of the first author and Hackney, Segal   Remark 3.13. Below in §9 we will define (weak) Segal fibrations over an algebraic pattern, which give general classes of examples of algebraic patterns. As a special case, we will see that every ∞operad O in the sense of Lurie [23] has an algebraic pattern structure O ♭ such that a Segal O ♭ -object in an ∞-category C is precisely an O-monoid in C.

Morphisms of Algebraic Patterns
In this section we define morphisms of algebraic patterns, and consider when they are compatible with Segal objects. We then discuss some examples of such morphisms.
Definition 4.1. Let O and P be algebraic patterns. A morphism of algebraic patterns from O to P is a functor f : O → P such that f preserves both active and inert maps, and takes elementary object in O to elementary objects in P.
In general, morphisms of algebraic patterns do not necessarily interact well with Segal objects. We therefore isolate the class of morphisms that preserve Segal objects under restriction: Definition 4.2. A morphism of algebraic patterns f : O → P is called a Segal morphism if it satisfies the following condition: Remark 4.3. The condition depends only on the restriction of F to P el , so we could equivalently have considered functors P el → S that occur as restrictions of Segal P-spaces.
Remark 4.4. In practice, a morphism f is a Segal morphism because the functor O el X/ → P el f (X)/ is coinitial, in which case we say that f is a strong Segal morphism. However, the more general definition allows for the following characterization: Proof. It is immediate from the definition that (1) is equivalent to (2) and that (3) implies (2). It remains to check that (2) implies (3). Suppose F : P → C is a Segal P-object; we need to show that f * F is a Segal O-object, i.e. that for all X ∈ O the natural map is an equivalence in C. Equivalently, we must show that for any C ∈ C, the map of spaces is an equivalence, which is true since Map(C, F ) is a Segal P-space.
Remark 4.6. One might feel that the Segal property is sufficiently fundamental that it should be included as part of the notion of a morphism of algebraic patterns. However, more general morphisms also turn out to be occasionally useful. For example, the identity functor of F * viewed as a functor F ♭ * → F ♮ * is a morphism of patterns, but is not a Segal morphism, and we will see later in §6 that it induces a functor from Segal F ♭ * -objects to Segal F ♮ * -object that can be viewed as a right Kan extension along id F * . Proof. Since f * restricts to a functor on Segal objects, for F ∈ Seg P (C) and G ∈ Seg O (C) we have a natural equivalence which implies the claim.
Remark 4.8. Below in §7 we will give conditions on a morphism f such that the left Kan extension functor f ! preserves Segal objects, and so gives a left adjoint to f * without localizing.
We now consider some examples of morphisms of patterns: There is a functor |-| : ∆ op → F * which takes an object [n] to |[n]| := n and a morphism α : This functor gives a Segal morphism of algebraic patterns ∆ op,♮ → F ♮ * as well as ∆ op,♭ → F ♭ * . Example 4.10. There is a functor τ n : ∆ n,op → Θ op n , defined inductively by setting τ 0 := id and This functor gives a Segal morphism of algebraic patterns ∆ n,op,♮ → Θ op,♮ n as well as ∆ n,op,♭ → Θ op,♭ n .
The previous examples are special cases of the following: Let f : Φ → Ψ be an operator morphism between perfect operator categories, as defined in [1, Definition 1.10]. As discussed in [1, §7] this induces a functor Λ(f ) : Λ(Φ) → Λ(Ψ) between the corresponding Leinster categories, and it is easy to check that this preserves the inert and active morphisms, and hence gives morphisms of algebraic patterns Λ(Φ) ♮ → Λ(Ψ) ♮ and Λ(Φ) ♭ → Λ(Ψ) ♭ . The latter is evidently a Segal morphism, since where the seond isomorphism is part of the definition of an operator morphism.

The ∞-Category of Algebraic Patterns
In this section we construct the ∞-category of algebraic patterns, and describe limits and filtered colimits in this ∞-category. As a first step, we consider the ∞-category of ∞-categories equipped with a factorization system: Definition 5.1. We define Fact to be the full subcategory of Fun(Λ 2 2 , Cat ∞ ) spanned by those cospans that describe factorization systems, i.e. those such that the functors C L , C R → C are essentially surjective subcategory inclusions, and Fun L, Proposition 5.2. The ∞-category Fact is closed under limits and filtered colimits in Fun(Λ 2 2 , Cat ∞ ). In particular, the ∞-category Fact has limits and filtered colimits, and the forgetful functor to Cat ∞ preserves these.
This will follow from the following observation: In the ∞-category Fun(∆ 1 , Cat ∞ ), the full subcategories of subcategory inclusions, essentially surjective subcategory inclusions, and full subcategory inclusions, are all closed under limits and filtered colimits.
is a monomorphism of spaces for all x, y ∈ C. A subcategory inclusion F is essentially surjective if the map C ≃ → D ≃ is an equivalence, and a full subcategory inclusion if the maps Map C (x, y) → Map D (F x, F y) are equivalences for all x, y ∈ C. Since mapping spaces and the underlying space of a limit (or filtered colimit) in Cat ∞ are computed as limits (or filtered colimits) of spaces, it suffices to observe that equivalences and monomorphisms are closed under limits and filtered colimits in S. This is obvious for equivalences, and for monomorphisms it follows from the characterization of these by [22,Lemma 5.5.6.15] as the morphisms f : X → Y such that the diagonal X → X × Y X is an equivalence, since filtered colimits commute with finite limits and limits commute.
Proof of Proposition 5.2. It follows from Lemma 5.3 that cospans of subcategory inclusions are closed under limits and filtered colimits in Fun(Λ 2 2 , Cat ∞ ). Since limits commute, the ∞-category Fun L,R (∆ 2 , -), viewed as a functor Fun(Λ 2 2 , Cat ∞ ) → Cat ∞ , preserves limits, which implies that objects such that the natural map Fun L,R (∆ 2 , -) → Fun(∆ {0,2} , -) is an equivalence are also closed under limits. The same holds for filtered colimits, since the objects mapped out of in the definition of Fun L,R (∆ 2 , -) are compact, and filtered colimits commute with finite limits in Cat ∞ .
Definition 5.4. We now define the ∞-category AlgPatt of algebraic patterns as the full subcategory of the fibre product Fact × Cat ∞ Fun(∆ 1 , Cat ∞ ) (where the pullback is over ev 0 : Fact → Cat ∞ and ev 1 : Applying Lemma 5.3 again, now in the case of full subcategory inclusions, we get:

. The full subcategory AlgPatt is closed under limits and filtered colimits in
In particular, AlgPatt has limits and filtered colimits, and the forgetful functor to Cat ∞ preserves these.
Example 5.6. For any pair of algebraic patterns O, P we have a cartesian product pattern O × P. For this we have an equivalence for any O × P-complete ∞-category C. To see this, observe that a right Kan extension along O el × P el → O × P can be computed in two stages in two ways, by first doing the right Kan extension Example 5.7. The pattern ∆ op,♭ can be described as the pullback ∆ op,♮ × F ♮ * F ♭ * using the map ∆ op,♮ → F ♮ * from Example 4.9 and the identity of F * viewed as a morphism of patterns F ♭ * → F ♮ * .
is the algebraic pattern defined in Example 3.5. The underlying category Θ is equivalent to that introduced by Joyal [18] to give a definition of weak higher categories. It is easy to see that in this case we have an equivalence so that Segal Θ ♮ -spaces model (∞, ∞)-categories. In particular, the canonical functor Seg Θ ♮ (S) → Seg Θ n,♮ (S) gives the underlying (∞, n)-category of an (∞, ∞)-category.

Right Kan Extensions and Segal Objects
Our goal in this section is to give a sufficient criterion on a morphism of algebraic patterns f : O → P such that right Kan extension along f preserves Segal objects. Definition 6.1. We say that a morphism f : O → P of algebraic patterns has unique lifting of active morphisms if for every active morphism φ : where ι is inert. The morphism α has an essentially unique inert-active factorization, and since f is compatible with this factorization we see that the full subcategory of objects where α is active is cofinal. By uniqueness of factorizations a morphism in this subcategory is required to be an equivalence, hence this is an ∞-groupoid, and so ( * ) is equivalent to this ∞-groupoid being contractible. But an object in this subcategory gives an inert-active factorization of φ, and we see that it is equivalent to the ∞-groupoid of lifts of the active part of φ to an active morphism in O.
Remark 6.4. We emphasize that the condition of unique lifting of active morphisms is far from a necessary one. Indeed, the functor f * will preserve Segal objects if and only if its left adjoint f * preserves Segal equivalences. In [7] the latter condition was checked for a certain morphism τ : ∆ 1,op F → Ω op , which clearly does not have unique lifting of active morphisms.
Proof of Proposition 6.3. By Lemma 6.2, the condition that f has unique lifting of active morphisms implies that for any functor F : O → C, the Beck-Chevalley transformation takes an object C ∈ C to the simplicial object i * C given by (i * C) n ≃ n i=0 C, with face maps corresponding to projections and degeneracies given by diagonal maps. This clearly satisfies the Segal condition. More generally, the inclusion Θ op,♮ n−1 ֒→ Θ op,♮ n has unique lifting of active morphisms for all n ≥ 1.

Left Kan Extensions and Segal Objects
In this section we will give conditions under which left Kan extension along a morphism f preserves Segal objects in C. In contrast to the case of right Kan extensions, this requires strong assumptions on both f and the target ∞-category C. Part of the condition is a uniqueness requirement on lifts of inert morphisms, which we consider first: Proof. This follows by the same argument as for Lemma 6.2, with the roles of active and inert morphisms reversed.
Unique lifting of inert morphisms allows us to functorially transport active morphisms along inert morphisms, in the following sense: is cocartesian if and only if ω is inert.
Proof. We first show that such a morphism with ω inert is cocartesian. This means that given a morphism O → X in O and a commutative diagram where the arrows labelled I are inert and those labelled A are active, there exists a unique lift Since f is compatible with the factorization system, we see that the unique inert-active factorization of Thus, there are unique diagrams which give the required unique factorization (since any other factorization through (O ′ , f (O ′ ) → P ′ ) must induce these by uniqueness of inert-active factorizations).
It remains to observe that cocartesian morphisms exist, i.e. that given with ω inert. This again follows from unique lifting of inert morphisms, which ensures that the inert-active factorization of f (O) → P → P ′ gives such a diagram. Since cocartesian morphisms are unique when they exist, this shows that a morphism as above is cocartesian if and only if ω is inert.
Straightening this cocartesian fibration, we get: Remark 7.5. Suppose f : O → P has unique lifting of inert morphisms. Then for every active where the first functor takes α : , and the second is induced by the forgetful functor X → O int . In particular, we can restrict to P el P/ and compose with the functor for the corresponding cartesian fibration. Using this functoriality we can now state the conditions we require of a morphism of algebraic patterns: Definition 7.6. A morphism of algebraic patterns f : O → P is extendable if the following conditions are satisfied: (1) The morphism f has unique lifting of inert morphisms.
(2) For P ∈ P, let L P denote the limit of the composite functor ǫ P : P el Remark 7.7. We have used the limit in condition (2) as this seems the most natural choice in Definition 7.10; we could also have used the lax limit instead, provided the same change is made in Definition 7.10. In the cases of interest the lax limit actually agrees with the usual limit, as it will either be a finite product or a limit of ∞-groupoids, so the distinction turns out not to matter in practice.
whenever either limit exists in C. If Φ is a Segal O-object, this implies that the following "relative Segal condition" holds: We now turn to the requirements we must make of our target category, for which we need the following notion: Definition 7.10. Consider a functor K : I → Cat ∞ with corresponding cocartesian fibration π : K → I. Let L be the limit of K, which we can identify with the ∞-category of cocartesian sections Fun cocart I (I, K). We then have a functor p : I × L → K adjoint to the forgetful functor Fun cocart I (I, K) → Fun(I, K); the composite π • p is moreover the projection L × I → I. This gives a commutative diagram which for any ∞-category C (with appropriate limits and colimits) determines an equivalence of functors between functor ∞-categories This induces a mate transformation from which we in turn obtain, by moving adjoints around, a natural transformation For a functor F : K → C we can interpret this as a natural morphism We say that I-limits distribute over K-colimits in C if this morphism is an equivalence for any functor F . Definition 7.11. Let f : O → P be an extendable morphism of algebraic patterns. We say that an ∞-category C is f -admissible if C is O-and P-complete, the pointwise left Kan extension f ! : Fun(O, C) → Fun(P, C) exists, and P el P/ -limits distribute over ǫ P -colimits for all P ∈ P, where ǫ P is the functor from Definition 7.6(2). In other words, if C is f -admissible then for every P ∈ P and every functor Φ, the natural map is an equivalence.
Having made these definitions, we can now state our result on left Kan extensions: Proof. Given Φ ∈ Seg O (C), we must show that f ! Φ is a Segal object, i.e. that the natural map is an equivalence. We have a sequence of equivalences which completes the proof.
Having identified conditions under which f ! preserves Segal objects, we now turn to the question of when these conditions hold. For extendability, we will see some general classes of examples below in §9; here, we will discuss two classes of examples where f -admissibility holds. The starting point is the following examples of distributivity of limits over colimits: Definition 7.13. We say an ∞-category C is ×-admissible if it has finite products and the cartesian product preserves colimits in each variable.
Lemma 7.14. Suppose C is ×-admissible. Then finite products distribute over all colimits in C.
Proof. For any functors F i : I i → C (i = 1, . . . , n) whose colimits exist we have Proposition 7.15. Let C be a presentable ∞-category and write t : S → C for the unique colimitpreserving functor taking * to the terminal object * C of C. Consider a functor K : I → S and suppose the following conditions hold: (1) t preserves I-limits.
(2) The functor C /t(S) → lim S C ≃ Fun(S, C) induced by taking pullbacks along * C ≃ t( * ) → t(S), is an equivalence for S = lim I K(i) and S = K i for all i ∈ I. Then I-limits distribute over K-colimits in C.
Proof. Condition (2) implies that we have a commutative diagram of right adjoints Passing to left adjoints, we get the commutative triangle from which we see that under the equivalence of (2) the colimit of a diagram S → C is given by the source of the corresponding morphism to t(S). Given F : S → C, it follows that we have pullback squares for s ∈ S. Now consider a functor F : K → C, where K → I is the left fibration corresponding to K. We have a commutative square where L := lim I K(i). Here the bottom horizontal map can be identified with the natural map This is an equivalence by assumption (1). The equivalence of assumption (2) then implies that the top horizontal map is an equivalence if and only if it induces an equivalence on fibres over each map t(l) : * C → t(L) for l ∈ L. Using the pullback squares above and the fact that limits commute, we see that the map on fibres at (k i ) i ∈ L is the identity This argument applies to C being S, or more generally any ∞-topos, giving: Remark 7.18. The assumption of ×-admissibility can be slightly weakened: It is enough to assume that the cartesian product in C preserves colimits of shape O act /E in each variable for all E ∈ P el . Corollary 7.19. Suppose X is an ∞-topos, and f : O → P is an extendable morphism of algebraic patterns such that the ∞-category P el P/ is finite for all P ∈ P (or arbitrary if X is the ∞-topos S). Then X is f -admissible, and the left Kan extension restricts to a functor

Free Segal Objects
Suppose O is an algebraic pattern, and C an O-complete ∞-category. Restricting Segal objects to the subcategory O el gives a functor where the second functor is an equivalence with inverse the right Kan extension functor i O, * . If C is presentable, using Proposition 4.7 this means the left adjoint F O is given by In this section we will first show that this adjunction is monadic, and then specialize the results of the previous section to j O to get conditions under which the free Segal objects are described by a formula in terms of limits and colimits.
Monadicity is a special case of the following observation: Proof. We first prove (i). One direction amounts to f being a Segal morphism, which is true by assumption. To prove the non-trivial direction, observe that for Φ : Here the second morphism is an equivalence since f is a Segal morphism, and if f * Φ is a Segal Oobject then the composite morphism is an equivalence. Thus the first morphism is an equivalence, and so Φ satisfies the Segal condition at every object of P in the image of f ; since f is essentially surjective this completes the proof.
Using the Barr-Beck theorem for ∞-categories [23, Theorem 4.7.3.5], to prove (ii) it suffices to show that f * detects equivalences, that Seg P (C) has colimits of f * -split simplicial objects, and these colimits are preserved by f * . Since f is essentially surjective it is immediate that f * detects equivalences. Consider an f * -split simplicial object p : ∆ op → Seg P (C). Letp : (∆ op ) ⊲ → Fun(P, C) denote the colimit of p in Fun(P, C). Since f * , viewed as a functor Fun(P, C) → Fun(O, C), is a left adjoint, f * p is the colimit of f * p in Fun(O, C). On the other hand, since f * p extends to a split simplicial diagram, and all functors preserve colimits of split simplicial diagrams, we see that the colimit of f * p in Seg O (C) is also the colimit in Fun(O, C). In particular, f * p (∞) lies in Seg O (C). By (i) this implies thatp(∞) is in Seg P (C). This completes the proof, since the colimit of p in Seg P (C) is the localization ofp(∞), which is already local.
Applying this to j O , we get: Let O be an algebraic pattern and C a presentable ∞-category. Then the freeforgetful adjunction Now we apply the results of the previous section to j O to understand when the free algebras are simply given by the left Kan extension j O,! . It is convenient to first introduce some notation: is an equivalence for any functor Φ : O el → C, provided either limit exists, and This in particular implies the following "generalized Segal condition": If Φ is a Segal object, then for any active morphism φ :

Remark 8.7. In practice, condition (2) holds because the map O el (φ) → O el
O/ is coinitial. However, with the more general formulation we get the following characterization of the extendable patterns: ∼ the top horizontal morphism is an equivalence. Hence we get an equivalence on fibres at each active morphism (φ : X → O) ∈ Act O (O), which we can identify with the natural map Using the description of F ′ as a right Kan extension we get F, as required.
. This functor is given by Combining this with the equivalence Seg O int (C) ≃ Fun(O el , C) given by right Kan extension along i O , we can reformulate this as: We end this section with some examples of extendable patterns: Example 8.13. The algebraic patterns F ♭ * and F ♮ * are extendable. In the former case, we recover the familiar formula for free commutative monoids: In the latter case, we get which describes a free commutative monoid on X → Y in the slice over Y .
Example 8.14. The algebraic patterns ∆ op,♭ and ∆ op,♮ are extendable. In the former case, we get the expected formula for free associative monoids: while in the latter case we get the formula for free ∞-categories: Example 8.15. More generally, the algebraic pattern Θ op,♮ n is extendable for every n; the conditions are checked in [17], giving a formula for free (∞, n)-categories. (On the other hand, the pattern Θ op,♭ n is not extendable for n > 1.) Example 8.16. The algebraic pattern Ω op,♮ is extendable; the conditions are checked in [10, §5.3], giving a formula for free ∞-operads. (On the other hand, the pattern ∆ op,♮ F is not extendable.)

Segal Fibrations and Weak Segal Fibrations
In this section we first consider Segal fibrations over an algebraic pattern, which are the cocartesian fibrations corresponding to Segal objects in Cat ∞ , and then generalize these to the class of weak Segal fibrations; for the pattern F ♭ * , these objects are respectively symmetric monoidal ∞-categories and symmetric ∞-operads in the sense of [23]. Our main goal is to show that extendability can be lifted from a base pattern to morphisms between (weak) Segal fibrations. Combined with our previous results this allows us, for example, to reproduce (in the cartesian setting) Lurie's formula for operadic Kan extensions along morphisms of symmetric ∞-operads.  Proof. The inert and active morphisms form a factorization system by [23, Proposition 2.1.2.5], so we have defined an algebraic pattern structure on E. To see that π is a Segal morphism it suffices to show that for X ∈ E X the induced functor is coinitial. But this functor is clearly an equivalence, since for each inert morphism X → E with E elementary there is a unique cocartesian morphism with source X lying over it.
We now show that we can lift extendibility along Segal fibrations: where f is an extendable morphism of algebraic patterns, p : E → O and q : F → P are Segal fibrations, and F preserves cocartesian morphisms. Then F is extendable. Moreover, if C is fadmissible and either (i) P el P/ -limits distribute over η-colimits in C for all functors η : P el P/ → Cat ∞ and all P ∈ P, or (ii) p and q are left fibrations, and P el /P -limits distribute over η-colimits in C for all functors η : P el /P → S and all P ∈ P, then C is F -admissible.
Proof. It is immediate from the definitions that F preserves inert and active morphisms. We now observe that F has unique lifting of inert morphisms. Given O ∈ E lying over O ∈ O, and an inert morphismǭ : F (O) → P in F, lying over ǫ : f (O) → P in P, there exists a unique inert morphism γ : O → O ′ such that f (γ) ≃ ǫ, since f is extendable. Since inert morphisms in E are cocartesian, there exists a unique inert morphism γ : O → O ′ lying over γ. Moreover, as F preserves cocartesian morphisms, the morphism F (γ) is the unique inert morphism over ǫ with source F (O), i.e. F (γ) ≃ǭ, and since cocartesian morphisms are unique,γ is the unique inert morphism that maps toǭ. Condition (3) in Definition 7.6 is immediate from f being extendable, using equivalences of the type E el X/ ≃ O el X/ . It remains to prove condition (2). For P ∈ F lying over P ∈ P andǭ : P → P ′ an inert morphism in F lying over ǫ : P → P ′ , we have a functor which fits in a commutative square We claim that here the vertical functors are cocartesian fibrations, and the top horizontal functor preserves cocartesian morphisms. The functor is a fibre product of cocartesian fibrations along functors that preserve cocartesian morphisms, hence it is again a cocartesian fibration. We can write E act /P as a pullback is a pullback of a cocartesian fibration and so is itself cocartesian. Moreover, a morphism in E act /P is cocartesian if and only if its image in E is cocartesian (since the functor F /P → F detects cocartesian morphisms, by [ Putting this together with the equivalence F P/P ∼ − → lim α∈P el P/ F E/α ! P (and similarly for F P ) we get i.e. the functor we get on fibres is indeed an equivalence, which completes the proof that F is extendable.
For admissibility, observe that since lim α∈P el O act /E is a cocartesian fibration, if we compute the colimit of a functor Φ over its source in two stages using the left Kan extension along this functor, we get from which we see that F -admissibility follows from f -admissibility plus either (i) or (ii).
induced by the cocartesian morphisms over inert maps, is an equivalence.
is cartesian.
Proof. Combining condition (2) in Definition 8.5 (the "generalized Segal condition") with the argument of Remark 9.7 shows that for any active morphisms φ : Y → X, ψ : Using this, we see that the functor E act /F → lim α∈O el q(F )/ E act /α ! F induces an equivalence on mapping spaces, and hence is fully faithful. To see that it is essentially surjective, consider the commutative square of ∞-groupoids we want to show that the top horizontal morphism is an equivalence. The bottom horizontal morphism is an equivalence by assumption, since O is extendable; it therefore suffices to show the map on fibres over φ : ∼ the map on fibres over each object of F ≃ O is an equivalence, hence the top horizontal morphism is an equivalence. Since limits commute, it follows that we have an equivalence which completes the proof. Act O (X) n , where Act O (X) n is the space of morphisms to X in O lying over the unique active morphism n → 1 in F * . If C is a cocomplete ×-admissible ∞-category, then our formula for the free Segal O-object monad T O gives: If O ≃ 1 is contractible, we can identify the space O(n) of n-ary operations with the fibre of Act O (X) → Act F * ( 1 ) ≃ BΣ n , and so rewrite this as the familiar formula Remark 9.19. Our description of free algebras differs from what are called "free algebras" in [23], because Lurie defines these to be given by operadic Kan extension along the inclusion O× F * F int * → O where the source is the subcategory containing all morphisms in O lying over inert morphisms in F * , not just the cocartesian ones. (This amounts to specifying the unary operations in advance.) Example 9.20. Since the algebraic pattern ∆ op,♭ is also extendable, the analogues of Examples 9.17 and 9.18 also hold for non-symmetric ∞-operads.
is a morphism of generalized symmetric ∞-operads. Then the previous example does not say that we can compute free Segal P ♭ -objects on Segal O ♭ -objects, as f ! generally will not restrict to a functor between these. In the definition of extendability, condition (1) is still automatic (as the inert morphisms in F ♮ * and F ♭ * are the same), while condition (3) reduces to F ♭ * being extendable. Thus the morphism f ♭ : O ♭ → P ♭ is extendable if and only if for all P over n in F * the functor is cofinal, where ρ i : n → 1 is as in the introduction.

Polynomial Monads from Patterns
In this section we introduce the notion of polynomial monad on an ∞-category of presheaves, and prove that the free Segal O-space monad for an extendable pattern O is polynomial. Moreover, we show that this is compatible with Segal morphisms of algebraic patterns, yielding a functor AlgPatt Seg ext → PolyMnd between the subcategory of AlgPatt consisting of extendable patterns and Segal morphisms, and an ∞-category of polynomial monads. We start by introducing some terminology: Proof. The equivalence of (1) and (2)  Remark 10.5. For ordinary categories, our notion of polynomial monads is the same as the strongly cartesian monads considered in [5]. For monads on ∞-categories of the form S /X for X ∈ S, we recover the polynomial monads studied in [10] (see Theorem 2.2.3 there), which is our reason for adopting this terminology.
where the composite lands in the subcategory Seg O int (S), and it suffices to observe that this composite is accessible and preserves weakly contractible limits. All three functors involved are accessible and except for j O,! they preserve limits. It therefore remains to show that j O,! preserves weakly contractible limits. By Lemma 7. Next, we show that the multiplication transformation is cartesian. To see this it suffices to show that the square on fibres over (Y O F (X)) f we can identify with the colimit over the fibre of the constant functor with value F (Y ). The square of fibres is therefore which is indeed cartesian. The value of the unit transformation . To see that the unit transformation is cartesian we must show that for F → G the square is cartesian. It again suffices to consider the square of fibres over (X which is cartesian as required. Remark 10.7. We can regard polynomial monads as being the monads in an (∞, 2)-category whose objects are presheaf ∞-categories, whose morphisms are local right adjoints, and whose 2-morphisms are cartesian transformations. The natural morphisms between polynomial monads are then the lax morphisms of monads in this (∞, 2)-category. If T is a polynomial monad on S I and S is a polynomial monad on S J , then these correspond to commutative squares for some functor f : I → J, such that the mate transformation is cartesian. Noting the contravariance here, this motivates the following definition of an ∞-category of polynomial monads: Definition 10.8. Consider the pullback We write PolyMnd op for the subcategory of this pullback whose objects are the monadic right adjoints of polynomial monads, and whose morphisms are commutative squares whose mate transformations are cartesian. Remark 10.9. Note that since U T detects pullbacks, the mate transformation above is cartesian if and only if the transformation T f * → f * S obtained by composing with U T is cartesian.
Next, we observe that any Segal morphism between extendable patterns gives a morphism of polynomial monads: Proposition 10.10. Suppose f : O → P is a Segal morphism between extendable patterns. Then the mate transformation Proof. We have to show that for every morphism Φ → Ψ the commutative square is cartesian in Seg O (S). Since Seg P int (S) has a terminal object it suffices to consider Ψ ≃ * , in which case we obtain the commutative square after evaluating at an object E ∈ O el . To show that this square is cartesian, it now suffices to observe that for every point (X → E) ∈ Act O (E), the map on fibres is the identity Φ(f X) → Φ(f X).
Definition 10.11. We let AlgPatt Seg ext denote the subcategory of AlgPatt whose objects are the extendable patterns and whose morphisms are the Segal morphisms.

Patterns from Polynomial Monads: Existence
In the previous section we saw that the free Segal space monad for any extendable pattern was a polynomial monad. In this section we will prove that, conversely, every polynomial monad on a presheaf ∞-category arises in this way, by constructing a canonical algebraic pattern from a polynomial monad. This result amounts to an ∞-categorical version of Weber's nerve theorem [30] in the setting of ordinary categories; our proof was particularly inspired by that of Berger, Melliès, and Weber [5]. We also show that the canonical patterns are compatible with morphisms of polynomial monads, yielding a functor P : PolyMnd → AlgPatt Seg ext such that PM ≃ id.
We begin by defining generic morphisms with respect to a local right adjoint functor, and extend some basic observations about them from [29] to the ∞-categorical setting. This allows us to define the canonical pattern, which we then prove recovers the monad. Definition 11.1. Suppose F : C → D is a local right adjoint functor between presentable ∞categories. Let L * : D /F ( * ) → C be the left adjoint to F / * : C → D /F ( * ) ; we will abusively write L * D for the value of L * at an object D → F ( * ). For any morphism D φ − → F (C) in D, we can view φ as a morphism in D /F ( * ) via the map F (q) : F (C) → F ( * ), where q is the unique morphism C → * . We say φ is F -generic (or just generic if F is clear from context) if the adjoint morphism there exists a unique morphism γ : C → A such that F (γ) • φ ≃ ψ and the equivalence in the square arises by combining this with the canonical equivalence F (α) • F (γ) ≃ F (αγ) ≃ F (β) induced by * being terminal. This is the version of the definition considered in [29].
Remark 11.3. For any morphism φ : D → F (C), if ψ : L * D → C is the adjoint morphism, we can write φ as a composite , where η D is the unit of the adjunction L * ⊣ F / * . This is the unique factorization of φ as a generic morphism followed by a morphism in the image of F ; we will often refer to this as the generic-free factorization of φ. Proof. Since φ is a cartesian transformation, we have natural cartesian squares This means we can write F / * as the composite But then the left adjoint L * ,F of F / * is the composite where L * ,G denotes the left adjoint to G / * . Given f : D → F (C), this means the adjoint morphism L * ,F D → C is the same as the adjoint morphism L * ,G D → C for the composite D → F (C) → G(C).
Proof. The functor (GF ) / * factors in two steps as The left adjoint is therefore also computed in two steps; to find the morphism adjoint to G(f )g we first get the commutative diagram which is an equivalence as required.
Definition 11.6. Suppose I is a small ∞-category and T is a polynomial monad on the functor ∞-category S I . We define W(T ) int,op to be the full subcategory of S I spanned by the objects X that admit a generic morphism I → T X with I ∈ I op (regarded as an object of S I through the Yoneda embedding). We write W(T ) op for the full subcategory of Alg T (S I ) spanned by the free T -algebras on the objects of W(T ) int .
Remark 11.7. From the definition of generic morphisms it follows that we can equivalently describe the objects of W(T ) int,op as those of the form L * I for some I ∈ I op and some morphism I → T * in S I .
, and active if the adjoint morphism Y → T X in S I is generic.
Proposition 11.9. Proof. Since T is a polynomial monad, the unit transformation id → T is cartesian and so by Lemma 11.4 the unit map I → T I is generic for all I ∈ I. Hence I → T I → T * is a generic-free factorization, where the second map is the image under T of the unique map I → * . This shows that I is in W(T ) int,op , which proves (i).
To prove (ii), observe that since X is in W(T ) int,op , we have a generic morphism I → T X with I in I op . But then by Lemma 11.4 and Lemma 11.5 the composite is generic, where the last morphism uses the multiplication T T → T , which is by assumption a cartesian transformation. Hence Y is in W(T ) int,op .
The same argument shows that the composite of active morphisms in W(T ) is again active, so both active and inert morphisms are closed under composition. The uniqueness of factorizations is then just the uniqueness of Remark 11.3. Remark 11.10. Given a polynomial monad T on S I we can identify W(T ) int with the subcategory of W(T ) containing only the inert morphisms.
Definition 11.11. We give W(T ) an algebraic pattern structure using the inert-active factorization system we just defined, by setting W(T ) el := I ⊆ W(T ) int .
We have extracted a canonical algebraic pattern from our polynomial monad T . Our goal is now to show that this pattern describes the same algebraic structure as the monad. The next proposition gives the key input needed to prove this.  This proves (i). Since i : I → A is fully faithful, it follows that i * is also fully faithful, which proves (ii). To prove (iii), since ν A is fully faithful it suffices to show that the composite is a colimit diagram. But this is now a Yoneda cocone for A op , which is always a colimit in S A .
Proof of Proposition 11.13. (i) and (ii) follow from Proposition 11.9(i) and Lemma 11.14. To prove (iii), since colimits in functor categories are computed objectwise, it suffices to show that for every X ∈ W(T ) int and Φ ∈ S I , the morphism is an equivalence. Let E → (W(T ) int,op ) /Φ be the left fibration for the functor (W(T ) int,op ) /Φ → S taking Y to Map S I (X, T Y ); then we have a pullback square so that an object of E is a pair (Y → Φ, X → T Y ). By [22,Proposition 3.3.4.5], the space colim Y ∈(W(T ) int,op ) /Φ Map S I (X, T Y ) is equivalent to the space E obtained by inverting all morphisms in E, and the morphism we are interested in is the map of spaces induced by the functor of Proposition 4.1.1.3] a morphism of spaces that arises from a cofinal functor of ∞-categories is an equivalence, so it suffices to show that the functor E → Map S I (X, T Φ) is cofinal. Since every functor to an ∞-groupoid is a cartesian fibration, to prove this we may apply [22,Lemma 4.1.3.2], which says that a cartesian fibration with weakly contractible fibres is cofinal. It thus suffices to check that the fibres E φ at φ : X → T Φ are weakly contractible. But the fibre E φ is the ∞-category Since T is a local right adjoint, this ∞-category has an initial object, corresponding to the generic-free factorization X → T Y → T Φ, as Y also lies in W(T ) int by Proposition 11.9(ii), so E φ is indeed weakly contractible, as required.
Corollary 11.15. Suppose T is a polynomial monad on S I , and let j T denote the inclusion W(T ) int → W(T ). Then the commutative square is cartesian, and the mate transformation Proof. We want to apply [10, Proposition 5.3.5] to conclude that the square is cartesian. All the requirements for this are clearly satisfied, with one exception: We must show that the mate transformation j T,! ν W(T ) int → ν W(T ) F T is an equivalence, i.e. is given by an equivalence when evaluated at every object Φ ∈ S I . We first consider the case of X ∈ W(T ) int,op ⊆ S I . Then ν W(T ) int X is the presheaf on W(T ) int represented by X, hence j T,! ν W(T ) int X is represented by j T X ≃ F T X, and so j T,! ν W(T ) int X ∼ − → ν W(T ) F T X, as required. Now let Φ ∈ S I be a general object. Since j * T detects equivalences, it suffices to show that the evaluation of the transformation T at Φ is an equivalence. We know from Lemma 11.14(iii) and Proposition 11.13(iii) that Φ is the colimit of the diagram (W(T ) int,op ) /Φ → S I taking X → Φ to X, and this colimit is preserved by the functors ν W(T ) int and ν W(T ) int T . Since j * T j T,! preserves colimits (being itself a left adjoint), we have a commutative square where the vertical morphisms are equivalences. Moreover, the top horizontal morphism is an equivalence, since it is the colimit of equivalences j * The bottom horizontal morphism is therefore also an equivalence, which completes the proof.
To prove (ii), we must show that if X is in W(T ) int,op , so that there is a generic morphism I → T X with I ∈ I op , then f ! X is in W(S) int,op . By (i), the composite f ( To show part (iii), note that we have equivalences Alg T (S I ) ≃ Seg W(T ) (S) and Alg S (S J ) ≃ Seg W(S) (S), and so these ∞-categories are in particular presentable. Since U S detects equivalences, preserves limits, and is accessible, and f * preserves both limits and colimits, it follows that Φ is accessible and preserves limits. By the adjoint functor theorem this implies that Φ has a left adjoint Ψ, as required.
From our commutative square of right adjoints we now get an equivalence ΨF T ≃ F S f ! . By definition the ∞-categories W(T ) op and W(S) op consist of free algebras on objects of W(T ) int,op and W(S) int,op , respectively, so it follows from (ii) that Ψ takes W(T ) op to W(S) op , and preserves inert morphisms. Using (i) we also see that Ψ preserves active morphisms. Standard manipulations of cocartesian fibrations now imply that the canonical patterns determine a functor, giving: Corollary 11.20. There is a functor P : PolyMnd → AlgPatt Seg ext that takes a polynomial monad T to the canonical pattern W(T ), and an equivalence MP ≃ id.

Patterns from Polynomial Monads: Uniqueness
Given an extendable algebraic pattern O, our work in the last two sections gives a canonical pattern O := W(T O ) that describes the same monad on Fun(O el , S). Our goal in this section is to investigate the relationship between the patterns O and O. We will show that, if O is "slim" (a mild technical assumption that holds in all examples) then there is a natural morphism of patterns O → O. Moreover, we will see that this map is an equivalence if and only if O is saturated, meaning that the presheaves Thus restricting to slim extendable patterns there is a natural transformation id → PM, and we will observe that this exhibits P as a right adjoint to M. We conclude that PolyMnd is a localization of the ∞-category AlgPatt Seg slim,ext of slim extendable patterns and Segal morphisms, and that there is an equivalence between PolyMnd and the full subcategory of saturated slim extendable patterns.
We begin with some key observations about the localization of the Yoneda embedding for O, which we denote as follows: This equivalence is natural with respect to inert morphisms, i.e. we have a commutative square corresponds under the equivalences to the composite where the equivalence follows from the inert-active factorization system and the second map from the definition of Λ int O X as the localization of Map O int (X, -). Proof. From the commutative square of functors in Lemma 12.3 we get for all O ∈ O a commutative square where the right-hand map can be identified with where the description of F O Λ int O X as a left Kan extension implies that the right-hand map is given on Putting these two diagrams together we therefore obtain natural commutative squares , for every active morphism φ : O → Y , where the right vertical map is the canonical one from the component of the colimit at φ. Taking colimits over Act O (Y ) we therefore get a commutative square Here the inert-active factorization system on O implies that the left vertical map is an equivalence, since its fibre at a morphism ψ : X → Y can be identified with the space of inert-active factorizations of ψ, and this completes the proof.
We can now prove that, in the appropriate sense, active morphisms are generic: . Then the generic-free factorization of η is Proof. We first check that this factorization exists. By Lemma 12.4 the morphismφ adjoint to Λ(φ) corresponds to the point in where the second morphism is the canonical one from the component of the colimit at φ. We therefore have a commutative diagram * where the outer triangle corresponds to the desired factorization Now we must show thatφ is generic, so suppose we have a commutative square This amounts to an equivalence between p and the image * But since the last map arises from T ψ, there is a commutative diagram which tells us that ψ must be the morphism Λ int O X → Φ obtained by localizing the unique natural transformation y int O X → Φ that takes id X to the point p. Thusφ satisfies the universal property of generic morphisms described in Remark 11.2. By the uniqueness of generic-free factorizations, this completes the proof.
Such a generic morphism is determined by a morphism Λ int O E → T O * , and from Proposition 12.6 we see that the generic-free factorizations of such morphisms yield precisely the objects of O • . This proves (i), from which (ii) follows using Lemma 12.3. Finally, as active morphisms in O are defined to be those adjoint to generic morphisms, (iii) follows from the identification of such generic morphisms with active morphisms in O in Proposition 12.6.
Remark 12.11. If O is a slim extendable pattern, then Corollary 12.10 says that O has the same objects and active morphisms as O, but the inert morphisms might be different.
Definition 12.12. We say a slim extendable pattern O is saturated if for every X ∈ O the canonical diagram O el,⊳ X/ → O int is a limit diagram, i.e. the object X is the limit of the inert maps to elementary objects.
where the first map is the Yoneda embedding. Since the composition of the first two morphisms is an equivalence by (4), the second map is an equivalence. The last map is an equivalence because Λ int O Y is a local object and y int O X → Λ int O X is a local equivalence. Hence, we have for every object X ∈ O int , which then implies that y int Example 12.14. Let T be a polynomial monad on S I . By Corollary 11.17 and the previous lemma, the algebraic pattern W(T ) is extendable and saturated. By Definition 11.6 every object X ∈ W(T ) admits a generic map T X → E with E ∈ W(T ) el , and so W(T ) is also slim. Omitting notation for the horizontal inclusions, we have equivalences . In other words, Λ O X is indeed the presheaf represented by X ∈ O, as required.   : F ♭,int,op * → S takes n to a finite set n with n elements, and an inert morphism n → m to the map m → n that takes i ∈ m to its unique preimage φ −1 (i) ∈ n. Thus inert morphisms correspond bijectively to injective morphisms of finite sets, and the functor is not fully faithful. The canonical pattern F ♭ * ⊆ Seg F ♭ * (S) op consists of the free commutative monoids on finite sets. By [12,Theorem A.4] this can be identified with the ∞-category Span(F) whose objects are finite sets and whose morphisms are spans of finite sets, with F * → F * identifying F * with the subcategory where the morphisms from I to J are spans I ← K → J with the backward map injective.