Operads, operadic categories and the blob complex

We will show that the Morrison-Walker blob complex appearing in Topological Quantum Field Theory is an operadic bar resolution of a certain operad composed of fields and local relations. As a by-product we develop the theory of unary operadic categories and study some novel and interesting phenomena arising in this context.


INTRODUCTION
The blob complex was introduced by S. Morrison and K. Walker in [14]. It associates to an n-dimensional manifold M, equipped with a system of fields C containing an ideal of local relations U, the blob complex B * (M, C), which is a chain complex whose salient feature is the The initial impulse for the present work was a seminar given by K. Walker at MSRI, Berkeley, in the winter of 2020. The second author noticed a striking similarity between the diagrams drawn by Kevin on the board, and pictures representing elements of free operads over graphrelated operadic categories that can be found in [4,Section 5]. This inspired the idea that the blob complex might be the bar resolution of an operad over a suitable operadic category.
That hope indeed turned out to be true; there even exist two related but non-equivalent ways to interpret the blob complex within operad theory. The first interpretation produces a com-  Figure 4 at the end of this paper.
Disclaimer. The present work does not bring anything new to the theory of blob complexes per se, neither it adds anything to the explicit calculations given in [14]. The free, acyclic resolution of the skein module in Theorem B on page 45 might however pave way for the study of derived TQFT invariants.
Novelties. The operadic category of blobs is unary, meaning that the cardinalities of all its objects are one. The blob complex thus represents an interesting, highly nontrivial example of an unary operadic category and justifies careful analysis of operads, operadic modules, fibrations and various versions of the bar construction in this context. Several interesting and new phenomena were discovered en route.
Conceptual understanding of the relationship between the decorated version of the unital operadic category of blobs and the un-decorated one in Section 9 inspired the notion of partial [December 29, 2022] [blob.tex] discrete operadic fibrations, partial operads and the associated partial Grothendieck construction, given in Section 4. The concrete partial operad that arose in this context is unital in a weak, unexpected sense, formalized in Definition 30 by introducing pseudo-units. Pseudo-unitality is a new, nontrivial concept even in the realm of traditional algebra, as Example 21 shows. We also introduce several versions of the 'standard' unitality condition for operads over operadic categories that are not equipped with the chosen local terminal objects required in [1, Section 1].
While free modules over classical operads have simple structure, cf. e.g. [ where the structure depends on the shape of the operadic category, as illustrated in Example 72.
A structure result can however still be obtained under the condition of rigidity introduced in Definition 70, which has no analog in the standard operad theory. We believe that all the above notions admit generalizations to non-unary operadic categories.
The present paper has two parts. Part  Requirements and conventions. We will assume working knowledge of operads; suitable ref- erences are the monograph [13] and the overview [12]. Operadic categories and related notions were introduced in [1], but all necessary material from that paper is recalled in Sections 1 and 3.
Some preliminary knowledge of [14] may ease reading Part 2.
Categories will be denoted by typewriter letters such as C,O,Q, &c, operads and their modules written in script, e.g. P, S, M, &c. From Section 6 on, all algebraic objects will live in the monoidal category R-Mod of graded modules over a unital commutative associative ring R.
Chain complexes will be non-negatively graded, with differentials of degree −1. By a quasiisomorphism we mean a morphism of chain complexes that induces an isomorphism of homology. Preprints [3,4] are under permanent revision, so we indicated explicitly which concrete versions we referred to.
Acknowledgment. The first author wishes to express his sincere gratitude to Joachim Kock for many illuminating discussions on décollage comonad and operadic categories from simplicial point of view. The second author is indebted to Benoit Fresse for pointing to the results of his impressive book [6] that were relevant to our work.

OPERADIC CATEGORIES AND OPERADS
Our immediate aim is to rephrase the definitions of operadic categories and their operads as given in [1,Section 1] to the particular, unary case when the cardinality functor is constant and equals 1. We believe that this would make this article independent of [1].
Unary operadic categories will appear in Definition 4 as categories equipped with fiber functors; Propositions 8 and 10 then describe them also as algebras for a certain monad. Operads over unary operadic categories are introduced in Definition 13. In this section we however, unlike in [1], do not require the existence of chosen local terminal objects in operadic categories, neither we assume units of operads. A refined analysis of these additional structures is given in Section 2.

Remark 1.
The approach to unary operadic categories presented in this section has a lot of overlaps with the material from [7] concerning the use of the décollage comonad D recalled in (6). But in contrast with [7], where D-coalgebra structure is fixed and the fiber-functor structure is imposed on top of it, we begin with imposing a fiber-structure functor and adding some pieces of D-coalgebra structure only when it is necessary. This allows us to analyze subtler unitality properties of operadic categories. To describe the action of F c on a morphism Φ : b → a in O/S given by the commutative diagram The verification that the above rules define a functor is straightforward.
To expand Definition 3, introduce the following notation and terminology. Given a map f : X → S, we call F S ( f ) the fiber of f and denote it simply by F ( f ). The fact that F ∈ O is the fiber of f will also be expressed by writing F ⊲ X f → S. For a morphism F in O/S given by the diagram Then the commutativity of (3) on objects is expressed by the equality of the fibers in the diagram In words, the fiber of a map equals the fiber of the induced map between its fibers.
To expand the commutativity of (3) on morphisms, notice that in the above notation, dia- In the situation of (1), diagram (3) requires that Since all operadic categories and operadic functors in this article will be strict, we will for brevity omit this adjective. If not indicated otherwise, by an operadic category we will always mean unary and nonunital one.

Remark 5.
Recall that a simplicial set is a collection S • = {S n } n≥0 of sets together with maps d i : S n → S n−1 , 0 ≤ i ≤ n, n ≥ 1, and that satisfy the identities: In the above display, O n consists of chains of arrows of 0. The operator d 0 acts on (5) by removing T 0 , d n removes T n . The operator d i with 0 < i < n replaces the part T i −1 The operator s i replaces T i in (5) by the identity automorphism 1 : The extra structure on O given by Lemma 2 is equivalent to adding to the nerve of O a sequence of additional face operators which satisfies all standard simplicial identities except for the top one, that is, the composite The functor D : Cat → Cat has a natural structure of a comonad in the category of small categories called décollage comonad, cf. [7]. We briefly describe its structure morphisms. Its counit To understand the comultiplication δ : D(A) → D 2 (A) we observe that the objects of D 2 (A) are commutative triangles Explicitly, the fiber of a morphism g : S → T in A ⊂ T(A) is g again, but interpreted as an object The operadic category T(A) is an operadic subcategory of D(A ⊙ ), where A ⊙ is the result of formally adjoining a terminal object ⊙ to A, which means adding to the morphisms of A the unit endomorphism of ⊙ and one new morphism X ! → ⊙ for any object X ∈ A. The inclusion (10) sends an object X ∈ A to X ! → ⊙ ∈ D(A ⊙ ) and objects X → Y of D(A) into the corresponding objects of D(A ⊙ ). The image of (10) covers all objects of D(A ⊙ ) except ⊙ ! → ⊙. Inclusion (10) will be used in Section 9 in our description of the operadic categories of blobs. Proposition 10. The assignment A → T(A) gives rise to a (unital) monad in Cat whose algebras are (non-unital) operadic categories with small sets of objects.
Proof. Direct verification. Remark 11. Comparing Propositions 8 and 10, one may wonder why two very different monads have the same algebras. The explanation lies in the unitality of T versus the non-unitality of D.
As a simple example of this phenomenon, consider the nonunital monadT in Set that sends a set X to the coproduct n≥2 X ×n , and the unital monad T given by TX := n≥1 X ×n . Both T and T have the same algebras, namely associative (non-unital) monoids. Ideologically, T is obtained fromT by freely adjoining the monadic unit. The relation between T(A) and D(A) is of the same nature.
commutes. The meaning of the symbols in that diagram is explained by the following instance of (4): We will see later in this paper that operads over unary operadic categories share many features with associative algebras, but also ones that do not have analogs in classical algebra. We believe that they in their own rite might present an interesting theme of research.
Example 15. Let A be a small category and D(A) the associated operadic category (6).
The multiplication is required to be associative in the obvious sense. Intuitively, A is a non-unital Mor(A)-graded associative algebra in V. The unital version of this example has a nice categorical interpretation, cf. Example 25 below.
commutes for any pair A f → B g → C of composable morphisms in A. Intuitively, M is a left Ob(A)graded A-module.

THE SHADES OF UNITALITY
In this section we discuss sundry versions of unitality for operadic categories and their operads, with or without the presence of chosen local terminal objects. This refinement of definitions given in [1] is required in Part 2 by the applications to the blob complex.   And, of course, right and left unitality together mean that the diagram is a simplicial set. To see why, assume that U ∈ A is a local terminal object of A. By definition, the fiber of a morphism a → U is a → U again, but now being an object D  More generally, an arbitrary set X with a multiplication X × X → X given by the projection to the first factor and pseudo-units e t := t for t ∈ X is a pseudo-unital monoid. The associated operadic category O X is the chaotic groupoid generated by X .
Remark 23. Let us try to 'categorify' pseudo-unital monoids of Example 21 by assembling the is expressed by the diagram in which µ is the multiplication in A and π 1 the projection to the second factor. Similarly, (14b) can be expressed via the diagrams where ∆ is the diagonal a → a × a, a ∈ A. Since the diagrams above involve the projection and diagonal, pseudo-unitality admits a categorification only inside a cartesian monoidal category.
We say that P is left unital if, for any T ∈ O, the diagram in which U c is the fiber of the identity automorphism 1 : T → T , commutes.
commutes. Finally, P is unital if it is both left and right unital.
If the background monoidal category is the cartesian category Set of sets, the family (15)  Exercise 19 tells us that D(A) is a unital operadic category with the chosen local terminal objects The left (resp. right) unitality of A is then expressed by the left (resp. right) diagram below: which are required to commute for any morphism f : c → B of A.
We notice that unital D(A)-operads in V are the same as lax 2-functors A → ΣV, with A considered as a 2-category with trivial 2-cells, and ΣV the 2-category with one object and V as the category of morphisms. In particular, if A is the chaotic groupoid Chaos(I ) on the set I , then We say that P is left unital if, for any T ∈ O, the diagram For an arbitrary morphism f : T → S of O, the axioms of unary operadic categories provide the with f T the induced map between fibers. The right unitality requires that the induced diagram for all c ∈ A. The the right unitality of (A, M) is just the right unitality of A. Notice that when V is the cartesian category of sets and A the terminal D(A)-operad, i.e. when if A(f ) is the one-point set for each morphism f of A, unital T(A)-operads are the same as presheaves over A. Remark 29. Notice that we do not claim that the left (resp. right) unitality of Definition 26 alone implies the left (resp. right) unitality of Definition 24; the second part of the proposition is true only for the (two-sided) unitality. This shall be compared with the obvious fact that, while an associative algebra admits at most one two-sided unit, it might have several left or right units.
An immediate implication of the proposition is that, for operads over unital unary operadic categories, the two definitions provide equivalent notions of (two-sided) unitality.
Since the fiber of each identity automorphism in a unital unary operadic category is a chosen local terminal object, each U T in Definition 26 equals U c for some c ∈ π 0 (O) uniquely determined by T . Therefore the family (15) determines a family (17) The commutativity of (20) follows from the left unitality of η U and the right unitality of η T . It implies that the composite of the leftmost arrows equals the composite of the rightmost arrows, The family (17) thus determines a family (15) by η c := η U c that satisfies the left and right unitality of Definition 24. For instance, (16) is fulfilled with η T in place of η c by (18), but η T = η c as proven above. The right unitality is discussed similarly.
The following definition extends the concept of pseudo-unitality of associative monoids introduced in Example 21 to operads. The background monoidal category will be crucially the cartesian category of sets, cf. Remark 23.
Definition 30. Let S be an O-operad in Set equipped with a family of elements for any t ∈ S(U T ) and ϕ ∈ S(F ). Finally, S is pseudo-unital if it is both left and right pseudounital. In this case we call the elements of the collection (21) the pseudo-units of S.
Proposition 31. Assume that S is an O-operad in the category of sets, left (resp. right) unital in the sense of Definition 26. Then it is left (resp. right) pseudo-unital.
Proof. The monoidal unit of the category Set is the one-point set {⋆}. The family (17) determines a family as in (21) by e t := η T (⋆). Notice that this e t depends only on U T , not on a concrete t ∈ S(U T ). It is simple to verify that if S is left (resp. right) unital in the sense of Definition 26, then {e t } are left (resp. right) pseudo-units of S.
Example 32. Let O be an unital unary operadic category. The operad with (T ) the empty set for each T ∈ O, is a pseudo-unital operad, which is however not unital. This shows that pseudounitality is less demanding than unitality even when O is unital. Below is a less trivial example.
The unary version of the operadic Grothendieck construction [1, page 1647] associates to a unital Set-valued operad S over a unary unital operadic category O, cf. Definition 24, a unary unital operadic category O S together with an operadic functor p : O S → O as follows. Objects of O S are elements t ∈ S(T ) for some T ∈ O. Given s ∈ S(S) and t ∈ S(T ), a morphism σ : is the fiber of the identity automorphism T → T and ⋆ is the only element of the monoidal unit The categorical composition in O S is given as follows. Assume that a ∈ S(A), b ∈ S(B) and The composite ψφ in O S is defined to be the pair (x, g f ), where x := γ f C (ω, y) and f C is as in diagram (12).
It turns out that the functor p : Any discrete operadic fibration induces and isomorphism π 0 (Q) Proof. Given a discrete operadic fibration p : Q → O, the corresponding Set-operad S has the components as in (24). The pseudo-unit e t ∈ S(U T ) associated to t ∈ S(T ) is, by definition, the fiber of the identity automorphism t → t in Q. To verify that this recipe is the inverse of the Likewise, there are one-to-one correspondences between -pseudo-unital associative monoids, -pseudo-unital -operads, and -discrete operadic fibrations of operadic categories over .
In particular, the chaotic groupoid generated by X is the Grothendieck construction ⊙ X of the pseudo-unital monoid X discussed in Example 22, with the fibers given by the domain functor.

PARTIAL OPERADS, PARTIAL FIBRATIONS
The Grothendieck construction used in (47) (12), Definition 41. Let S be a partial O-operad as in Definition 40 equipped with a family of elements We say that S is left pseudo-unital if, for any T ∈ O and t ∈ S(T ), γ 1 (e t , t ) is defined and equals t .
We moreover require that, for any diagram as in (22) and elements ρ ∈ S(R), c ∈ S(C ) for which We say that S is right pseudo-unital if, in the situation of diagram (19), γ f T (ϕ, e t ) is defined for any t ∈ S(U T ) and ϕ ∈ S(F ), and equals ϕ. Example 42. Partial pseudo-unital operads over the terminal unital operadic category are partial pseudo-unital monoids. We define them as partial associative monoids A equipped with a family {e b ∈ A | b ∈ A} such that the product z e t is defined for each z, t ∈ A and equals z and, if t b is defined, then e tb t is defined and equals t , for each t , b ∈ A. Such partial pseudo-unital monoids are, of course, partial versions of pseudo-unital monoids introduced in Example 21.
The Grothendieck construction recalled in Section 3 works even when S is only a partial unital Set-valued operad. The objects of the modified category O S are elements t ∈ S(T ), T ∈ O, as before, but the pair (ǫ, f ) with f :

is defined (and equals s).
Let us verify that (25) guarantees that the categorical composition is defined for all pairs of morphisms of O S whose targets and domains match as usual. Assume that φ : a → b and ψ : b → c are as in the paragraph on page 19, Section 3, where the composition in O S is described.
Since γ g (y, c) is defined and equals b, is defined and, thus, In particular, γ f C (ω, y) must be defined, and we define the composite ψφ to be the pair ( where e t is as in (26); notice that The sets {L( f )} f are such that (i) for any (ε, s) ∈ L(f ) there exists a unique lift σ as in (23), where u t is the fiber of the identity Denote the lift σ of (ε, s) ∈ L(f ) in item (i) above by ℓ( f , ε, s). Consider the diagram (12) in O and elements y ∈ p −1 (Y ), r ∈ p −1 (C ) and ε ∈ p −1 (F ). We require that Equivalence (28)

OPERADIC MODULES
The inputs of the classical bar resolution [9, Section X.2] are an associative algebra Λ and its (left) Λ-module C . In Section 7 we generalize the input data to an operad P and its suitably defined P-module M. Operadic modules are the content of the present section; its floor plan is similar to that of Section 1.
While P is, as before, defined over an operadic category O, P-modules live over a categorical 'module' M over O. This feature has no analog in the classical algebra. The word 'module' in the rest of this paper might thus mean either a categorical module over an operadic category, or a module over an operad. We believe that the concrete meaning will always be clear from the context. which are associative, i.e. ( f g )α = f (g α) for α and g as above and f ∈ C(T,R), and unital, meaning that 1 S α = α for each α ∈ L(L,S) and the identity automorphism 1 S ∈ C(S,S).
Right C-modules as well as C-bimodules can be defined analogously, but we will not need them here. Conversely, let L be a category equipped with a map of sets γ : π 0 (L ) → L 0 and a discrete opfibration t : L → C. We claim that these data determine a categorical left C-module L with Indeed, any category is the coproduct of its connected components, in particular Now for L ∈ L 0 and S ∈ C we define the set of arrow L(L,S) as the set of objects of α ∈ c∈γ −1 (L) L c such that t (α) = S. The action of C is defined using the lifting property of the opfibration t .
Translated to the language of simplicial sets this construction amounts to the following. The simplicial nerve of the category D C (L), cf. Remark 5, consists of the sets L n , n ≥ 1, of all possible composable chains where L ∈ L 0 and S 0 , . . . , S n−1 ∈ C. The functor d 0 induces a diagram of sets In this diagram all usual simplicial identities hold, the bottom and the shifted top simplicial sets satisfy Segal conditions and, moreover, all (commutative) diagrams involving top horizontal face operators are pullbacks. Conversely, any diagram with the above properties is the 'nerve' of a categorical left module.
Remark 47. The rule (α, g ) → g α does not look as a left action, one would expect (α, g ) → αg instead. This unpleasing feature is due to the bad but favored convention of writing 'α followed by g ' as g α.
Example 48. Given a category C and a set S, one has the chaotic C-module Chaos(S,C) with exactly one arrow L → T for every L ∈ S and T ∈ C. A concrete example will be given in Section 9.
[December 29, 2022] [blob.tex] Example 49. If both C and L have just one object, the resulting structure is the standard left module over an associative unital algebra.
Example 50. Each category is a left module over itself. More generally, any functor F : D → C determines a left module L(F ) over C whose set of objects are objects of D and whose set of arrows L(d,c) equals C (F (d ), c), for d ∈ D, c ∈ C. Still more generally, any functor F : D op ×C → Set determines a left C-bimodule by similar formulas. Such functors are also known as profunctors, distributors or bimodules from D to C.
Example 51. If L is a C-module and c ∈ C, then there exits the left 'overmodule' L/c over C/c whose objects are arrows α : L → c in L. Arrows from α to g : Definition 52. We denote by MOD the category whose objects are pairs (C, L) consisting of a category C and its left module L.
commutes for an arbitrary morphism f : S ′ → T ′ of the category C ′ . In commutes for any c : S → T .
We use similar notation and terminology as for operadic categories. That is, given an arrow α : M → S in M, we call G S (α) the fiber of α and denote it simply by G (α). The fact that G = G (α) where α : L → X is an arrow in M , g : X → S is a morphism of O and β = g α, we denote by α S the induced arrow G (β) → F (g ) between the fibers. The module analog of diagram (4) associated to (31) reads The new face operators satisfy all usual simplicial relations.
is required to commute. The symbols in that diagram are explained by the following instance of (32): in which η T is as in (17) commutes for each α : X → L with fiber G. We denote by Mod M (P) the corresponding category.

FREE MODULES
In this section we study the structure of free operadic modules. The main result, Proposition 71, requires a certain rigidity property that has no analog in the classical algebra. The base monoidal category will be from this point on the category R-Mod of (graded) vector spaces over a commutative unital ring R though any closed monoidal category would do as well.
To warm up, we recall the following simple classical facts. Let E be a vector space and Λ a non-unital associative algebra. ThenF Remark 66. Let us act on both sides of the equality e ⊕ 0 = 0 ⊕ (1 ⊗ e) in the denominator of (33) by some λ ∈ Λ. By the definition of the left Λ-action, we get the equality 0⊕(λ⊗e) = 0⊕(λ·1⊗e), which implies the relation The assumption that 1 is also a right, not only the left unit of Λ guarantees that the 'unexpected' relation in (34) is satisfied automatically.
Free modules in the operadic context have a similarly simple structure only when the following unary version of the weak blow-up axiom [3, Section 2], abbreviated WBU, is fulfilled.

Weak blow-up (category version).
For each morphism f ′ : X ′ → S in 0 with fiber F ′ , and another morphism φ : F ′ → F ′′ , the left diagram in (WBU) below can be uniquely completed to the diagram in the right hand side so that φ will became the map between the fibers induced by ϕ:    It is not difficult to verify that the above formula defines the required morphism in Mod M (P), and that such a morphism is unique.
Let us discuss the unital version of the above constructions, assuming that that the operad   Let O be as before, and let M have one object ⋆ but no arrow. A unital P-module is just a vector space. The free unital P-module generated by E is thus E again, while the right hand side of (36) is trivial. This shows that (36) need not to hold without the rigidity assumption.
Theorem 73. The augmented chain complex β * (Λ,C ) ǫ −→ C is an acyclic resolution of C via free left Λ-modules.
Let 1 ∈ Λ be the unit of Λ. For each n ≥ 1 define linear operators s j : β n (Λ,C ) → β n+1 (Λ,C ), The following statement is also classical.  Its nth differential ∂ n is the sum n 0 (−1) i d i , with d i acting on an element of P(T 0 )⊗P(F 1 )⊗· · · ⊗P(F i )⊗P(F i +1 )⊗· · · ⊗P(F n )⊗M(N ) as follows. If 1 ≤ i ≤ n −1, replace first the piece in the tower (37) by where F ′ is the fiber of f i f i +1 . This situation gives rise to the following instance of (4): The operation d i now replaces p i ⊗ p i +1 in (39) by γ(p i +1 , p i ) ∈ P(F ′ ), where γ is the operadic composition induced by the subdiagram F i +1 ⊲ F ′ → F i of (40). To define d 0 , we cut the left end of the tower (37) to and replace t 0 ⊗ p 1 in (39) by γ f 1 (p 1 , t 0 ) ∈ P(T 1 ). To define d n , we replace We finally replace p n ⊗ m in (39) by the composite ν(n, p n ) ∈ M(N ′ ) associated to the subdiagram N ⊲ N ′ → F n of (41).
Proof. An exercise on the axioms of operads and their modules. Then modify T M by inserting the identity automorphism of T j to it, which results in the tower with the fiber sequence (T 0 , F 1 , . . . , F j ,U T j , F j +1 , . . . , F n , N ). Then s j (u) ∈ β n+1 (P, M) is defined as  Let us move to the issue of acyclicity of the bar resolution. For an object Υ ∈ M denote by Υ/O the category whose objects are arrows α : Υ → X , X ∈ O, and morphisms α ′ → α ′′ are commuta- where the horizontal arrow is a morphism of O. It will turn out that the bar resolution is acyclic at objects Υ of M with the property that The terminality in (P1) means that each arrow α : Υ → X in M is uniquely left divisible: By (P2), the fiber of the unique arrow X ! → ⊙ is X .
Assume that P is left unital the sense of Definition 26 and M a unital P-module as in Definition 63. Since the fiber of the identity 1 : ⊙ → ⊙ is ⊙, by the left unitality of P there exists a morphism η ⊙ : 1 → P(⊙) for which the diagram commutes. Likewise, the unitality of M implies the commutativity of We are finally able to formulate the operadic version of Theorem 73. Recall that we assume To construct h n for n ≥ 0, we consider a tower T Υ as in (37) with M = Υ and a related element u ∈ β n (P, M)(Υ) in (39). Since Υ ∈ π 0 (⊙), all T i 's and T 0 in particular belong to π 0 (⊙), so there exists a unique ! : T 0 → ⊙, hence T Υ can be uniquely extended to with the associated fiber sequence (⊙, T 0 , F 1 , . . . , F n , N ). We finally define h n (u) to be 1 ⊙ ⊗ u in the component of β n+1 (P, M)(Υ) corresponding to the tower h(T Υ ). The desired property of the contracting homotopies for h, h 0 , h 1 , . . . constructed above is easy to verify.
We note that the right unitality of P only is sufficient for the acyclicity of β * (P, M)(Υ) ǫ → M(Υ).
Theorem A has the following obvious such that the diagram in which the horizontal arrows are the augmentations, commutes.
Proof. The component of β * Φ * (P), Ψ * (M) (M) corresponding to the tower in (37) is mapped to the component It is simple to check that the morphism thus constructed has the desired properties.

BASIC NOTIONS
We will deal with manifolds, their boundaries, embeddings, &c. The precise meanings of these nouns will depend on the setup in which we choose to work -manifolds could be topological, smooth, piecewise linear, with some additional structures, &c. Since all constructions below are of combinatorial and/or algebraic nature, we allow ourselves to be relaxed about the nomenclature; compare the intuitive approach in the related sections of [15]. We reserve d for a non-negative integer.
Definition 83. A system of fields is a rule that to each manifold X of dimension ≤ d +1 assigns a set C(X ). This assignment should satisfy properties listed e.g. in [14,Section 2]. We will in particular need the following: (ii) Let X ′ ⊔ Z X ′′ be a manifold obtained by glueing manifolds X ′ and X ′′ along a common piece Z of their boundaries, and c ′ ∈ C(X ′ ), resp. c ′′ ∈ C(X ′′ ) be fields whose restrictions to Z agree. Then c ′ and c ′′ can be glued into a field c ′ ⊔ Z c ′′ ∈ C(X ′ ⊔ Z X ′′ ) which restricts to the original fields on X ′ resp. X ′′ .
We will denote by C(X ; c) the subset of C(X ) consisting of fields that restrict to c ∈ C(Z ). We will always be in the situation when Z in (ii) is closed, so we will not need 'glueing with corners' described in [14,Section 2]. We allow Z to be empty, in which case we denote the result of the glueing by c ′ ⊔ c ′′ ∈ C(X ′ ⊔ X ′′ ). Standard examples of fields are C(X ) the set of maps from X to some fixed space B, or C(X ) the set of equivalence classes of G-bundles with connection over X .
Definition 84. Let X be a (d +1)-dimensional, not necessarily connected, manifold, with the (possibly empty) boundary ∂X and the interiorX . A blob in X is the image D of the standard , and X \D is a (d +1)-dimensional manifold with the boundary ∂X ∪ ∂D, or (ii) D is one of the connected components of X .
A configuration of blobs in X is a nonempty unordered finite set D = {D 1 , . . . , D r } of pairwise disjoint blobs in X .
We will sometimes use D also to denote the union D := r i =1 D i if the meaning is clear from the context. It is a manifold with the boundary ∂D := r i =1 ∂D i and the interiorD := r i =1D i , and X \D is a (d +1)-dimensional manifold with the boundary ∂X ∪ ∂D. If some of the blobs in D happen to be some of the components of X , then the corresponding components of X \D are d -dimensional embedded spheres, interpreted as degenerate (d+1)-dimensional manifolds with empty interiors.
In the rest of this paper we assume that the space of fields on codimension zero submanifolds of X is enriched in R-Mod. This is the case e.g. when the fields come from a (d+1)-category whose If X has a non-empty boundary ∂X and b ∈ C(∂X ), one has the obvious restricted version

BLOBS VIA UNARY OPERADIC CATEGORIES
We are going to introduce various operadic categories and operadic modules together with the related operads and their modules, arising from the blobs and fields in Section 8. Our aim is to describe the associated bar resolutions, cf. the second half of Section 7. In Section 10 we show that they are quasi-isomorphic to the original blob complex in [15]. The notation is summarized at the end of this section.
Let us fix a connected (d +1)-dimensional, not necessarily closed, non-empty manifold M. If its boundary ∂M is non-empty, some constructions below will depend on a fixed field b on ∂M.
We will however often omit such a boundary condition from the notation.  Let blob be the category blob with a terminal object ∅ formally added. In this particular case, ∅ can be viewed as the empty configuration of blobs, whence the notation. Inclusion (10) describes the tautological operadic category Blob := T(blob) as the subcategory of Blob := D(blob) whose objects are inclusions i ′ : D ′ ← D, where D is allowed to be empty. If this is so, we identify i ′ with D ′ ∈ blob. Morphisms of Blob then arise as diagrams in (45)  We will tacitly assume that all inclusions of decorated blobs satisfy the above condition. Denoting by blob(C) the category blob(C) extended by the empty blob, the tautological operadic category Blob(C) := T(blob(C)) becomes the full subcategory of Blob(C) := D(blob(C)) whose objects are 'extended' morphisms i ′ : (D ′ ; c ′ ) ← (D; c) in blob(C) with D allowed to be empty, and morphisms the diagrams with D allowed to be empty.
It turns out that Blob(C) is the partial operadic Grothendieck construction, in the sense of Section 4, over its un-decorated version. Explicitly The pseudo-unit e t in (26) associated to a field t ∈ S(T ) is the restriction e t := t | ∂D ′ ∈ S(U T ).
Proposition 87. The isomorphism (47) holds for the partial pseudo-unital operad S defined above. The natural projection Blob(C) → Blob that forgets the decorating fields is thus a partial discrete operadic Grothendieck fibration.
The proposition is easy to check. The subspace L(f ) in (27) associated to the partial discrete operadic Grothendieck fibration Blob(C) → Blob equals when f is the morphism (46).
We will also need modules arising from blobs and fields. Let us denote by m the left blob- the span of fields in C(D \D ) that restrict to the field c ⊔ c 1 ⊔ · · · ⊔ c r on ∂D ∪ ∂D, cf.   Warning. The symbol ' ' in the above display is not an input color, but indicates that the set of inputs is empty.
Proposition 89. The structure F c defined above is a unital colored R-linear operad.
Proof. Let x ∈ F c (D; c) (D 1 ; c 1 ) · · · (D r ; c r ) and Then the operadic composite (54) x(x 1 , . . . , is the field obtained by glueing the fields x, x 1 , . . . , x r along the boundaries of the balls D 1 , . . . , D r . The color-matching guarantees that this glueing is possible. The image of the glueing with the left F c -action assigning to each m ∈ M c (D 1 , c 1 ) · · · (D r , c r ) and to x i 's as in the proof of Proposition 89 the element m(x 1 , . . . , x n ) ∈ M c (D 1 1 ; c 1 1 ) · · · (D 1 k 1 ; c 1 k 1 ) · · · (D r 1 ; c r 1 ) · · · (D r k r ; c r k r ) , given by the glueing of fields as before. In the rest of this section, by 'colors' we mean the colors The right hand side of (51) then becomes Notice that the two tensor products in the curly brackets of (51) have been absorbed by one tensor product, thanks to the convenient definition of the operad F c . The normalized variant is obtained by assuming that in the 'big' tensor product either i D ≥ 2, or i D = 1 but D 1 D = D. As in the paragraph following formula (51) we interpret the product (55) as the space of all vertex-decorations of a planar rooted tree with n + 2 levels and edges colored by fields, such that the root is decorated by M c and the other vertices with F c in such a way that the output and the inputs of the decorations match the colors of the adjacent edges.  Figure 2. We must however be careful, since the fields (= decorations of the vertices) are assigned degree +1, cf. (57), so we are in fact dealing with the equivalence classes of forests in Figure 2 with vertices linearly ordered compatibly with the partial order given by the distance from the soil (= root). We identify a forest F ′ with the forest ǫ · F ′′ , where ǫ ∈ {+1, −1} is the signum of the permutation that brings the order of vertices of F ′ to the order of vertices of F ′′ . The differential contracts the edges, one at a time, and decorates the new vertex thus created by the glued field. Notice that this description is practically identical with the definition of the blob complex in [14]. of the augmented bar construction. This should be compared to the explicit description of the initial terms of the blob complex given on pages 1500-1502 of [14].
We finally arrive at are decorated trees equipped with levels. The chain map φ * (L, P, R) sends a given decorated tree to the sum, with appropriate signs, of all decorated trees with levels whose underlying non-leveled decorated tree equals the given one. Fresse then proved in [6,Theorem 4.1.8] that φ * (L, P, R) is a quasi-isomorphism. Although he assumed simple connectivity, his theorem holds without this assumption, which expresses the folklore fact that the space of levels of a given tree is a contractible groupoid, cf. also [8]. Combining this with isomorphisms (56) and (58)    The vertical map φ * is Fresse's levelization morphism, ℓ * is the map in Theorem C. The remaining maps are either natural isomorphisms, or augmentations, or inclusions, or projections. The vertical isomorphism between row (i) and (ii) comes from Proposition 92, the two vertical isomorphism between rows (iii) and (iv) are that of Proposition 90.