Matriarch: Mathematics Supplement

Transcription

1 Matriarch: Mathematics Supplement Ravi Jagadeesan, Tristan Giesa, David I. Spivak, Markus J. Buehler May 7, 2015 Abstract In this paper, we describe the mathematical foundations of the software package Matriarch presented in our recent paper [2]. Contents 1 Introduction Connection between the Matriarch program and the operad Mat S Plan of the supplementary document Notation and conventions Signed attachment and enumeration The category of signed attachment Counts and enumerators Bond structure Primitive terminal types Oriented rigid bodies Bonding operad Realizing Bond S as signed attachment Bonded blocks, the Kan extension The counter Axis structure Axis twisters The axis operad Axes and operations Building points and operations Realizing AxTyp as signed attachment Axis blocks The counter

2 5 Materials architecture The ambient category of operads: 2-generation The operad of materials architecture Building blocks Functoriality The Set -valued algebra functors The forgetful functor and the forgetful natural transformation The universal reversal operad and the tautological natural transformation The universal bonding category and the universal reversal category inclusion The fibered product functor and the outer fibered product natural transformation Parameter spaces Proof of Theorem Introduction In [2], we discuss a Python program for material architecture, based on operads. The idea is that a sequence of protein building blocks can be arranged according to various instructions to form a new protein building block. The operad Mat S, called the operad of materials architecture, establishes the set of building block types and the permissible building instructions. The actual amino acid arrangements are encoded in a Mat S algebra, BuildBlk: Mat S Set. This supplementary document contains the precise mathematical definitions of the operad Mat S and the algebra Buildblk. To the extent possible, this follows the notation of the user s guide to the Matriarch software, [1]. The user s guide [1] provides many examples of the behavior of the operations defined in this article: we have only provided examples in this document where they are not already presented in the user s guide. 1.1 Connection between the Matriarch program and the operad Mat S Given building instructions for which the inputs and the output building blocks have compatible bond types, one can compose building instructions into a program. Sometimes, two programs of basic building instructions might return the same result if they are given the same inputs many situations in which this happens are described in Supplement S These are formalized as equations 2

3 in the operad. For experts, forming a program corresponds to composing a tree of morphisms in the operad, while applying a building instruction to a sequence of building blocks corresponds to evaluating the image of the building instruction under an appropriate set-valued functor at the sequence of building blocks. Building block replacement, as defined in [3], can be performed by altering the arguments passed to a building instruction, while more complex structures can be studied by changing the building instruction itself. 1.2 Plan of the supplementary document We will begin each section with a diagram explaining the important content of that section. We will then proceed by explaining each operad, functor, and natural transformation in the opening diagram. Our goal is to build the operad Mat S of materials architecture, and the building block algebra BuildBlk: Mat S Set. We construct Mat S, which is roughly the theory of bonded amino acids arranged in space, as a fiber product. Namely, we construct operads Bond, which governs the operations of forming bonds, AxTyp, which governs spacial operations, and Att, the common aspects of the two. Mat S AxTyp Bond S Att We will then construct the algebra BuildBlk as a fibered product as well. The construction of the auxilliary operads and their algebras are given in Sections 2, 3, and 4. The constructions of Mat S and BuildBlk are given in Section 5. In Section 6, we discuss the functoriality of (Mat S, BuildBlk) with respect certain parameters. 1.3 Notation and conventions We assume familiarity with the language of categories (see [5, 6]) and operads. Whenever we speak of operads, we mean symmetric colored operads (see [4]). We will work in ZFC with the Grothendieck Universe axiom. Let U 0 U 1 U 2 be a sequence of universes. We call elements of U 0 small, elements of U 1 large, and elements of U 2 very large. Let Set denote the (large) category of small sets, let Cat denote the (large) category of small categories, and let CAT denote the (very large) category of large categories. It will be evident which categories and operads are small and which ones are not (most of the time, we deal only with small categories and small operads, except for operads related to Set and in Section 6). 3

4 By Set we mean the operad underlying the symmetric monoidal category (Set,, { }). We write Set + to refer to the operad underlying the coproduct symmetric monoidal structure on the category of sets, and we write (Set + ) op to denote the operad underlying its opposite. For example, Hom (Set+ ) op(x, Y ; Z) = Hom Set(Z, X Y ). We denote the free monoid monad by List: Set Set. Given elements l 1, l 2 List(S), we abuse notation to write l 1 l 2 for the concatenation of l 1 and l 2. For a nonnegative integer n, let [n] = {1, 2,..., n}. The absence of any natural transformation arrows indicates that a diagram (strictly) commutes. 2 Signed attachment and enumeration Count Enum Set + π Count Att Count Set Our first goal in this section is to define an operad Att; almost everything in this paper happens over Att. Our next goal is to define a functor Count: Att Set, which basically encodes addition of natural numbers (coproducts of enumerators): almost all Set -valued algebras lie over a pullback of Count. In this sense, (Att, Count) is the base over which materials architecture happens. We then define something like an assembly theory for the enumerators of building blocks, namely a functor Enum: Count Set The category of signed attachment, Att Definition Let Ob Att = {A}. We define its morphisms by presentation, as follows: generators: a morphism combine Hom Att (A, A; A); a morphism reverseorbs Hom Att (A; A); relations: combine (combine, id) = combine (id, combine); 4

5 reverseorbs reverseorbs = id; combine σ (reverseorbs, reverseorbs) = reverseorbs combine where σ is the transposition in the symmetric group S 2, which acts on the 2-ary morphisms in a (symmetric) operad. We call Att the operad of signed attachment. 2.2 Counts and enumerators: Count and Enum Definition We now define an algebra Count: Att Set as follows: Count(A) = N; Count(combine)(x, y) = x+y; Count(reverseOrbs)(x) = x Even though reverseorbs has no effect in the algebra Count, it will help control the morphisms in the fiber product operad Mat S. Intuitively, reverseorbs plays a role in Bond S and AxTyp, and Att needs to capture all common parts of those two operads. In defining the algebra of axis blocks (building blocks with only an axis structure), namely AxBlk: AxTyp Set, we will need to handle the role of reverseorbs more carefully. In particular, we will need to encapsulate the fact that reverseorbs reverses the order of the ORBs that comprise a building block. To this end, we will define a functor Enum. Definition Define a functor Enum: Count (Set + ) op as follows. On objects, let Enum(n) = [n]. On morphisms, define Enum as follows: (Action of reverseorbs) Let n reverseorbs(k) = n + 1 k. (Action of combine) Let m, m N. Denote elements of Enum(m) Enum(m ) as k for elements of Enum(m) and k for elements of Enum(m ). Let { m,m combine(k) = k if k m (k m) if k > m. It is not difficult to verify that the relations defining Att are respected, so that Enum is in fact well-defined and functorial. 3 Bond structure Orb Neg S Set I S Bond S BondBlk:=(I S )! Orb 5

6 C C N N N C N N C C C N C C N N N C N C C C N Figure 1: A building block with terminals connecting to its involuted partner. In this section we define an operad Bond S that encodes attachment of bond structures via an algebra BondBlk: Bond S Set. This algebra is constructed as the left Kan extension of the Orb functor. This functor is defined on a category Neg S, which encodes reversibility of oriented rigid bodies. 3.1 Primitive terminal types, S Let S = (S, κ) be a set with involution κ: S S. We will typically consider the free monoid List(S) on S, and denote multiplication as concatenation, e.g., the product of s 1 and s 2 would be denoted s 1 s 2. For any natural number ξ N and element s S, we may write s ξ to denote the ξ-fold product of s. The monoid List(S) comes equipped with a canonical involution List(κ), which by abuse of notation, we often denote by κ: List(S) List(S). Example In the case of polypeptides, we let S = {C, N}, where C stands for carboxyl and N stands for amine, and let κ: S S be the function such that κ(c) = N and κ(n) = C; it is an involution because κ κ = id S. The free monoid associated to S consists of all words in S, e.g., CCNNNC, an example of which is shown in Figure 1. This is to be the negative terminal of a building block. Applying the involution we get κ(ccnnnc) = NNCCCN. This is to be the positive terminal of the building block. 3.2 Oriented rigid bodies, Neg S and Orb The space of oriented rigid bodies will have the structure of a Neg S -algebra Orb: Neg S Set, because oriented rigid bodies have bond types and can have 6

7 their orientations reversed. Definition Let S be a set. We define a category, called the reversal category for S, denote Neg S. The set of objects in Neg S is Ob Neg S := S S. For all X, Y S, there is a (non-identity) morphism Composition is defined as X,Y reverseorbs Hom((X, Y ), (Y, X)). Y,XreverseOrbs X,Y reverseorbs = (X,Y ) id. The morphisms X,Y reverseorbs are the only non-identity morphisms in Neg S. Example In the case of polypeptides, Neg S has four objects, given by elements of {C, N} 2. Let AA = {Ala,...} denote the set of amino acid names. The algebra Orb is defined as Orb(N, N) = Orb(C, C) = Orb(C, N) = Orb(N, C) = AA Orb(reverseOrbs) = id. We can recover the set ORB of oriented rigid bodies in the sense of [1, Section 4.2.1] as Ob Orb. 3.3 Bonding operad, Bond S Definition Let S = (S, κ) be a set with involution. We define an operad, called the bonding operad on S, denoted Bond S. The set of objects in Bond S is Ob Bond S := List(S) List(S). Morphisms in Bond S are presented as follows: generators: for all (X, Y ) Ob Bond S, a unary morphism X,Y reverseorbs Hom BondS ((X, Y ), (Y, X)); 7

8 for all Z 0 = (X 0, Y 0 ), Z 1 = (X 1, Y 1 ) Ob Bond S and ξ {0, 1}, a binary morphism Z 0,Z 1 combine NoBond,ξ Hom BondS (Z 0, Z 1 ; (X 0 X ξ 1, Y ξ 0 Y 1 )); for all Z 0 = (X 0, Y 0 ), Z 1 = (X 1, Y 1 ) Ob Bond S such that X 1 = κ(y 0 ), a binary morphism X 0,Y 0,Y 1 combine Bond,0 Hom BondS (Z 0, Z 1 ; (X 0, Y 1 )); We abuse notation and suppress the lower-left-hand subscripts, X, Y, Z 0, Z 1 in morphisms, e.g., denoting Z0,Z 1 combine NoBond,1 by combine NoBond,1. All relations are taken when and only when the composites are defined. relations: (Associativity) for all b, b {Bond,NoBond} and ξ {0, 1} with (Bond, 1) / {(b, ξ), (b, ξ)}, combine b,ξ (id, combine b,ξ) = combine b,ξ (combine b,ξ, id) ; (( 1) 2 = 1) reverseorbs reverseorbs = id; (Anti-involutionarity of reverseorbs) for all b {Bond,NoBond} and ξ {0, 1} with (b, ξ) (Bond, 1), combine σ b,ξ (reverseorbs, reverseorbs) = reverseorbs combine b,ξ where σ is the transposition in the symmetric group S 2, which acts on the 2-ary morphisms in a (symmetric) operad; (Forgetful forgetfulness) combine NoBond,0 (combine NoBond,0, id) (id, combine NoBond,0, id) =combine NoBond,0 (combine NoBond,0, id) (id, combine NoBond,1, id)). Let Bond f S be the free operad on the generators of Bond S, and denote by π Bond : Bond f S Bond S the canonical quotient. The subscript Bond (resp. NoBond) in combine Bond,ξ (resp. combine NoBond,ξ ) controls indicates that a bond is formed (resp. not formed) between the first and second arguments at their interface. The subscript ξ = 0 in combine b,ξ indicates that the left (resp. right) terminal of the output building block is the left terminal 8

9 of the first argument (resp. right terminal of the second argument), such as in attach and. For example, the morphism combine Bond,0 represents the building instruction called attach in the User s Guide [1, Section 4.3.4], and the morphism combine NoBond,0 represents the building instruction of [1, Section 4.3.7]. If ξ = 1, then the left (resp. right) terminal of the output is the composite of the left (resp. right) terminals of the arguments. For example, the morphism combine NoBond,1 represents the building instruction called overlay in the User s Guide [1, Section 4.3.6]. Definition The reversal category inclusion for S, denoted I S : Neg S Bond S is defined as follows. On objects, I S : S S List(S) List(S) is the unit inclusion, sending an element to a one-entry list. On morphisms, X,Y reverseorbs is sent to X,Y reverseorbs. Recall that (Set ) Neg S parameterizes possible sets of ORBs with primitive terminal type set S. Pushing forward and pulling back by I S will let us identify ORBs with basic building blocks (in the sense of [1, Section 4.2.2]) and build more complicated building blocks from the basic building blocks. 3.4 Realizing Bond S as signed attachment: the functor V S The functor V S will be needed in the definition of Mat S as a fibered product. Intuitively, V S interprets Bond S as signed attachment, because Att should be thought of as the universal operad of signed attachment. Definition Define V S : Bond S Att on objects by sending all Z Ob Bond S to the unique object A Ob Att. On generating morphisms, define V S by V S (reverseorbs) = reverseorbs; for all b {Bond, NoBond} and ξ {0, 1} with (b, ξ) (Bond, 1), let V S (combine b,ξ ) = combine. The fact that if relations in Bond S push forward to relations in Att under V S guarantees that V S is well-defined on morphisms and is a functor. 3.5 Bonded blocks, the Kan extension BondBlk Definition Define the building block algebra BondBlk as the Kan extension BondBlk := (I S )! Orb. 9

10 We will now give an explicit construction of BondBlk. More precisely, we will construct a functor BondBlk : Bond S Set and an injective natural transformation λ : BondBlk BondBlk with λ injective. The functor BondBlk will use the bond structure construction of the User s Guide [1, Section 4.2.1]. We can then interpret elements of BondBlk(Z) as bond structures satisfying certain coherence conditions for all bond types Z List(S) List(S). Given a set T, let e L, e R denote the projections of T T onto the first and second components, respectively. When Z is a bond type (i.e. an element of List(S) List(S), we obtain the left and right terminal types of Z as e L (Z) and e R (Z), respectively. Definition Let Orb: Neg S Set be a functor, and let (X, Y ) Ob Bond S. Suppose that X = x 1 x 2 x k and Y = y 1 y 2 y m with x i S and y i S for all i. Let ORB = Ob Orb be the set of objects in the Grothendieck category of elements of Orb, and let bondt ype: ORB S S be the projection of ORB to S S = Ob Neg S. A bonded block of bond type (X, Y ) for Orb is the data of a nonnegative integer n (we call [n] the enumerator of the block) a function sig : [n] ORB, called the signature a subset B [n] [n], called the set of bonds two subsets L, R [n], called the interface structure satisfying the following conditions: (Interface compatibility) We have L = k and R = m. Writing L = {l 1 < < l k ) and R = {r 1 < < r m }, we have e L (l i ) = x i and e R (r i ) = y i for all i. (Bond compatibility) If (i, j) B, then e L (bondt ype(j)) = κ(e R (bondt ype(i)). We are now prepared to define give protein-like bonded blocks the structure of an algebra for Bond S. We will use the notation n, n, n 1 to denote the values of n for bonded blocks Γ, Γ, Γ 1, respectively, and similarly for other sub-scripts and super-scripts on Γ and the other data defining a bonded block. Definition Let Orb: Neg S Set be a functor. Define a functor BondBlk : Bond f S Set on objects by taking (X, Y ) to the set of bonded blocks of type (X, Y ). Define BondBlk on the generators of Bond f S as follows. 10

11 (Action of reverseorbs) Let (X, Y ) Ob Bond S, and let Γ be a building block of type (X, Y ). Define Γ = BondBlk(reverseOrbs)(Γ) by n = n; sig (k) = reverseorbs(sig Γ (n + 1 k)); (i, j) B (n + 1 j, n + 1 i) B (L, R ) = (n + 1 R, n + 1 L). (Action of combine NoBond,ξ ) Let (X 1, Y 1 ), (X 2, Y 2 ) Ob Bond S, and let Γ i be a building block of type (X i, Y i ) for i = 1, 2. Define by Γ = BondBlk (combine)(γ 1, Γ 2 ) n = n 1 + n { 2 sig sig 1 (k) if k n 1 (k) = ; sig 2 (k n 1 ) if k > n 1 B = B 1 ((n 1, n 1 ) + B 2 ); { L L 1 if ξ = 0 = L 1 (n 1 + L 2 ) if ξ = 1 ; { R n 1 + R 2 if ξ = 0 = L 1 (n 1 + R 2 ) if ξ = 1. (Action of combine Bond,0 ) Work in the notation of the previous bullet point, and assume that X 2 = κ(y 1 ). Let R 1 = (r 1, r 2,..., r m ) and let L 2 = (l 1, l 2,..., l m ). Define Γ = BondBlk (combine)(γ 1, Γ 2 ) by n = n 1 + n { 2 sig sig 1 (k) if k n 1 (k) = ; sig 2 (k n 1 ) if k > n 1 B = B 1 ((n 1, n 1 ) + B 2 ) {(r i, l i + n 1 ) i [m]} L = L 1 and R = n 1 + R 2. The interface compatibility of Γ 1, Γ 2 and the condition X 2 = κ(y 1 ) ensure the bond compatibility of Γ. It is not difficult to verify from the definition of a bonded block that BondBlk is indeed a functor from Bond S to Set. The following proposition is similarly elementary, and hence we omit the proof. 11

12 Proposition The functor BondBlk respects the relations of Bond S and hence descends to a functor BondBlk : Bond S Set. We are now ready to compare BondBlk to BondBlk via a natural transformation φ: BondBlk BondBlk. Definition Define a natural transformation λ: Orb I S (BondBlk ) as follows. Let Z Ob Neg S and let x Orb(Z). Define Γ = λ Z (x) by n Γ = 1; sig Γ (1) = x; B Γ = ; L Γ = R Γ = {1}. We abuse notation and write λ(x) for λ Z (x). We call λ the basic bonded block natural transformation, we say that a bonded block ( Z an element of Ob BondBlk ) is basic if it is in the image of λ, and we call λ(x) the basic bonded block associated to x. In the language of the User s Guide [1, Sections and 4.2.3], the basic building block λ(x) is simply the bond structure of chain( x ). Definition Let φ: BondBlk BondBlk be the adjugate of λ under the adjunction (I S )! I S (recall that BondBlk := (I S)! Orb). Conjecture The natural transformation φ: BondBlk BondBlk is injective. Assuming Conjecture 3.5.7, we can interpret elements of BondBlk(Z) as bonded blocks of bond type Z satisfying additional coherency conditions that guarantee that they can be built from basic bonded blocks. 3.6 The counter, BondBlk V S Count We will actually define natural transformations γ S : BondBlk VS Count and δ S : BondBlk VS Count that are compatible with the natural transformation φ. Definition Define the natural transformation γ S : Orb IS V S Count by sending every element to 1. The natural transformation BondBlk VS Count is defined to be the adjugate δ S of γ S. The following proposition is not difficult to verify from definition of an adjugate and gives intuition for the natural transformation δ S. Proposition The natural transformation δ S : BondBlk VS Count is the restriction of the natural transformation δ S : BondBlk VS that sends a bonded block to the size of its enumerator (i.e. sends Γ to n Γ ). 12

13 4 Axis structure CountAxes ζ 2 Count Enum (Set + ) op (4.1) Set + Pts ζ 1 Axes π Count Axes π Axes Set AxTyp U Att Count Set CountAxes (ζ 2 Enum,ζ 1 Pts) (Set + ) op Set + (4.2) ζ 1 Axes Hom Att U π Axes AxTyp AxBlk Set Count In this section the goal is to define an operad AxTyp and also an algebra AxBlk: AxTyp Set. We give an explicit construction of the operad AxTyp, which governs building instructions that effect axis structures, in Section 4.2. The systematic construction of AxBlk will require the construction of several auxiliary operads. Recall from [1, Section 3.2.1] that an axis structure consists of an axis, together with additional data indexed by an enumerator S. First we need to define the action of building instructions on building block axes, which we do by defining the AxTyp-algebra Axes. Then we define a functor Pts: Axes Set +, which encodes how a fixed ORB transforms under building instructions. We can then consider ORBs indexed by the enumerator Enum by forming the pullback operad CountAxes and taking the left Kan extension as in Diagram Axis twisters When dealing with clutch directions, it will be useful to make the identifications S 1 R 2 R 3, where the second inclusion is the inclusion of the first two factors. We make the usual identification R/2πZ = S 1 via θ exp(2πiθ). Let p z : R 3 R denote the projection onto the z axis, and let Tt z denote translation by t in the positive z-direction in R 3. 13

14 Definition A curved axis consists of functions f, d: R 0 R 3 such that f is differentiable, with f (x) = 1, and d(x) is a unit normal to f (x) for all x R 0. Definition A total clutch is a function d: R >0 S 1. Definition An axis twister is a pair W = (W map, W new ), where W map is a curved axis and W new is a total clutch. 4.2 The axis operad, AxTyp Definition We define an operad AxTyp, with one object, Ob AxTyp = {A}. Morphisms in AxTyp are presented as follows: generators: a unary morphism reverseorbs Hom AxTyp (A; A); for all axis twisters W, a unary morphism twist W Hom AxTyp (A; A); for all s R 0, a unary morphism pad s Hom AxTyp (A; A); for all ρ {Sep, UnSep}, a binary morphism combine ρ Hom AxTyp (A, A; A); for all g Euc, a unary morphism moveorbs g Hom AxTyp (A; A); relations: (Associativity) for ρ {Sep, UnSep}, combine ρ (id, combine ρ ) = combine ρ (combine ρ, id) : (( 1) 2 = 1) reverseorbs reverseorbs = id; (Anti-involutionarity of reverseorbs) combine σ Sep (reverseorbs, reverseorbs) = reverseorbs combine Sep where σ is the transposition in the symmetric group S 2, which acts on the 2-ary morphisms in a (symmetric) operad; some trivial identities: for 0 R 0, 1 Euc, pad 0 = moveorbs 1 = id. 14

15 for all s, s R 0, pad s+s = pad s pad s ; for all s R 0, pad s combine UnSep = combine UnSep (pad s, pad s ); for all s R 0, pad s combine Sep = combine Sep (pad s, id); for all s R 0 and all g Euc, pad s moveorbs g = moveorbs g pad s ; for all g, h Euc, moveorbs g moveorbs h = moveorbs gh. Let AxTyp f denote the free operad on the generators of AxTyp, and let π AxTyp : AxTyp f AxTyp be the canonical quotient. 4.3 Axes and operations, Axes: AxTyp Set Definition An axis is a pair (l, θ) with l R >0 and θ S 1. We call l the length and θ the torsional angle of an axis. Definition Define a functor Axes: AxTyp f Set as follows. On objects, let Axes(A) be the set of all axes. Define Axes on the generators of AxTyp f as follows. (Action of reverseorbs) Define Axes(reverseOrbs) = id. (Action of twist W ) Suppose that W map = (f map, d map ) and W new = d new. Let W = (W map, W new ). Let (l, θ) be an axis, and let { } l = max 0, max p z(f(t)). t [0,l] Define (Action of pad s ) Define (Action of combine 0 ) Define (Action of combine 1 ) Define Axes(twist W )(l, θ) = (l, d new (l )). Axes(pad s )(l, θ) = (l + s, θ). Axes(combine 0 )((l, θ), (l, θ )) = (max{l, l }, θ). Axes(combine 1 )((l, θ), (l, θ )) = (l + l, θ + θ ). (Action of moveorbs g ) Let Axes(moveOrbs g ) be the identity. Proposition The functor Axes: AxTyp f Set respects the relations of AxTyp and hence descends to a functor Axes: AxTyp Set. 15

16 4.4 Building points and operations, Pts: Axes Set + Definition A building point is a quadruple P = (p, N 1, N 2, π) of a point p R 3, orthogonal unit tangent vectors N 1, N 2 T p R 3. Let BP denote the set of building points. Note that BP has an obvious action of Euc. Definition Define a functor Pts: Axes Set + as follows. On objects, send an axis L = (l, θ) Ob Axes to Pts(L) = BP [0, l]. Define Pts on generating morphisms as follows: (Action of reverseorbs) Let C = (l, θ) be a building block axis. Let T be the Euclidean automorphism of R 3 that sends (0, 0, l) to (0, 0, 0) and has Jacobian cos θ sin θ 0 dt = sin θ cos θ Define Pts( C reverseorbs)(p, α) = (T (p), l α). (Action of twist W ) Let W = (W map, W new ) be an axis twister, let W map = (f, d), and let C = (l, θ) be a building block axis. For t [0, l], let T t denote the Euclidean automorphism that sends the points (0, 0, t) to f(t) and sends the tangent vector z (resp. x ) to f (t) (resp. d(t)). Define Pts( C twist W )(P, α) = (T α (P ), max {0, p z (f(α))}). (Action of pad s ) Let C = (l, θ) be a building block axis. Define Pts( C pad s ) to be the obvious injection BP [0, l] BP [0, l + s]. (Action of moveorbs) For all building block axes C, define Pts( C moveorbs g )(P, α) = (g(p), α). (Action of combine Sep ) Let C 1 = (l 1, θ) and C 2 = (l 2, θ 2 ) be building block axes. Let M be the matrix cos θ sin θ 0 M = sin θ cos θ Let T 1 be the identity Euclidean automorphism and let T 2 denote the composite of translation by l 1 units in the positive z-direction followed multiplication by M. Define Pts( C1,C 2 combine Sep ((P, α), C i )) = (T i (C i ), δ i,2 l 1 + α). 16

17 (Action of combine UnSep ) Let C 1 = (l 1, θ) and C 2 = (l 2, θ 2 ) be building block axes. Let Pts( C1,C 2 combine UnSep ) be the coproduct of the obvious injections BP [0, l 1 ] BP [0, max{l 1, l 2 }] and BP [0, l 2 ] BP [0, max{l 1, l 2 }]. It is not difficult to verify that Pts respects the relations between the generators (coming from the relations in AxTyp), and hence indeed defines a functor from Axes to Set Realizing AxTyp as signed attachment: the functor U The functor U : AxTyp Set will be needed in the definition of Mat S as a fibered product. Definition Define U : AxTyp Att on objects to be the unique function, and on generating morphisms by U(reverseOrbs) = reverseorbs; for all axis twisters W, U(twist W ) = id; for all s R 0, U(pad s ) = id; for all ρ {Sep, UnSep}, U(combine ρ ) = combine; for all g Euc, U(moveOrbs) = id. The fact that if relations in AxTyp push forward to relations in Att under U guarantees that U is well-defined on morphisms and is a functor. 4.6 Axis blocks, AxBlk Definition Every operad and functor in Diagram 4.1, other than the operad CountAxes and the projections ζ 1 and ζ 2, has been defined. Define CountAxes to be the fiber product of operads CountAxes := Axes Att Count, completing Diagram 4.1, and let ζ 1, ζ 2 be the canonical projections of CountAxes onto Axes and Count, respectively. The key property of CountAxes is the following obvious lemma. 17

18 Lemma The projection ζ 1 : CountAxes Axes exhibits CountAxes as π Axes U Count. Hence, the projection π Axes ζ 1 exhibits CountAxes as Axes U Count. We need the functor Hom: (Set + ) op Set + Set, where each of these is an operad. Definition Define the functor AxBlk: AxTyp Set as the left Kan extension AxBlk := (π Axes )! (ζ 1 )! (Hom (ζ 2Enum, ζ 1Pts)). Thus we have the following diagram CountAxes (ζ 2 Enum,ζ 1 Pts) (Set + ) op Set + ζ 1 Axes Hom π Axes AxTyp AxBlk Set Our goal is to give a concrete description of AxBlk. To do so, we will define AxBlk in Definition and then exhibit a canonical isomorphism in Proposition Definition Define a functor AxBlk : AxTyp f Set as follows. On the unique object A Ob AxTyp, let ( ) AxBlk (A) = Hom Set ([n], BP [0, l]). On morphisms, let: (l,θ) Ob Axes n=1 (Action of reverseorbs) Let n N, Define r : [n] [n] by r(k) = n + 1 k. For (l, θ) Ob Axes, n N, and ω : [n] BP [0, l], let AxBlk (reverseorbs)((l, θ), n, ω) = ( Axes(reverseOrbs)(l, θ), n, Pts ( (l,θ)reverseorbs ) ω r ) ; (Action of twist W, pad s, and moveorbs) Let f be among twist W, pad s, and moveorbs. Define AxBlk (f)((l, θ), n, ω) = ( Axes(f)(l, θ), n, Pts ( (l,θ)f ) ω ). 18

19 (Action of combine ρ ) Let n, n be positive integers. Let r : [n+n ] [n] [n ] be defined by { k in the first summand if k n r(k) = k n in the second summand if k > n. Define AxBlk (combine ρ )(((l, θ), n, ω), ((l, θ ), n, ω )) =(Axes(combine ρ ) ( (l, θ), (l, θ )), n + n, Pts ( (l,θ),(l,θ )combine ρ ) (ω ω ) r ) The following proposition is not difficult to verify from the defining relations of AxTyp, and so we omit the proof. Proposition The functor AxBlk : AxTyp f Set respects the relations of AxTyp and hence descends to a functor AxBlk : AxTyp Set. We will now compare the functors AxBlk and AxBlk via the universal property of the Kan extension. Let δ denote the canonical isomorphism δ : Hom(Hom (ζ 2Enum, ζ 1Pts), ζ 1π AxesAxBlk) Hom(AxBlk, AxBlk ) coming from the adjunction (π Axes )! (ζ 1 )! ζ 1π Axes. Definition Define ν (l,θ),n : Hom Set ([n], BP [0, l]) AxBlk (A) by ν (l,θ),n (ω) = ((l, θ), n, ω). Noting that the functor π Axes ζ 1, being a pullback of a faithful functor, is faithful, we obtain a description of the morphisms in CountAxes. The following lemma is then not difficult to verify. Lemma The functions ν (l,θ),n define a natural transformation ν : Hom (ζ 2Enum, ζ 1Pts) ζ 1π AxesAxBlk. We can then obtain the desired isomorphism between AxBlk and AxBlk. Proposition The natural transformation η = δ(ν) is a natural isomorphism from AxBlk to AxBlk. Proof. Immediate from Lemma and the explicit description of F! for F a Grothendieck fibration. 19

20 4.7 The counter, AxBlk U Count The natural transformation AxBlk U Count comes from the universal property of the left Kan extension. Indeed, by Lemma 4.6.2, the functor (π Axes )! (ζ 1 )! induces an equivalence of categories from (Set ) CountAxes to the slice category (Set ) AxTyp /U Count. We can describe the natural transformation more concretely using Proposition Lemma The pullback of the canonical natural transformation AxBlk U Count along δ(ν) 1 sends ((l, θ), n, ω) to n. 5 Materials architecture A Mat S S AxTyp B S U Bond S Att V S AxBlk BuildBlk B S BondBlk A S AxBlk A S U Count BondBlk Count Set We put everything together in this section to define the operad Mat S and its algebra BuildBlk. The operad Mat S is defined as a pullback of operads defined in previous sections. However, for technical reasons, this limit is not taken in the category Oprd of small operads. First we define an appropriate ambient category of operads, as described in Section 5.1. We then complete the construction of Mat S as the pullback, and then describe (almost) explicitly in Section 5.2. Finally in Section 5.3, we define the algebra BuildBlk as a fibered product, and then provide an explicit construction of it. 5.1 The ambient category of operads: 2-generation The construction of limits is poorly behaved in Oprd with regard to operads defined by presentation, and hence we must define Mat S as a fibered product in a different ambient category of operads. Definition We call a (symmetric) operad 2-generated if its morphisms are generated by its unary and binary morphisms. Let the category of 2-generated operads, denoted by Oprd 2, be the full subcategory of Oprd on the 2-generated small (symmetric) operads. 20

21 When dealing with operads whose morphisms are presented in terms of unary and binary morphisms, the categorical product in Oprd 2 yields more physically natural categories than the categorical product in Oprd. The key property of Oprd 2 is the following theorem. Theorem Let U : Oprd 2 Oprd be the inclusion. Then U has a right adjoint TwoGen: Oprd Oprd 2 which is given on objects by sending M to the suboperad of M on Ob M generated by the unary and binary morphisms of M. Therefore, Oprd 2 is a coreflexive full subcategory of Oprd. Proof. Define TwoGen on objects by sending an operad M to the suboperad of M generated by unary and binary morphisms, so that there is a natural inclusion ɛ M : TwoGen(M) M. On morphisms, if f : M M is a functor, then we get a composite f : TwoGen(M) M M. However, any morphism in the image of f is a composite of unary and binary morphisms and therefore lies in Hom TwoGen(M ). Thus, we get a functor TwoGen(f) such that the diagram TwoGen(M) ɛ M M TwoGen(f) TwoGen(M ) ɛ M f M commutes. Such an TwoGen(f) is unique because ɛ M is a monomorphism (i.e., is faithful). Defining TwoGen(f) by this universal property makes it clear that TwoGen is a functor. It remains to show that TwoGen is right adjoint to U. We do this by giving a unit and counit. By definition of TwoGen, the morphisms ɛ define a natural transformation U TwoGen id Oprd. Because TwoGen U = id MultiCat2, we can take as counit η the identity natural transformation on id MultiCat2. It is clear that TwoGenɛ ηtwogen = id TwoGen and ɛu Uη = id U, and therefore U TwoGen. Because the counit is a natural isomorphism, Oprd 2 is a coreflexive subcategory of Oprd. The key consequence of Theorem is that TwoGen preserves limits, and thus we can evaluate limits in Oprd 2 by evaluating limits in Oprd and then applying TwoGen. 5.2 The operad of materials architecture, Mat S The following definition formalizes the interpretation of Att as the common part of Bond S and AxTyp. 21

22 Definition Define Mat S as the fibered product Bond S AxTyp in Att Oprd 2. Let B S and A S be the projections of Mat S onto the first and second factors, respectively. For sake of concreteness, we will also explicitly construct an operad similar to Mat S. Definition Define the operad mat S as follows. Let Ob mat S = List(S) List(S). Morphisms in mat S are presented as follows. generators: for all (X, Y ) Ob mat S, a unary morphism (X,Y )reverseorbs Hom mats ((X, Y ); (Y, X)); for all Z Ob mat S and all axis twisters W, a unary morphism Ztwist W Hom mats (Z; Z); for all Z Ob mat S and all g Euc, a unary morphism ZmoveOrbs g Hom mats (Z; Z). for all Z 0 = (X 0, Y 0 ), Z 1 = (X 1, Y 1 ) Ob mat S, ξ {0, 1} and ρ {Sep, UnSep}, a binary morphism Z 0,Z 1 combine NoBond,ξ,ρ Hom mats Hom mats (Z 0, Z 1 ; (X 0 X ξ 1, Y ξ 0 Y 1 )); for all Z 0 = (X 0, Y 0 ), Z 1 = (X 1, Y 1 ) Ob mat) S such that X 1 = κ(y 0 ) and all ρ {Sep, UnSep}, a binary morphism Z 0,Z 1 combine Bond,0,ρ Hom mats Hom mats (Z 0, Z 1 ; (X 0, Y 1 )); We abuse notation and suppress the lower-left-hand subscripts, X, Y, Z 0, Z 1 in morphisms, e.g., denoting Z0,Z 1 combine NoBond,1,UnSep by combine NoBond,1,UnSep. All relations are taken when and only when the composites are defined. relations: (Associativity) for all b, b {Bond, NoBond and ξ {0, 1} with (Bond, 1) / {(b, ξ), (b, ξ)}, and ρ {Sep, UnSep}, combine b,ξ,ρ (id, combine b,ξ,ρ) = combine b,ξ,ρ (combine b,ξ,ρ, id). 22

23 (( 1) 2 = 1) reverseorbs reverseorbs = id; (Anti-involutionarity of reverseorbs) for all b {Bond,NoBond} and ξ {0, 1} with (b, ξ) (Bond, 1), combine σ b,ξ,sep (reverseorbs, reverseorbs) = reverseorbs combine b,ξ,sep where σ is the transposition in the symmetric group S 2, which acts on the 2-ary morphisms in a (symmetric) operad; (Forgetful forgetfulness) for all ρ 1, ρ 2, ρ 3 {Sep, UnSep}, combine NoBond,0,ρ1 (combine NoBond,0,ρ2, id) (id, combine NoBond,0,ρ3, id) =combine NoBond,0,ρ1 (combine NoBond,0,ρ2, id) (id, combine NoBond,1,ρ3, id)). some trivial identities: for 0 R 0, 1 Euc, pad 0 = moveorbs 1 = id. for all s, s R 0, pad s+s = pad s pad s ; for all s R 0, b {Bond, NoBond}, and ξ {0, 1} with (b, ξ) (Bond, 1), pad s combine b,ξ,unsep = combine b,ξ,unsep (pad s, pad s ); for all s R 0, b {Bond, NoBond}, and ξ {0, 1} with (b, ξ) (Bond, 1), pad s combine b,ξ,sep = combine b,ξ,sep (pad s, id); for all s R 0 and all g Euc, pad s moveorbs g = moveorbs g pad s ; for all g, h Euc, moveorbs g moveorbs h = moveorbs gh. Let mat f S be the free operad on the generators of mat S, and let π mat : mat f S mat S be the canonical quotient. We need to compare Mat S to mat S. Definition Define a functor CompBond S : mat f S Bond S as follows. On objects, let CompBond S be the identity endomorphism of List(S) List(S). On morphisms, define CompBond S as follows: CompBond S ( Z reverseorbs) = Z reverseorbs CompBond S ( Z twist W ) = id Z CompBond S ( Z moveorbs g ) = id Z CompBond S ( Z0,Z 1 combine b,ξ,ρ ) = Z0,Z 1 combine b,ξ. 23

24 Definition Define a functor CompAx S : mat f S AxTyp as follows. On objects, let CompBond S be the unique map from Ob mat f S to Ob AxType = {A}. On morphisms, define CompBond S as follows: CompBond S ( Z reverseorbs) = Z reverseorbs CompBond S ( Z twist W ) = Z twist W CompBond S ( Z moveorbs g ) = Z moveorbs g CompBond S ( Z0,Z 1 combine b,ξ,ρ ) = Z0,Z 1 combine ρ. It is not difficult to verify the following lemma. Lemma The functors CompBond S and CompAx S respect the defining relations of mat S and therefore descend to functors CompBond S : mat S Bond S and CompAx S : mat S AxTyp. Furthermore, we have U CompAx S = V S CompBond S as functors from mat S to Att. We are now ready to compare mat S and Mat S using the universal property that defined Mat S. Note that mat S is manifestly 2-generated. Definition Let Comp: mat S Mat S be the unique functor making the diagram mat S CompAx S CompBond S Comp A S Mat S AxTyp B S U Bond S V S Att commute, coming from the definition of Mat S as a fibered product in Oprd 2. Theorem The functor Comp S is surjective on objects and full. Proof. Surjectivity on objects is obvious. We just need to prove fullness. By Theorem and the description of limits in the Oprd as taking limits of object and morphism sets, it suffices to prove that the image of Comp S contains all pairs (f, g) Hom Bond S Hom AxTyp such that V S f = Ug and f, g have arity 1 or 2. Let (f, g) be such a pair, and let u = V S f(= Ug). The definition of mat S ensures that if f, g are each a generator or an identity, then (f, g) is actually the image of a generator morphism or an identity of mat S under Comp S. We will divide into cases based on the arity of f, g to prove that (f, g) lies in the image of Comp S in general. 24

25 Case 1: f and g have arity 1. It suffices to find a positive integer n and morphisms f 1,...,, f n, g 1,..., g n such that f i, g i are all unary, each f i, g i is either an identity or a generator, and f = f n f 1 g = g n g 1. and V S f i = Ug i for all indices i. This will imply the case of the theorem in question, because each pair (f i, g i ) is guaranteed to be the image of a generator or an identity of mat S under Comp S. Given an operad M, let M 1 denote the underlying category of M (i.e., the suboperad of M on all the objects consisting of only the unary morphisms). Because Bond S does not contain any nullary morphisms, the category Bond 1 S is generated by the unary generators of Bond S. It follows that Bond 1 S = Neg S (where Neg S S is identified as a subcategory of Bond S via the faithful functor I S ), because the image of I S contains all the unary generators of Bond S and Bond S does not have any null-ary morphisms. Suppose that f Hom BondS ((X, Y ); (Z, W )). It follows that (Z, W ) {(X, Y ), (Y, X)}. The definition of the functor V S ensures that (Z, W ) = (X, Y ) if and only if V S f = id (and (Z, W ) = (Y, X) otherwise). Because AxTyp does not contain any 0-arity morphisms, we can write g = g n g 1 with each g n a unary generator of AxTyp. Let u i = Ug i, and let f i = reverseorbs if u i = reverseorbs and let f i = id otherwise, with argument types chosen so that the domain (resp. codomain) of f n f 1 is (X, Y ) (resp. (Z, W )). It is possible to choose such argument types because (Z, W ) = (X, Y ) if u n u 1 = reverseorbs. and (Z, W ) = id if u n u 1 = reverseorbs. Because each u i is either reverseorbs or id, we know that V S f i = u i for all i, so that V S (f n f 1 ) = Ug = V S f. Note that the functor V S : Neg S = Bond 1 S Att 1 is faithful, because both the source and the target are generated by morphisms reverseorbs with the same relations. It follows that f = f n f 1, as desired. Case 2: f and g have arity 2. It suffices to find morphisms f 1,1, f 1,2, f 2, f 3, g 1,1, g 1,2, g 2, g 3 such that f I, g I are unary for I 2 and binary generators for I = 2, f = f 3 f 2 (f 1,1, f 1,2 ) g = g 3 g 2 (g 1,1, g 1,2 ) 25

26 and V S f I = Ug I for all indices I. Indeed, the previous case shows that (f I, g I ) lies in the image of Comp for I 2, and the morphisms f 2, g 2 are generators, so that (f 2, g 2 ) is the image of a binary generator of mat S. Permuting the arguments of f, g if necessary, write and g = g 3 g 2 (g 1,1, g 1,2 ) f = f 3 f 2 (f 1,1, f 1,2 ) with g 2 = combine ρ, f 2 = combine τ b,ξ for some τ S 2, and all other morphisms f I, g I unary. This is possible because AxTyp (resp. Bond S ) contains no nullary morphisms and the morphisms combine ρ (resp. combine b,ξ ) are the only binary generators of AxTyp (resp. Bond S ). We claim that it is possible to choose f I such that τ is the identity in S 2. Let σ S 2 be the involution. Suppose that f = f 3 f 2 (f 1,1, f 1,2) with f 2 = combine σ b,ξ and all other morphisms f I unary. Let f 3 = f 3 reverseorbs, let f 1,i = reverseorbs f 1,i, and let f 2 = (f 2) σ. The antiinvolutionarity of reverseorbs in Bond S ensures that which implies the claim. f = f 3 f 2 (f 1,1, f 1,2 ), Therefore, we can and will assume that τ is the identity element of S 2. We will eliminate τ from the notation. We are now in the situation that V S f 3 combine (V S f 1,1, V S f 1,2 ) = V S f = Ug = Ug 3 combine (Ug 1,1, Ug 1,2 ). We claim that Ug I = V S f I for all I. This is clear for I = 2. Note that Ug I, V S f I {reverseorbs, id} for I {(1, 1), (1, 2), 3}, because there are only two unary morphisms in Att. Therefore, there are only finitely many (at most 64) cases to check. We omit the straightforward verification. The theorem follows. In light of Theorem 5.2.7, any building instruction (i.e., any morphism in Mat S ) can be expressed as a composite of primitive building instructions (i.e., images of the generators of mat S under Comp S, possibly with arguments permuted using the symmetric structure on the operad mat S ). This means that the building instructions described in the User s guide [1] indeed generate the operad 26

27 of materials architecture (in the case in which S = {C, N} and κ is the unique non-trivial automorphism of S). We have not discussed the faithfulness of Comp S. This is difficult, because the trick of taking a fibered product in the category of two-generated operads does not elucidate a generating set of relations for Mat S in the way that it controlled a generating set of morphisms for Mat S. Open Question What is a complete set of relations for mat S that would make Comp S an isomorphism of operads? 5.3 Building blocks, BuildBlk Notation Let C be a 2-generated and let A, B Ob Oprd 2 /C be operads over C with structure maps A: A C and B : B C, respectively. Let F : C Set be an algebra, and let D Ob (Set ) A /A F and E Ob (Set ) B /B F be Set -valued algebras for A and B over the pullbacks of F. Let π A, π B denote the projections from A B to A, B, respectively, where the fibered product is C taken in Oprd 2. Define the outer fibered product of D and E over F as D E := π F AD πbe = πad πbe. πa A F πb B F We can also take outer fibered products of morphisms d, e, f, denoted by d f e, whenever A d = B e = f. Definition Define and BuildBlk = BondBlk Count AxBlk BuildBlk = BondBlk AxBlk. Count Assuming Conjecture 3.5.7, the natural transformation φ id id: BuildBlk BuildBlk is injective, so that we can interpret elements of Ob BuildBlk as elements of Ob BuildBlk satisfying certain coherence conditions. Definition We will use the notation of Definition A building block of bond type (X, Y ) for Orb is the data of a nonnegative integer n (we call [n] the enumerator of the block) a function sig : [n] ORB, called the signature a subset B [n] [n], called the set of bonds; 27

28 two subsets L, R [n], called the interface structure a building block axis C = (l, θ), called the axis; a function P : [n] BP, called the locator; a function α: [n] [0, l], called the axis projector satisfying the following conditions: (Interface compatibility) We have L = k and R = m. Writing L = {l 1 < < l k ) and R = {r 1 < < r m }, we have e L (l i ) = x i and e R (r i ) = y i for all i. (Bond compatibility) If (i, j) B, then e L (bondt ype(j)) = κ(e R (bondt ype(i)). Let BuildBlk 0 (X, Y ) denote the set of building blocks of bond type (X, Y ). Using the fact that limits in (Set ) Mat S can be computed pointwise, we obtain the following proposition Proposition For all (X, Y ) Ob Mat S, the maps of forgetting the signature, bonds and interface structure (resp. forgetting the axis, locator, and axis projector) induce a bijection BuildBlk 0 (X, Y ) BuildBlk (X, Y ). In light of Proposition 5.3.4, there is a canonical way to interpret elements of BuildBlk(X, Y ) as building blocks of bond type (X, Y ) assuming Conjecture Functoriality Orb MatBuildBlk Alg (6.1) π Orb Set Z/2 Mat π Alg Oprd Set Z/2 Bond I Neg Oprd 2 /Att χ Alg/Count AxTyp Att AxBlk Count Oprd 2 /Att Ξ Oprd Alg/Count ι (6.2) Alg Ξ Neg CAT 28 Alg

29 The construction of Mat S depends on the choice of a set with involution (S, κ), and the construction of BuildBlk additionally depends on the choice of a Neg S -algebra Orb: Neg S Set, where Neg S depends on (S, κ) as well. In this section, we will give organize the parameter space of triples P = (S, κ, Orb) into a category so that (Mat S, BuildBlk) depends functorially on P. The scientific application of this section is twofold: one can add user-defined amino acids by changing the functor Orb (see the User s guide [1, Section 4.2.5]), and one can add new bond types to S, e.g., S = {C, N, 5, 3 } to enable DNA bonding. The goal will be to construct Diagram 6.1. For a set with involution S = (S, κ), the operad Mat(S) is Mat S. For an algebra Orb: Neg S Set, the pair (Mat S, BuildBlk) is MatBuildBlk(S, Orb). The commutativity of Diagram 6.1 embodies the fact that the construction of Mat S does not depend on the choice of oriented rigid bodies (which is specified by the algebra Orb: Neg S Set). More precisely, we will prove the following theorem. Theorem 6.1. The functions (S, κ) Mat S and (S, κ, Orb) (Mat S, BuildBlk) are the object functions of functors Mat: Set Z/2 Oprd and MatBuildBlk: Orb Alg making Diagram 6.1 commute, where we identify Ob Alg = Ob (Set ) M. M Oprd We will apply the Grothendieck construction (for presheaves of categories) to Diagram 6.2 in order to construct the categories and functors involved in statement of Theorem 6.1. The commutativity of Diagram 6.1 will then be clear. Remark 6.2. For a functor F : C D, the functor F! : (Set ) C (Set ) D will appear more often than F. Put differently, The structure map π Orb : Orb Set Z/2 is an op-fibration instead of a fibration. There is a scientific reason for this occurrence. Given a set ORB of oriented rigid bodies over a set S with involution and a Z/2-equivariant function f : S T, it is natural to regard an oriented rigid body x ORB of type s as an oriented rigid body of type f(s). The pullback construction, given g : T S, it very unnatural. As a result, we use op-fibrations instead of fibrations and! instead of. Remark 6.3. One could prove functoriality by explicit computation. Our approach avoids explicit computation as much as possible and explains why the constructions of the previous section are automatically functorial. 29

30 6.1 The Set -valued algebra functors, Alg: Oprd CAT and Alg/Count: Oprd 2 /Att CAT Definition Define the covariant functor Alg : Oprd CAT defined by Alg(C) = (Set ) C Alg(F ) = F! for all C Ob Cat, for all F Hom Oprd. The functor Alg/Count allows us to deal with algebras over a pullback of Count (e.g., to form outer fibered products over Count). Definition Define the covariant functor Alg/Att : Oprd 2 /Att CAT defined by Alg/Count(C) = (Set ) C /C Count for all C Ob Oprd 2 /Att, Alg/Count(F ) = F! for all F Hom Oprd 2, where C : C Att denotes the structure map of an operad C Ob Oprd 2 /Att. The universal property of F! ensures that Alg/Count is well-defined on morphisms. The restriction of the domain of Alg/Count to Oprd 2 /Att Oprd/Att is not essential for the discussion of this subsection. It is useful when dealing with the fibered product functor AxTyp to ensure that we form fibered products Att of operads in Oprd 2 instead of in Oprd. 6.2 The forgetful functor and forgetful natural transformation, Ξ: Oprd 2 /Att Oprd and ι: Alg/Count Alg Ξ There is a forgetful functor Ξ: Oprd 2 /Att Oprd that is the composite Ξ: Oprd 2 /Att Oprd 2 Oprd that is the composite of the forgetful functor from the slice category Oprd 2 /Att to Att with the inclusion Oprd 2 Oprd. That is, Ξ forget the structure map to Att and the property of being 2-generated of an element of Ob Oprd 2 /Att. We will use Ξ to forget the fact that Mat S lies in Ob Oprd 2 /Att and regard Mat S as an operad. We will also want to forget the fact that BuildBlk has the structure of a functor over A S U Count = BS V S Count. To this end, let M Ob Oprd 2/Att and let M : M Att be the structure map of M. There is an forgetful functor ι M : Alg/Count(M) = (Set ) M /M Count (Set ) M = (Alg Ξ) (M). 30