aa r X i v : . [ m a t h . C T ] F e b ON COFIBRATIONS OF PERMUTATIVE CATEGORIES
AMIT SHARMA
Abstract.
In this note we introduce a notion of free cofibrations of permutative categories. Weshow that each cofibration of permutative categories is a retract of a free cofibration. Introduction A permutative category is a symmetric monoidal category whose associativity and unit naturalisomorphisms are identites. Permutative categories have generated significant interest in topology.An infinite loop space machine was constructed on permutative categories in [May78]. A K -theory(multi-)functor from a multicategory of permutative categories into a symmetric monoidal categoryof symmetric spectra, which preserves the multiplicative structure, was constructed in [EM06a].In [EM06b], the K -theory of [EM06a] was enhanced to a lax symmetric monoidal functor. It wasshown in [Man10] that permutative categories model connective spectra.Every symmetric monoidal category is equivalent (by a symmetric monoidal functor) to a per-mutative category. The category of symmetric monoidal categopries SMCAT does NOT have amodel category structure, however its subcategory of permutative categories and strict symmetricmonoidal functors
Perm can be endowded with a model category structure. The category
Perm isisomorphic to the category of algebras over the (categorical) Barrat-Eccles operad. Using this fact,the a model category structure follows from [BM07] and [Lac07, Thm. 4.5]. This model categorystructure is called the natural model category structure of permutative categories.The main objective of this note is to identify a class of cofibrations in the natural model category
Perm , called free cofibrations such that every cofibration in
Perm is a retract of a free cofibration.A useful property of free cofibrations is that cobase changes along a free cofibration preserve acyclicfibrations in the natural model category
Perm . This property allows us to prove that the naturalmodel category
Perm is left proper . Acknowledgments.
The author is thankful to Andre Joyal for proposing the idea of a free-cofibrationand also for many insightful discussions regarding this note.2.
Cofibrations in
Perm and left properness
In this note we define a class of maps called free cofibrations in the natural model category ofpermutative categories
Perm . We show that a strict symmetric monoidal functor is a cofibration in
Perm if and only if it is a retract of a free cofibration. Using this characterization of cofibrations wewill show that the natural model category
Perm is left proper. A characterization of cofibrationsin
Perm was formulated, purely in terms of object functions (which are monoid homomorphisms)of the underlying strict symmetric monoidal functor, in [Sha20]. In order to define free cofibrations,we will start by reviewing some basic notions of permutative categories:
Date : February 25, 2021.
Definition 2.1.
A symmetric monoidal category is called a permutative category or a strict sym-metric monoidal category if it is strictly associative and strictly unital.
Remark . A permutative category is an internal category in the category of monoids.We recall that the forgetful functor U : Perm → Cat has a left adjoint F : Cat → Perm . Definition 2.2.
A monoid M is called a free monoid if there exists a (dotted) lifting monoidhomomorphism whenever we have the following (outer) commutative diagram of monoid homomor-phisms: ∗ / / (cid:15) (cid:15) N p (cid:15) (cid:15) M / / > > ⑥⑥⑥⑥ Q where p is a surjective monoid homomorphism and ∗ is a zero object in the category of monoids. Definition 2.3. A free cofibration of permutative categories is a (strict symmetric monoidal)functor i : A → C whose object function is the inclusion Ob ( i ) : Ob ( A ) → Ob ( A ) ∨ M = Ob ( C ),where M is a free monoid and the coproduct is taken in the category of monoids.The next proposition presents the desired characterization of cofibrations: Proposition 2.4.
A strict symmetric monoidal functor F : C → D is a cofibration in Perm ifand only if it is a retract of a free cofibration by a map that fixes C .Proof. Let us first assume that F is a retract of a free cofibration i : E → M . We observe thatthe object function of a free cofibration has the left lifting property with respect to all surjectivemonoid homomorphisms, therefore each free cofibration is a cofibration in Perm . A retract of acofibration is again a cofibration. Thus, F is a cofibration in Perm .Conversely, let us assume that F is a cofibration in Perm . We have the following (outer)commutative diagram in the category of monoids Ob ( C ) i / / Ob ( F ) (cid:15) (cid:15) Ob ( C ) ∨ F m ( Ob ( D )) p (cid:15) (cid:15) Ob ( D ) L ♠♠♠♠♠♠ Ob ( D )where F m ( Ob ( D )) is the free monoid generated by the set Ob ( D ), i is the inclusion into the co-product and p = Ob ( F ) ∨ ǫ , the summand ǫ : F m ( Ob ( D )) → Ob ( D ) is the counit of the reflection: F m : Set ⇄ Mon : U Since the right vertical homomorphism of monoids is surjective and F is a cofibration by assump-tion, therefore there exists a (dotted) lifting homomorphism L which makes the whole diagramcommutative. Thus Ob ( F ) is a retract of the inclusion i in the category of monoids. We will con-struct a strict symmetric monoidal functor I : C → E whose object function is the inclusion i andshow that F is a retract of I . We begin by constructing the category E : N COFIBRATIONS OF PERMUTATIVE CATEGORIES 3
The object set of E is Ob ( C ) ∨ F ( Ob ( D )). The morphism monoid of E is defined by the followingpullback square in the category of monoids:(1) M or ( E ) p / / p (cid:15) (cid:15) M or ( D ) ( s D ,t D ) (cid:15) (cid:15) ( Ob ( C ) ∨ F ( Ob ( D ))) × ( Ob ( C ) ∨ F ( Ob ( D ))) p × p / / Ob ( D ) × Ob ( D )We will denote the projection map p in the above cartesian square by ( s E , t E ). This pair will besource and target maps for the proposed category E . The projection map p in the above cartesiandiagram restricts to a map between the set of composable arrows in E and D : p c : M or ( E ) × s E = t E M or ( E ) → M or ( D ) × s D = t D M or ( D ) . Now we observe that the composite ( − ◦ D − ) ◦ p c factors through M or ( E ) as follows:(2) M or ( E ) × s E = t E M or ( E ) −◦ E − / / p c (cid:15) (cid:15) M or ( E ) p (cid:15) (cid:15) M or ( D ) × s D = t D M or ( D ) −◦ D − / / M or ( D )The map − ◦ E − in the above commutative diagram provides the composition of the category E .Finally, we define the symmetry natural transformation of E as follows:(3) γ Ez ,z := γ Dp ( z ) ,p ( z ) for each pair of objects z , z ∈ Ob ( E ). This defined a permutative category ( E, − ⊠ E − , γ E ), wherethe tensor product is uniquely determined by the monoid structures on Ob ( E ) and M or ( E ).The commutative diagrams (1), (2) and the definition of the symmetry natural transformation(3) together imply that there is a strict symmetric monoidal functor P : E → D whose objecthomomorphism is p and morphism homomorphism is p . Further P is surjective on objects andalso fully-faithful. This implies that P is an acyclic fibration in the natural model category Perm .Now we construct the free cofibration I : C → E mentioned above. The object homomorphismof I is the inclusion i : Ob ( C ) → Ob ( C ) ∨ F ( Ob ( D )). The morphism homomorphism of I is definedas follows: M or ( I ) := M or ( F ) . In other words, I ( f ) = F ( f ) for each morphism f ∈ M or ( C ). Now we have the following (outer)commutative diagram in Perm : C I / / F (cid:15) (cid:15) E P (cid:15) (cid:15) D L > > ⑦⑦⑦⑦ D Since F is a cofibration and P is an acyclic fibration in the natural model category Perm ,therefore there exists a (dotted) lifting arrow L which makes the entire diagram commutative. Thisimplies that F is a retract of the free cofibration I in the natural model category Perm . (cid:3) A. SHARMA z 3.
Left properness of the natural model category
Perm
In this section we show that the natural model category of permutative categories
Perm is leftproper. We recall that a model category is left proper if the cobase change of a weak-equivalencealong a cofibration is again a weak-equivalence. We will first show that the cobase change of aweak-equivalence along a free cofibration is a weak-equivalence. Using this intermediate result, wewill prove the left properness of
Perm .Let G : A → B be an acyclic fibration in Perm and i A : A → C be a free cofibration thereforethe object monoid of C can be written as a coproduct Ob ( A ) ∨ V , where V is a free monoid. Weobserve that the following commutative square is coCartesian: Ob ( A ) Ob ( i A ) / / Ob ( G ) (cid:15) (cid:15) Ob ( A ) ∨ V Ob ( G ) ∨ id (cid:15) (cid:15) Ob ( B ) i B / / F ( B ) ∨ V We will construct the following pushout square in
Perm : A i A / / G (cid:15) (cid:15) C (cid:15) (cid:15) B / / B ⊔ A C A strict symmetric monoidal functor G : A → B is an acyclic fibration in Perm if and onlyif there exists a unital symmetric monoidal section [Sha20, Cor. 3.5(3)] S : D → C such that GS = id D and a monoidal natural isomorphism ǫ S : SG ∼ = id . Let us fix such a section S : B → A and natural isomorphism ǫ S . Remark . The above characterization of acyclic fibrations implies that S : B → A is a left-adjoint-right-inverse of G : A → B . This means that ǫ S : SG ∼ = id A is a counit of an adjointequivalence whose unit η : GS = id B is the identity natural transformation. This further impliesthat Gǫ S · ηG = id G . In other words, for each a ∈ A , we have the following equality: G ( ǫ S ( a )) ◦ η ( G ( a )) = id G ( a ) . Since the unit natural transformation η is the identity, therefore Gǫ S = G . Remark . Let b , b be a pair of objects in B . Since ǫ S is a monoidal natural transformation,therefore we have the following commutative diagram: SG ( S ( b ) ⊗ S ( b )) ǫ S ( S ( b ) ,S ( b )) / / λ S ( b ,b ) (cid:15) (cid:15) S ( b ) ⊗ S ( b ) S ( GS ( b ) ⊗ GS ( b )) S ( b ) ⊗ S ( b )Thus we have shown that λ S = ǫ S S. N COFIBRATIONS OF PERMUTATIVE CATEGORIES 5
This further implies that Gλ S = Gǫ S S = GS = id B . The unital symmetric monoidal functor S gives us the following unital symmetric monoidalfunctor: S ∨ F ( C ; V ) : B ∨ F ( C ; V ) → A ∨ F ( C ; V ) , where F ( C ; V ) is the full permutative subcategory of C whose object set is the (free) monoid V .We observe that S ∨ F ( C ; V ) is a section of the strict symmetric monoidal functor G ∨ F ( C ; V ) i.e. ( G ∨ F ( C ; V )) ◦ ( S ∨ F ( C ; V )) = id . Moreover, we get a monoidal natural isomorphism ǫ S ∨ F ( C ; V ) : ( S ∨ F ( C ; V )) ◦ ( G ∨ F ( C ; V )) ∼ = id Hence the functor G ∨ F ( C ; V ) is an acyclic fibration in the natural model category Perm by[Sha20, Cor. 3.5(3)].We observe that the free cofibration i A factors as follows:(4) A / / ι A % % ❑❑❑❑❑❑❑❑❑❑ CA ∨ F ( C ; V ) i A,V ssssssssss where ι A : A → A ∨ F ( C ; V ) is the inclusion into the coproduct and i A,V : A ∨ F ( C ; V ) → C is theunique map induced by the inclusions i A : A → C and i V : F ( C, V ) → C Remark . The following commutative square is a coCartesian:(5) A ι A / / G (cid:15) (cid:15) A ∨ F ( C ; V ) G ∨F ( C ; V ) (cid:15) (cid:15) B ι B / / B ∨ F ( C ; V )We observe that the object monoid of C is the same as the object monoid of A ∨F ( C ; V ), namelythe coproduct ( Ob ( A )) ∨ V . This implies that for each c ∈ Ob ( C ) there is the following isomorphismin C : ( i A,V ◦ ( ǫ S ∨ F ( C ; V )))( c ) : ( S ∨ F ( C ; V )) ◦ ( G ∨ F ( C ; V ))( c ) ∼ = c, Now it follows from [Sha20, Prop. 2.7] that there exists a (uniquely defined) functor S C : C → C and a natural isomorphism δ C : id C ∼ = S C . The functor S C is defined on objects as follows: S C ( c ) := ( S ∨ F ( C ; V )) ◦ ( G ∨ F ( C ; V ))( c ) . The following lemma now tells us that S C is a unital symmetric monoidal functor and δ C is amonoidal natural isomorphism: Lemma 3.1.
Given a unital oplax symmetric monoidal functor ( F, λ F ) between two symmetricmonoidal categories C and D , a functor G : C → D , and a unital natural isomorphism α : F ∼ = G ,there is a unique natural isomorphism λ G which enhances G to a unital oplax symmetric monoidalfunctor ( G, λ G ) such that α is a monoidal natural isomorphism. If ( F, λ F ) is unital symmetricmonoidal then so is ( G, λ G ) . A. SHARMA
Proof.
We consider the following diagram: C × C F × F } } G × G ! ! −⊗ C − / / C G } } F ! ! α × α k s α + D × D −⊗ D − / / D This diagram helps us define a composite natural isomorphism λ G : G ◦ ( − ⊗ C − ) ⇒ ( − ⊗ D − ) ◦ G × G as follows:(6) λ G := ( id −⊗ D − ◦ α × α ) · λ F · ( α − ◦ id −⊗ C − ) . This composite natural isomorphism is the unique natural isomorphism which makes α a unitalmonoidal natural isomorphism. Now we have to check that λ G is a unital monoidal natural iso-morphism with respect to the above definition. Clearly, λ G is unital because both α and λ F areunital natural isomorphisms. We first check the symmetry condition [Sha20, Defn. 2.4 OL. 2]. Thiscondition is satisfied because the following composite diagram commutes G ( c ⊗ C c ) α − ( c ⊗ C c ) / / G ( γ C ( c ,c )) (cid:15) (cid:15) F ( c ⊗ C c ) F ( γ C ( c ,c )) (cid:15) (cid:15) λ F ( c ,c ) / / F ( c ) ⊗ D F ( c ) γ D ( F ( c ) ,F ( c )) (cid:15) (cid:15) α ( c ) ⊗ D α ( c ) / / G ( c ) ⊗ D G ( c ) γ D ( G ( c ) ,G ( c )) (cid:15) (cid:15) G ( c ⊗ C c ) α − ( c ⊗ C c ) / / F ( c ⊗ C c ) λ F ( c ,c ) / / F ( c ) ⊗ D F ( c ) α ( c ) ⊗ D α ( c ) / / G ( c ) ⊗ D G ( c )The condition [Sha20, Defn. 2.4 OL. 3] follows from the following equalities α D ( G ( c ) , G ( c ) , G ( c )) ◦ λ G ( c , c ) ⊗ D id G ( c ) ◦ λ G ( c ⊗ C c , c ) =( α ( c ) ⊗ D α ( c )) ⊗ D α ( c ) ◦ α D ( F ( c ) , F ( c ) , F ( c )) ◦ λ F ( c , c ) ⊗ D id F ( c ) ◦ λ F ( c ⊗ C c , c ) ◦ α − (( c ⊗ C c ) ⊗ C c ) =( α ( c ) ⊗ D α ( c )) ⊗ D α ( c ) ◦ id F ( c ) ⊗ D λ F ( c , c ) ◦ λ F ( c , c ⊗ C c ) ◦ F ( α C ( c , c , c )) ◦ α − (( c ⊗ C c ) ⊗ C c ) = id G ( c ) ⊗ D λ G ( c , c ) ◦ λ G ( c , c ⊗ C c ) ◦ G ( α C ( c , c , c )) . If F = ( F, λ F ) is a symmetric monoidal functor then so is G = ( G, λ G ) because (6) is a naturalisomorphism. (cid:3) The section S ∨ F ( C ; V ) provides us with a unital symmetric monoidal functor i A,V ◦ ( S ∨F ( C ; V )) : B ∨ F ( C ; V ) → C which we denote by S F . The unital symmetric monoidal functor S F N COFIBRATIONS OF PERMUTATIVE CATEGORIES 7 has the following Gabriel factorization: B ∨ F ( C ; V ) S F / / Γ S F & & ▼▼▼▼▼▼▼▼▼▼ CG ( S F ) ∆ < < ③③③③③③③③③ By lemma A.1 ( G ( S F ) , − (cid:26) − , γ ) is a permutative category structure. Also, by the same lemma, Γis a strict symmetric monoidal functor. Remark . The following diagram of unital symmetric monoidal functors is commutative: B ∨ F ( C ; V ) S ∨F ( C ; V ) (cid:15) (cid:15) S F / / CA ∨ F ( C ; V ) i A,V / / C S C O O The above commutative diagram implies that for each object z ∈ G ( S F ), λ S F ( z ) = λ S C ( z ).We claim that there exists a strict symmetric monoidal functor P : C → G ( S F ) such that thefollowing diagram, in Perm , is coCartesian:(7) A G (cid:15) (cid:15) i A / / C P (cid:15) (cid:15) B Γ / / G ( S F )where Γ = Γ S F ◦ ι B . The object function of the functor P is the monoid homomorphism Ob ( G ) ∨ V : Ob ( A ) ∨ V → Ob ( B ) ∨ V. For any pair of objects c , c ∈ Ob ( C ), we observe the following equality: G ( S F )( P ( c ) , P ( c )) = C ( S C ( c ) , S C ( c )) . Now we define the morphism function of P as follows: P ( f ) := S C ( f ) , where f is a morphism in C . The functoriality of P follows from that of S C .The object function of P is a monoid homomorphism therefore P ( c ⊗ C c ) = P ( c ) (cid:26) P ( c ), foreach pair of objects c , c ∈ Ob ( C ). The following commutative diagram shows that P ( f ⊗ C f ) = A. SHARMA P ( f ) (cid:26) P ( f ), for each pair of maps ( f , f ) ∈ C ( c , c ) × C ( c , c ): c ⊗ C c f ⊗ C f / / δ ∼ = (cid:15) (cid:15) c ⊗ C c δ ∼ = (cid:15) (cid:15) S C ( c ⊗ C c ) P ( f ⊗ C f ) / / S C ( c ⊗ C c ) S F ( P ( c ) (cid:26) P ( c )) P ( f ) (cid:26) P ( f ) / / λ S F (cid:15) (cid:15) S F ( P ( c ) (cid:26) P ( c )) λ S F (cid:15) (cid:15) S F ( P ( c )) ⊗ C S F ( P ( c )) P ( f ) ⊗ C P ( f ) / / S F ( P ( c )) ⊗ C S F ( P ( c ))Thus, we have defined a strict symmetric monoidal functor P which is fully faithful. Further, eachobject of G ( S F ) is isomorphic to one in the image of P . Thus,discussion P is an equivalence ofcategories. Proposition 3.2.
The commutative square (7) is coCartesian.Proof.
In order to show that (7) is coCartesian, it is sufficient to show that the following commu-tative square is coCartesian, in light of factorization (4) and remark 4: A ∨ F ( C ; V ) G ∨F ( C ; V ) (cid:15) (cid:15) i A,V / / C P (cid:15) (cid:15) B ∨ F ( C ; V ) Γ S F / / G ( S F )We will show that whenever we have the following (outer) commutative diagram, there exists aunique dotted arrow L which makes the whole diagram commutative in Perm : A ∨ F ( C ; V ) G ∨F ( C ; V ) (cid:15) (cid:15) i A,V / / C P (cid:15) (cid:15) R (cid:17) (cid:17) B ∨ F ( C ; V ) Γ S F / / T / / G ( S F ) L " " ❋❋❋❋❋ X Since Ob (Γ S F ) is the identity, therefore the object homomorphism Ob ( L ) has to be the same as Ob ( T ) in order to make the diagram commutative, therefore we define Ob ( L ) = Ob ( T ). Themorphism function of L is defined as follows: L z ,z := R S F ( z ) ,S F ( z ) : G ( S F )( z , z ) = C ( S F ( z ) , S F ( z )) → X ( L ( z ) , L ( z )) . for each pair of objects z , z ∈ Ob ( G ( S F )). This defines a functor L which makes the diagramabove commutative (in Cat ). In order to verify that L is a strict symmetric monoidal functor, it is N COFIBRATIONS OF PERMUTATIVE CATEGORIES 9 sufficient to show that for each pair of maps f : z → z , f : z → z in G ( S F ),(8) L ( f (cid:26) f ) = L ( f ) ⊗ X L ( f ) = R ( f ) ⊗ X R ( f ) . We recall that the map f (cid:26) f is defined by the following commutative diagram: S F ( z ) ⊗ C S F ( z ) f ⊗ C f / / S F ( z ) ⊗ C S F ( z ) S F ( z ⊗ B z ) f (cid:26) f / / λ S F O O S F ( z ⊗ B z ) λ S F O O Since R is a strict symmetric monoidal functor, therefore R ( f ⊗ C f ) = R ( f ) ⊗ X R ( f ). Now itsufficient to show that Rλ S F = id , in order to establish the equalities in (8). We observe that λ S F = i A,V ( λ S ∨F ( C ; V ) ). Since G ∨ F ( C, V ) is an acyclic fibration, it follows from remark 3 that G ∨ F ( C, V ) λ S ∨F ( C ; V ) = id . Since T ◦ G ∨ F ( C, V ) = R ◦ i A,V , it follows that R ( λ S F ) = id . Theuniqueness of the object functor of L is obvious. The uniqueness of the morphism homomorphismof L can be easily checked. (cid:3) The main objective of this section is to show that the natural model category
Perm is leftproper. The next lemma serves as a first step in proving the main result. The lemma follows fromthe above discussion:
Lemma 3.3.
In the natural model category
Perm a pushout of a weak-equivalence along a freecofibration is a weak-equivalence.Proof.
In light of the facts that each weak equivalence in a model category can be factored as anacyclic cofibration followed by an acyclic fibration and acyclic cofibrations are closed under cobasechange, it is sufficient to see that the cobase change of an acyclic fibration is a weak-equivalence.This follows from the dicussion above. (cid:3)
Now we state and prove the main result of this note:
Theorem 3.4.
The natural model category of permutative categories
Perm is a left proper modelcategory.Proof.
We will show that a pushout P ( F ; q ) of a weak equivalence F : A → D in Perm alonga cofibration q : A → B in Perm is a weak-equivalence. We consider the following commutative diagram: C P ( F ; l ) (cid:15) (cid:15) ❆❆❆❆❆❆❆❆ A q / / F (cid:15) (cid:15) r / / B l > > ⑥⑥⑥⑥⑥⑥⑥⑥ P ( F ; q ) (cid:15) (cid:15) B P ( F ; q ) (cid:15) (cid:15) D / / / / / / P ❅❅❅❅❅❅❅❅ P s ❅❅❅❅❅❅❅❅ P Since F is a cofibration therefore by proposition 2.4 there exists a free cofibration r : A → C such that F is a retract of r by a map that fixes A . The top left commutative square in the abovediagram is coCartesian. The map P ( F ; l ) is a pushout of F along the free cofibration r and thereforea weak-equivalence by lemma 3.4. Now the result follows from the observation that the diagonalcomposite P → P s → P , in the above diagram, is the identity map and the commutativity of theabove diagram. (cid:3) Appendix A. Gabriel Factorization of symmetric monoidal functors
In this appendix we construct a
Gabriel Factorization of a unital symmetric monoidal functorbeyween permutative categories. Our construction factors a unital symmetric monoidal functorinto an essentially surjective strict symmetric monoidal functor followed by a fully-faithful unitalsymmetric monoidal functor.
Lemma A.1.
Each unital symmetric monoidal functor F : C → D between permutative categoriescan be factored as follows: C F / / Γ F " " ❉❉❉❉❉❉❉❉ DG ( F ) ∆ F < < ③③③③③③③③ where Γ F is a strict symmetric monoidal functor which is identity on objects and ∆ is fully-faithful.Proof. We begin by defining the permutative category G ( F ). The object monoid of G ( F ) is thesame as Ob ( C ). For a pair of objects c , c ∈ Ob ( C ), we define G ( F )( c , c ) := C ( F ( c ) , F ( c )) . The Gabriel factorization of the underlying functor of F gives us the following factorization in Cat : C F / / Γ F " " ❉❉❉❉❉❉❉❉ DG ( F ) ∆ F < < ③③③③③③③③ N COFIBRATIONS OF PERMUTATIVE CATEGORIES 11
We will show that the functor Γ F is strict symmetric monoidal and ∆ F is unital symmetric monoidal.We define a symmetric monoidal structure on G ( F ) next which we denote by ( G ( F ) , (cid:26) , γ ). For anypair of objects c , c ∈ Ob ( G ( F )), we define c (cid:26) c := c ⊗ C c . For a pair of maps f : c → c and f : c → c , we define f (cid:26) f to be the following arrow: F ( c ⊗ C c ) λ F ∼ = (cid:15) (cid:15) f (cid:26) f / / F ( c ⊗ C c ) λ F ∼ = (cid:15) (cid:15) F ( c ) ⊗ D F ( c ) f ⊗ D f / / F ( c ) ⊗ D F ( c )It is easy to establish that − (cid:26) − is a bifunctor: Let f : c → c and f : c → c be another pairof arrows in G ( F ). Now we consider the following commutative diagram: F ( c ⊗ C c ) λ F ∼ = (cid:15) (cid:15) f (cid:26) f / / F ( c ⊗ C c ) λ F ∼ = (cid:15) (cid:15) f (cid:26) f / / F ( c ⊗ C c ) λ F ∼ = (cid:15) (cid:15) F ( c ) ⊗ D F ( c ) f ⊗ D f / / F ( c ) ⊗ D F ( c ) f ⊗ D f / / F ( c ) ⊗ D F ( c )The above diagram tells us that:( f (cid:26) f ) ◦ ( f (cid:26) f ) = ( f ◦ f ) (cid:26) ( f ◦ f )because the composite map in the bottom row of the above diagram namely ( f ⊗ D f ) ◦ ( f ⊗ D f ) isthe same as ( f ◦ f ) ⊗ D ( f ◦ f ). The tensor product − (cid:26) − on G ( F ) is strictly associative becausethe object set of G ( F ) is a monoid and the tensor product of morphisms is associative because thetensor product of morphisms in G ( F ) is inherited from that in D which is strictly associative. Thesymmetry natural transformation γ is defined on objects as follows: γ c ,c := F ( γ Cc ,c ) . Let f : c → c and f : c → c be a pair of maps in G ( F ). The following commutative diagramshows us that γ is a natural isomorphism: F ( c ⊗ C c ) γ c ,c , , λ F / / f ⊠ f (cid:15) (cid:15) F ( c ) ⊗ D F ( c ) γ DF ( c ,F ( c / / f ⊗ D f (cid:15) (cid:15) F ( c ) ⊗ D F ( c ) f ⊗ D f (cid:15) (cid:15) λ F / / F ( c ⊗ C c ) f ⊠ f (cid:15) (cid:15) F ( c ⊗ C c ) γ c ,c λ F / / F ( c ) ⊗ D F ( c ) γ DF ( c ,F ( c / / F ( c ) ⊗ D F ( c ) λ F / / F ( c ⊗ C c )which shows that γ is a natural transformation. The following equalities verifies the symmetrycondition: γ c ,c ◦ γ c ,c = F ( γ Cc ,c ) ◦ F ( γ Cc ,c ) = F ( γ Cc ,c ◦ γ Cc ,c ) = id. This defines a permutative category structure on the category G ( F ). Using the definition of thesymmetric monoidal structure on G ( F ), one can easily check that Γ F is a strict symmetric monoidalfunctor. (cid:3) References [BM07] Clemens Berger and Ieke Moerdijk,
Resolution of coloured operads and rectification of homotopy algebras ,2007.[EM06a] A. Elmendorf and Michael A. Mandell,
Rings, modules, and algebras in infinite loop space theory , Advancesin Mathematics (2006), 163–228.[EM06b] A. D. Elmendorf and M. A. Mandell,
Permutative categories, multicategories and algebraic K-theory , Alg.and Geom. Top. (2006), no. 4, 163–228.[Lac07] S. Lack, Homotopy-theoretic aspects of 2-monads. , Journal of Homotopy and Related Structures (2007),no. 2, 229–260 (eng).[Man10] M. A. Mandell, An inverse K-theory functor , Doc. Math. (2010), 765–791.[May78] J. P. May, The spectra associated to permutative categories , Topology (1978), no. 3, 225–228.[Sha20] A. Sharma, Symmetric monoidal categories and Γ -categories , Th. and Appl. of categories (2020), no. 14,417–512. Email address : [email protected]@kent.edu