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Home , Connected category, Coproduct, Polynomial functor

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In Theorem 4.8 we prove an analogous result characterizing Dirichlet polynomials:

Theorem 1.2. For a functor  : Finop → Fin, the following are equivalent: 1.  is a Dirichlet polynomial. 2.  is a sum of representables. 3.  sends connected colimits to limits – or equivalently,  preserves wide pushouts.

We will also show that Dir is equivalent to the arrow category of finite sets,

Dir ≃ Fin→, and in particular that Dir is an elementary topos. If one allows arbitary sums of functors represented by finite sets, one gets analytic functors in the covariant case—first defined by Joyal in his seminal paper on combina- torial species [Joy81]—and Dirichlet functors in the contravariant case—first defined by Baez and Dolan and appearing in Baez’s This Week’s Finds blog [BD]. Baez and Dolan also drop the traditional negative sign in the exponent (that is, they use =B where =−B usually appears), but also find a nice way to bring it back by moving to groupoids. Here, we drop the negative sign and work with finite sets to keep things as simple as possible. Similar considerations hold with little extra work for infinite Dirichlet series or power series, and even more generally, by replacing Fin with Set.

2 Polynomial and Dirichlet functors

Recall that a co-representable functor Fin → Fin is one of the form Fin(:, −) for a finite set : = {‘1’, ‘2’,..., ‘:’}. We denote this functor by y: and say it is represented by : ∈ Fin. Similarly, a (contra-) representable functor Finop → Fin is contravariant functor of the form Fin(−, :); we denote this functor by :y. The functors y− and −y are the contravariant and covariant Yoneda embeddings,

y: ≔ Fin(:, −) and :y ≔ Fin(−, :).

For example y3(2)  8 and 3y(2)  9. Note that the functor 0y  0 is not the initial object in Dir; it is given by

1 if B = 0 0y(B) = (0 if B ≥ 1.

0 The coefficient 00 of 1 = y in a polynomial % is called its constant term. We refer to y the coefficient zc ≔ 00 of 0 in a Dirichlet series  as its zero-content term. Rather y than having no content, the content of the functor zc·0 becomes significant exactly when it is applied to zero.

2 Example 2.1. The reader can determine which Dirichlet polynomial (y) ∈ Dir as in Eq. (1) has the following terms

y ··· 5 4 3 210 (y) ··· 96 48 24 12 6 7

Hint: its zero-content term is zc = 4. The set %(1) (resp. the set (0))has particular importance;it is the set ofpure-power terms y: in % (resp. the pure-exponential terms :y in ). For example if % = y2 +4y+4 and  = 2y + 4 + 4·0y then %(1) = (0) = 9.

Definition 2.2. A is a functor % : Fin → Fin that can be expressed as a sum of co-representable functors. Similarly, we define a Dirichlet functor to be a functor  : Finop → Fin that can be expressed as a sum of representable presheaves (contravariant functors):

%(1) (0) ?8 y % = y and  = (38 ) . (2) = = Õ8 1 Õ8 1 ( ) ( ) ( ) = % 1 Fin( ) ( ) =  0 Fin( ) That is, % - 8=1 ?8 ,- and  - 8=1 -, 38 as functors applied to - ∈ Fin. Í Í See Theorem 1.1 above for well-known equivalent conditions in Poly and Theo- rem 4.8 below for a Dirichlet analogue.

3 The categories Poly and Dir

For any small category , let Fin denote the category whose objects are the functors  → Fin and whose morphisms are the natural transformations between them.

Definition 3.1. The category of polynomial functors, denoted Poly, is the (skeletonof the) full of FinFin spanned by sums % of representable functors. The category op of Dirichlet functors, denoted Dir, is the (skeleton of the) full subcategory of Fin(Fin ) spanned by the sums  of representable presheaves.

While we will not pursue it here, one can take PolySet to be the full subcategory of functors Set → Set spanned by small of representables, and similarly for DirSet.

Lemma 3.2. The set of polynomial maps % → & and Dirichlet maps  →  are given by the following formulas:

Poly(%,&) ≔ &(?8) and Dir(,) ≔ (38). 8∈Ö%(1) 8∈Ö(0) Example 3.3. Let % = 2y2, & = y + 1, and let  = 2 · 2y and  = 1 + 0y. Then there are nine (9) polynomial morphisms % → &, zero (0) polynomial morphisms & → %, one (1) Dirichlet  → , and eight (8) Dirichlet morphisms  → .

3 Remark 3.4. Sums and products of polynomials in the usual algebraic sense agree exactly with sums and products in the categorical sense: if % and & are polynomials, i.e. objects in Poly, thentheircoproductistheusualalgebraicsum %+& of polynomials, and similarly their product is the usual algebraic product %& of polynomials. The same is truefor Dir: sums and products of Dirichlet polynomials in the usual algebraic sense agree exactly with sums and products in the categorical sense.

Formal structures. We review some formal structures of the categories Poly and Dir; all are straightforward to prove. There is an adjoint quadruple and adjoint 5- as follows, labeled by where they send objects = ∈ Fin, % ∈ Poly,  ∈ Dir:

=·0y %(0) ⊥ (0) ⊤ ⊥ = = Fin⊤ Poly Fin⊥ Dir (3) %(1) (1) ⊤ ⊥ =y =y

All five of the displayed functors out of Fin are fully faithful. For each : : Fin the functors % 7→ %(:) and  7→ (:) have left adjoints, namely = 7→ =y: and = 7→ =·:y respectively. These are functorial in : and in fact extend to two-variable adjunctions Fin × Poly → Poly and Fin × Dir → Dir. Indeed, for = ∈ Fin and %,& ∈ Poly (respectively , ∈ Dir), we have

Poly(=%,&)  Poly(%,&=)  Fin(=, Poly(%,&)), Dir(=,)  Dir(,=)  Fin(=, Dir(,)), where =% and = denote =-fold coproducts and %= and = denote =-fold products. Consider the unique function 0 → 1. The induced by it, denoted  : (1)→ (0), is equivalent to two natural transformations on Dir via the adjunctions in Eq. (3):

 =·0y → =, (1) −−→ (0), = → =y. (4)

The one labeled  is also (0!), where0!: 0 → 1 is the unique function of that type. The composite of two polynomial functors Fin → Fin is again polynomial, (% ◦ &)(=) ≔ %(&(=)); this gives a nonsymmetric monoidal structure on Poly. The monoidal unit is y. Day convolution for the monoidal structure provides symmetric monoidal structure ⊗ : Poly × Poly → Poly, for which the monoidal unit is y. This monoidal structure—like the Cartesian monoidal structure—distributes over + We can write an explicit formula for % ⊗ &, with %,& as in Eq. (2):

%(1) &(1) % ⊗ & = y?8 @9 (5) = = Õ8 1 Õ9 1 We call this the Dirichlet product of polynomials, for reasons we will see in Remark 4.1.

4 The Dirichlet monoidal structure is closed; that is, for any ,& : Poly we define

[,&] ≔ & ◦ (08y), (6) 8Ö:(1) for example [=y, y]  y= and [y= , y]  =y. For any polynomial  there is an (− ⊗ )⊣ [, −] adjunction Poly(% ⊗ ,&)  Poly(%, [,&]). (7) In particular we recover Lemma 3.2 using Eqs. (3) and (6). The cartesian monoidal structure on Poly is also closed, Poly(% × ,&)  Poly(%,&), and the formula for & is similar to Eq. (6):  & ≔ & ◦ (08 + y). 8Ö:(1) If we define the global sections functor Γ: Poly → Finop by Γ% ≔ Poly(%, y), or ex- Γ = = plicitly (%) [%, y](1) 8 ?8, we find that it is left adjoint to the Yonedaembedding

Î =7→y= Finop ⊤ Poly . Γ%←%

Each of the categories Poly and Dir has pullbacks, which we denote using “fiber product notation”  × . We can use pullbacks in combination with monad units % : % → %(1) and  :  → (0) arising from Eq. (3) to recover Eq. (2):

%(1) (0) = = % % ×%(1) ‘8’ and   ×(0) ‘8’. = = Õ8 1 Õ8 1 Remark 3.5. By a result of Rosebrugh and Wood [RW94], the category of finite sets is characterized amongst locally finite categories by the existence of the five left adjoints op to its Yoneda embedding : 7→ H: : Fin → FinFin . The adjoint sextuple displayed in (3) is just the observation that five of these six functors restrict to the subcategory Dir.

4 Poly and Dir in terms of bundles

There is a bijection between the respective object-sets of these two categories

 Ob(Poly) −→ Ob(Dir) = = :8 y y 7→ (:8) . (8) = = Õ8 1 Õ8 1 We call this mapping the Dirichlet transform and denote it using an over-line % 7→ %.  We willsee inTheorem 4.6 that this bijection extends to an equivalence Polycart Dircart between the of cartesian maps.

5 Remark 4.1. With the Dirichlet transform in hand, we see why % ⊗ & can be called the Dirichlet product, e.g. in Eq. (5). Namely, the Dirichlet transform is strong monoidal with respect to ⊗ and the cartesian monoidal structure × in Dir:

% ⊗ & = % × &.

Proposition 4.2. There is a one-to-one correspondence between the set of polynomials in one variable, the set of Dirichlet polynomials, and the set of ( classes of) functions : B → C between finite sets.

Proof. We already established a bijection % 7→ % between polynomials and finite Dirichlet series in Eq. (8). Given a finite Dirichlet series , we have a function  : (1)→ (0) as in Eq. (4). ≔ C y ≔ −1 And given a function : B → C, define  8=1(38) , where 38  (8) for each 1 ≤ 8 ≤ C. (N.B. Rather than constructing  from  by hand, one could instead use a  Í certain orthogonal factorization system on Dir.) It is easy to see that the roundtrip on Dirichlet series is identity, and that the round-trip for functions is a natural isomorphism. 

We will upgrade Proposition 4.2 to an equivalence Polycart ≃ Dircart between certain subcategories of Poly and Dir in Theorem 4.6. Example 4.3. Under the identification from Proposition 4.2, both the polynomial 2y3 + y2 + 3 and the Dirichlet series 2·3y + 1·2y + 3·0y correspond to the function

(1, 3) (2, 3) • •  ( )  (1, 2) (2, 2) (3, 2) 8  1 • • • (1, 1) (2, 1) (3, 1) • • •  (9)

  1 2 3 4 5 6 6 (0) • • • • • •

We can think of a function : B → C, e.g. that shown in (9), as a bundle of fibers −1(‘8’), one for each element ‘8’ ∈ C. In Definition 4.4 we define two different notions of morphism between bundles. We will see in Theorem 4.6 that they correspond to morphisms in the categories Poly and Dir. For any function ′ : B′ → C′ and function 5 : C → C′, denote by 5 ∗(′) the pullback function as shown ′ ′ B × ′ B B C y 5 ∗(′) ′ C C′ 5

Definition 4.4. Let : B → C and ′ : B′ → C′ be functions between finite sets. • ′ a bundle morphism consists of a pair ( 5 , 5♯) where 5 : C → C is a function and ∗ ′ 5♯ :  → 5 ( ) is a morphism in the slice category over C;

6 5 ♯ ♯ ′ ′ 5 ′ ′ B C × ′ B B B C × ′ B B C y C y 5 ∗′ ′ 5 ∗′ ′  C C′  C C′ 5 5

Figure 1: The categories Bun and Cont have the same objects, functions : B → C. ′ ♯ ′ Here a morphism ( 5 , 5♯):  →  in Bun and a morphism ( 5 , 5 ):  →  in Cont are shown.

• a container morphism consists of a pair ( 5 , 5 ♯) where 5 : C → C′ is a function and 5 ♯ : 5 ∗(′)→  is a morphism in the slice category over C. ♯ We say a bundle morphism ( 5 , 5♯) (resp. a container morphism ( 5 , 5 )) is cartesian ♯ if 5♯ (resp. 5 ) is an isomorphism. Define Bun (resp. Cont) to be the category for which an object is a function be- tween finite sets and a morphism is a bundle morphism (resp. container morphism); see Fig. 1. Denote by Buncart (resp. Contcart) the subcategory of cartesian bundle morphisms.

One may note that Bun is the Grothendieck construction of the self-indexing op Fin/(−) : Fin → Cat, while Cont is the Grothendieck construction of its point-wise op op opposite (Fin/(−)) : Fin → Cat. The name container comes from the work of Abbot, Altenkirch, and Ghani [AAG03; AAG05; Abb03] (see Remark 2.18 in [GK12] for a discussion of the precise relationship betweenthenotionofcontainerandthenotionofpolynomial and polynomial functor). Remark 4.5. By the of pullbacks, Bun ≃ Fin→ is equivalent (in fact isomorphic) to the category of morphisms and commuting squares in Fin. Further- more, Buncart is equivalent to the category of morphisms and pullback squares in ♯ Fin, and Buncart ≃ Contcart (as in both cases a cartesian morphism ( 5 , 5♯) or ( 5 , 5 ) is determined by 5 alone). Nextwe showthat Bun ≃ Dir is also equivalent to the category of Dirichlet functors, from Definition 3.1. Recall that a natural transformation is called cartesian if its naturality squares are pullbacks.

Theorem 4.6. We have equivalences of categories

Poly ≃ Cont and Dir ≃ Bun.

In particular, this gives an equivalence Polycart ≃ Dircart between the category of polynomial functors and cartesian natural transformations and the category of Dirichlet functors and cartesian natural transformations.

Proof. The functors %− : Cont → Poly and − : Bun → Dir are defined on each object, i.e. function : B → C, by the formula  7→ % and  7→  ≔ % as in Proposition 4.2. −1 For each 1 ≤ 8 ≤ C, denote the fiber of  over 8 by :8 ≔  (8).

7 For any finite set -, consider the unique map -!: - → 1. Applying %− and − - y to it, we obtain the corresponding representable: %-!  y and -!  - . We next check that there are natural

:8 Poly(%-!,%)  %(-) = -  Cont(-!, ), Õ C - Dir(-!,)  (-) = (:8)  Bun(-!, ). (10) = Õ8 1 In both lines, the first isomorphism is the Yoneda lemma and the second is a compu- tation using Definition 4.4 (see Fig. 1). Thus we define %− on morphisms by sending 5 :  → ′ in Cont to the “compose-with- 5 ” natural transformation, i.e. having -- component Cont(-!, 5 ): Cont(-!, ) → Cont(-!, ′), which is clearly natural in -. We define − on morphisms similarly: for 5 in Bun, use the natural transformation Bun(−!, 5 ). By definition, every object in Poly and Dir is a coproduct of representables, so to prove that we have the desired equivalences, one first checks that coproducts in Cont and Bun are taken pointwise:

(: B → C) + (′ : B′ → C′)  ( + ′): (B + B′) → (C + C′), and then that %+′ = % + %′ and +′ =  + ′ ; see Remark 3.4. By Remark 4.5, we know that Buncart ≃ Contcart, and we have just established the equivalences Poly ≃ Cont and Dir ≃ Bun. It thus remains to check that the latter equivalences identify cartesian natural transformations in Poly with cartesian morphisms in Cont, and similarly for Dir and Bun. For polynomial functors, we may refer to [GK12, Section 2]. Turning to Dirichlet functors, we want to show that for any 5 :  → ′ the square

51 (1) ′(1)

 ′ (11) (0) ′(0) 50 is a pullback in Set iff for all functions , : - → -′, the naturality square

5 ′ (-′) - ′(-′)

(,) ′(,) (12) (-) ′(-) 5- is a pullback in Set; we will freely use the natural isomorphism (-)  Bun(-!, ) from Eq. (10). The square in Eq. (11) is a special case of that in Eq. (12), namely for , ≔ 0! the unique function 0 → 1; this establishes the only-if direction.

8 Tocomplete the proof, supposethat Eq. (11) is a pullback, take an arbitrary , : - → -′, and suppose given a commutative solid-arrow diagram as shown:

, --′

(1) ′(1)

1 1

(0) ′(0)

We can interpret the statement that Eq. (12) is a pullback as saying that there are unique dotted arrows making the diagram commute, since -  Bun(-!,0!) and similarly for the other corners of the square in Eq. (12). So, we need to show that if the front face is a pullback, then there are unique diagonal dotted arrows as shown, making the diagram commute. This follows quickly from the universal property of the pullback. 

Corollary 4.7. Dir is an elementary topos.

Proof. For any finite category , the Fin is an elementary topos. The result now follows from Remark 4.5 and Theorem 4.6, noting that Dir ≃ Fin→. 

As we mentioned in the introduction, this all goes through smoothly when one drops all finiteness conditions. The general topos of Dirichlet functors is the category of (arbitrary) sums of representables Setop → Set, and this is equivalent to the arrow category Set→ and so is itself a topos. We conclude with the equivalence promised in Section 1.

Theorem 4.8. A functor  : Finop → Fin is a Dirichlet polynomial if and only if it preserves connected limits, or equivalently wide pullbacks.

= y Proof. Let (y) 8:(0)(38) , and suppose that is any . Then for any diagram - : → Fin, we have Í

colim -9 (colim -9) = (38) 8:Õ(0) -9  lim(38) 8:Õ(0) -9  lim (38) 8:Õ(0) = lim (-9) since connected limits commute with sums in any topos (in particular Set). Now suppose  : Finop → Fin is any functor that preserves connected limits; in particular, it sends wide pushouts to wide pullbacks. Every finite set - can be

9 expressed as the wide pushout

-

1 1 ··· 1 1

0 of its elements. Therefore, we have the following diagram:

(-)

(1) (1) ··· (1) (1)

(0)

That is, an element of (-) is a family of elements 0G ∈ (1), one for each G ∈ -, such that the (0!)(0G ) are all equal in (0). But this is just a bundle map, i.e.

(-)  Bun(-!,(0!)) where -!: - → 1 and (0!): (1) → (0). Thus by Theorem 4.6, the functor  is the Dirichlet polynomial associated to the bundle (0!). 

Acknowledgments

The authors thank Joachim Kock, André Joyal, and Brendan Fong for helpful com- ments that improved the quality of this note. Spivak also appreciates support by Honeywell Inc. as well as AFOSR grants FA9550-17-1-0058 and FA9550-19-1-0113. Jaz Myers appreciates support by his advisor Emily Riehl and the National Science Foundation grant DMS-1652600.

References

[AAG03] Michael Gordon Abbott, Thorsten Altenkirch, and Neil Ghani. “Categories of Containers”. In: FoSSaCS. 2003 (cit. on p. 7). [AAG05] Michael Abbott, Thorsten Altenkirch, and Neil Ghani. “Containers: Con- structing strictly positive types”. In: Theoretical Computer Science 342.1 (2005). Applied Semantics: Selected Topics, pp. 3–27. issn: 0304-3975 (cit. on p. 7). [Abb03] Michael Gordon Abbott. “Categories of Containers”. PhD thesis. Univer- sity of Leicester, Aug. 2003 (cit. on p. 7).

10 [BD] John Baez and James Dolan. This Week’s Finds 300. Accessed: 2020-02-16. url: http://math.ucr.edu/home/baez/week300.html (cit. on p. 2). [GK12] Nicola Gambino and Joachim Kock. “Polynomial functors and polynomial monads”. In: Mathematical Proceedings of the Cambridge Philosophical Society 154.1 (Sept. 2012), pp. 153–192. issn: 1469-8064 (cit. on pp. 1, 7, 8). [Joy81] André Joyal. “Une théorie combinatoire des séries formelles”. In: Advances in Mathematics 42.1 (1981), pp. 1–82. issn: 0001-8708 (cit. on p. 2). [RW94] Robert Rosebrugh and R. J. Wood. “An Adjoint Characterization of the ”. In: Proc. Amer. Math. Soc 122 (1994), pp. 409–413 (cit. on p. 5).

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