Every Rig with a One-Variable Fixed Point Presentation is the Burnside Rig of a Prextensive Category

We extend the work of Schanuel, Lawvere, Blass and Gates in Objective Number Theory by proving that, for any L(X)∈ℕ[X]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${L(X) \in \mathbb {N}[X]}$\end{document}, the rig ℕ[X]/(X=L(X))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb {N}[X]/(X = L(X))}$\end{document} is the Burnside rig of a prextensive category.

R. Gates, was able to show that a similar result holds, not only for the Tree equation, but for a vast number of other polynomial fixed point equations. (Namely, for polynomials that are not constant and have a non-zero constant term.) The objective of the present paper is to remove all restrictions on the polynomial involved in the fixed point equation and to begin to point the way toward multi-variable extensions.
We now describe the contents of the paper in some technical detail. The reader will be assumed to be familiar with (pr)extensive and distributive categories and their Burnside rigs as discussed in [16] and [14]. If C is a small distributive category, then its Burnside rig will be denoted by BC. We also need to assume that the reader is familiar with algebraic theories [13] and some topos theory including the construction of classifying toposes as in Section D3.1 of [10].
Is R the Burnside rig of a distributive category? If L(X) is constant the answer is 'yes' because in this case R = N, which is the Burnside rig of the category of finite sets. As recalled above, the main result in [16] shows that the answer is positive for L(X) = X + 1 + X = 2X + 1; indeed, in this case R is the Burnside rig of the category of bounded polyhedra. Using a different technique, [3] proves that the answer is also positive for L(X) = 1 + X 2 . A related but different approach is used in [8] to show that the answer is positive if L(X) is not constant and L0 = 0.
We show that the answer is positive for all L(X) ∈ N[X]. Our proof is a combination of the techniques used in [3] and [8], synthesising the use of toposes and of calculi of fractions. (We will only deal with one-variable presentations, but it must be mentioned that it is also proved in [16] that N[X, Y ]/(X = 2X + 1, Y = X + 1 + Y, Y 2 = 2Y 2 + Y ) is the Burnside rig of the category of unbounded polyhedra.) Let T be a small category with finite products and let T be the induced topos of presheaves. Denote the (finite-)coproduct completion of T by FamT . Since T has products, FamT is prextensive (see paragraph before Proposition 4.6 in [5]) and the essentially unique functor FamT → T making the following diagram commute is fully faithful and preserves products and coproducts. (Of course, the diagonal map is the Yoneda embedding and the horizontal one is the universal inclusion of its domain into the coproduct completion.) Now let J be a Grothendieck topology on T and j : Sh(T , J ) → T be the induced subtopos. The inverse image of j may be precomposed with the inclusion FamT → T to obtain a functor FamT → Sh(T , J ). Since both the top and right functors in the square of Definition 1.1 preserve finite products and coproducts, the category Fam(T , J ) is prextensive and the inclusion Fam(T , J ) → Sh(T , J ) preserves products and coproducts. In particular, we can apply the above to the case where T is a free algebraic theory. Let us quickly recall the basic definitions and introduce some notation.
Fix a small version fSet of the category of finite sets. An (algebraic) theory is a small category T with finite products together with a bijective-on-objects functor T : fSet op → T that preserves finite products [13]. When there is no risk of confusion we will omit the functor T and simply say that T is a theory. It is convenient to use 'exponential notation' so that if f : A → B is a map in fSet then T f : T B → T A is the corresponding map in T . In particular, we will write T instead of T 1 and, for each element i : 1 → I in fSet, we may let π i = T i : T I → T . A morphism of theories is a functor F : T → T that preserves finite products and makes the following diagram commute. Let Th be the category of theories.
Fix a natural numbers object N in Set and consider it as a discrete category. An object P in the topos Set N may be thought of as a 'signature' such that for each n ∈ N, P n is the set of 'operations of arity n'. Now fix an inclusion N → fSet sending each n ∈ N to a finite set n of cardinality n. This inclusion induces a functor U : Th → Set N such that for any T in Th and n ∈ N, (U T )n = T (T n , T ). It is well-known that the functor U has a left adjoint F : Set N → Th. (See Section II.2 in [13].) For any P in Set N , F P is the free theory determined by P .
Any polynomial L(X) ∈ N[X] determines a 'signature' ∈ Set N such that if m is the coefficient of degree n then n = m. The free theory determined by will be denoted by L. For every n ≥ 0, each f ∈ n induces a map in L that we denote by f : T n → T . Let J L be the least Grothendieck topology on L such that T is covered by the sieve generated by the family (f : T n → T | n ∈ N, f ∈ n). Our main result (Theorem 15.2) shows that: if L(X) ∈ N[X] is not constant, the canonical BL → N[X]/(L(X) = X) extends to a unique iso B(Fam(L, J L )) → N[X]/(L(X) = X) of rigs. We will also present the geometric theory classified by Sh(L, J L ).
The purpose of Sections 2 to 5 is, roughly, to show that the more complicated part of the proof of the main result may be confined, as in Blass' paper [3], to a site with finite products (Proposition 5.7). The notion of semi-saturated subcategory (admitting a calculus of (right) fractions) plays here the most important role.
In Blass' paper, the more difficult part of the proof takes place in a site whose underlying category is the free algebraic theory generated by a constant and a binary operation. A key Lemma (see p. 16 in [3]) shows that the relevant covering sieves may be characterized as those that contain all constants. Of course, this must change since the theories we consider may lack constants. Yet, there is a sense in which the same idea works. We try to capture the essence of the idea via the notion of ample family introduced in Section 6 in the context of forts.
Sections 7 to 11 culminate in the definition of ranked (algebraic) theory and the result that free theories are ranked in a canonical way. Intuitively, ranked theories form a class of algebraic theories where most of Blass' argument makes sense. We decided to formulate such a concept because all the attempts to mimic Blass' proof in an arbitrary free theory described using syntactic terms led to calculations that were impossible to read. Fortunately, all that is needed about free theories may be expressed as in Proposition 11.5. Unfortunately, this requires some new auxiliary concepts such as that of rigged theories but, altogether, we believe that the abstract formulation is better than the alternative using strings of symbols.
Sections 12 to 14 mimic Blass' proof. In particular, the notion of development is the natural analogue of Blass' notion in our more general context.
In Section 15 we combine our results on calculi of fractions (inspired by Gates' work) with our generalization of Blass' ideas. This combination proves the main result (Theorem 15.2). In Section 16 we give a presentation of the theory classified by the topos used to prove the main result.

Definition 2.1
A bijective-on-objects subcategory → X is said to admit a calculus of (right) fractions if: CF1 Every cospan as on the left below can be completed to a commutative square as on the right above.
CF2 For every commutative diagram as on the left below there exists an s : W → X in such that the diagram on the right above commutes.
For the rest of the section fix a bijective-on-objects subcategory → X admitting a calculus of fractions. Notice that if every map in is mono then condition (CF2) is trivially satisfied.
The category of fractions X [ −1 ] has the same objects as X and, as arrows X → Y , equivalence classes of spans with s ∈ . Two such spans (f, s) and (g, t) are equivalent if there is a commutative diagram with sa = tb ∈ . The equivalence class determined by (f, s) will be denoted by f s . The obvious functor from X to X [ −1 ] sending f to f id is many times denoted by P : X → X [ −1 ] and it is universal among functors from X sending all maps in to isos.
u if and only if there is a w : W → U such that uw : W → X is in and the following diagram Proof The condition f id = g u means that there is a commutative diagram The following result is essentially that appearing in Section I.3.5 of [7]. and only if  there is a diagram as below with α and β in and such that both triangles inside the square commute.
Let F : X 0 → X be a full subcategory and 0 → X 0 be a bijective-on-objects subcategory admitting a calculus of fractions such that, for every f in 0 , Ff is in . Then there exists a unique functor G : such that the following diagram commutes.

Lemma 2.4 If for every
Proof To prove that G is full, let A and B be in X 0 and f s : FA → FB in X [ −1 ] with s : X s → FA in . By hypothesis there exists g : FA → X s such that sg = F t for some t : A → A in 0 as in the following diagram and, since F is full, there exists b : Assume that G f s = G g t so there exists a commutative diagram as on the left below such that (F s)a = (F t)b ∈ . Let us call this map c : X → FA. By hypothesis there exists a map k : FA → X such that ck = F h for some h : A → A in 0 . Since F is full there are a : A → A s and b : A → A t in X 0 such that F a = ak and F b = bk and the diagram on the right above commutes. We also have that (F s)(F a ) = F h = (F t)(F b ) and, since F is faithful, a and b witness that f s = g t . An important ingredient in the proof of our main result is a weakening of the following concept.

Definition 2.5
The subcategory → X (admitting a calculus of fractions) is called saturated if for every map f in X , f id an iso implies that f ∈ .
We end this section with a relevant example of a saturated subcategory. Let j be a (Lawere-Tierney) topology in a topos E and let L : E → Sh j E be the associated-sheaf functor. We assume that the reader is familiar with the relation between Lawvere-Tierney topologies and closure operators, and with the concept of (j -)dense monos.
The next result follows from the material in Section 3.4 in [9].

Proposition 2.7 The subcategory → E of bidense morphisms admits a calculus of fractions and the composite Sh
In other words, Sh j E is a category of fractions.

Proper Families
Let C be a small category. We assume that the reader is familiar with the notion of Grothendieck topology but we recall the related notion of basis. Definition 3.1 A basis (for a Grothendieck topology) on C is a function assigning to each object U of C a collection KU of families (f i : U i → U | i ∈ I ) of maps in C (called Kcovering families) such that: Let K be a basis on C such that every K-covering family is finite.
We now extend this definition to maps with arbitrary codomain in FamC. Each object in FamC is of the form i∈I U i for a finite set I and, for each i ∈ I , U i an object in C. Any map with codomain i∈I U i is of the form i∈I g i : i∈I X i → i∈I U i . (Notice that we are not requiring the X i 's to be in C. ) We say that such a map is selected if g i : X i → U i is basic for every i ∈ I .

Lemma 3.2
The selected maps form a bijective-on-objects subcategory that we denote by K → C. If every K-covering family is finite then condition (CF1) in Definition 2.1 is satisfied.
Proof The first condition for bases implies that all identities are selected. The third condition for bases implies that selected maps are closed under composition. The second condition for bases, together with finiteness, implies that (CF1) holds.
In order to establish a sufficient condition for K → C to admit a calculus of fractions we introduce the following.
The following result gives two alternative formulations.

Lemma 3.4 For any family
of maps in C the following are equivalent: 1. The family F is monic. 2. The following two conditions hold: (a) the map f i : C i → C is mono for each i ∈ I and (b) for every i, j ∈ I the existence of a commutative diagram in C implies that i = j .
If, moreover, I is finite then the above are also equivalent to: Proof The equivalence between the first two items is left for the reader. We prove that the last two items are equivalent when I is finite. So let f = [f i | i ∈ I ] : i∈I C i → C in FamC. Since every object of FamC is a finite coproduct of objects in C, f is mono if and only if for every object D in C, and maps g, h : Since D is connected in FamC, g = in i g i for some i ∈ I and g i : D → C i and, similarly, h = in j h j for some j ∈ I and h j : D → C j . So, the equality fg = f h simply means that the square below commutes. It follows that the map f is mono in FamC if and only if the family F is monic in the sense of Definition 3.3.
The next somewhat ad-hoc terminology will prove efficient.

Definition 3.5
A family (f i : C i → C | i ∈ I ) of maps in C will be called proper if it is finite and monic.
We can exhibit our source of calculi of fractions coming from bases.

Lemma 3.6
If every K-covering family is proper then the subcategory K → FamC admits a calculus of fractions such that every map in K is mono.
Proof Follows from Lemma 3.2 and the fact that if every K-covering family is monic then every map in K is mono.
Let Sh(C, K) → C be the associated topos of sheaves and a : C → Sh(C, K) be the associated sheaf functor. Adapting the notation in Definition 1.1 from topologies to bases, the full image of FamC → C → Sh(C, K) will be denoted by Fam(C, K) → Sh(C, K).

Proposition 3.7 If every K-covering family is proper then there is full and faithful
Proof By Proposition 2.7, Sh(C, J ) coincides with C[ −1 ] where → C is the subcategory of bidense maps. So it is enough to check that Lemma 2.4 is applicable to FamC → C.
First we need to check that the full inclusion FamC → C sends maps in K to dense monos. This follows from the fact that the inclusion FamC → C preserves monos and coproducts, and the fact that Yoneda sends K-covers to families of maps generating a dense subobject. It remains to prove that the condition in the statement of Lemma 2.4 holds. It is enough to restrict to bidense maps X → C in C with codomain in C. Its image s : S → C is a dense subobject. Since K is a basis, the sieve S must contain the maps in a K-cover In other words, the induced map f : i∈I C i → C factors through S → C and, since the domain of f is projective in C, f factors through the bidense X → C. That is, we obtain a map i∈I C i → X such that the composite i∈I C i → X → C is in K .
We will need the following closure property. Proof If f * G exists then it is clearly a finite family of monos. To prove that it is monic Assume that the diagram on the left below commutes for some i, j ∈ I . Then the diagram on the right above commutes and, since G is monic, i = j . So f * G is monic by Lemma 3.4.
For example, in an algebraic theory, pullbacks along projections always exist.

Semi-Saturation
Let → X be a bijective-on-objects subcategory admitting a calculus of fractions. The second paragraph of page 9 in [3] says that very explicit bijections are determined by two families of patterns (p i ) i∈I and (q i ) i∈I such that, for each i, "the same labels occur in p i as in q i ". This condition motivates the following.
The next result provides a useful alternative formulation.

Lemma 4.2 The subcategory → X is semi-saturated if and only if for every map
Proof One direction is trivial, for the other assume that f s : Clearly, if is saturated in the sense of Definition 2.5 then it is semi-saturated.

Definition 4.3
A bijective-on-objects subcategory → X is said to admit a calculus of dense monos if it admits a calculus of fractions, every map in is mono and for every diagram of monos as below Our source of semi-saturated subcategories is based on the following.

Proposition 4.4 If → X admits a calculus of dense monos then it is semi-saturated.
Proof Assume that f id : X → Y is an iso. Then there is a diagram as in the statement of Lemma 2.3 with α and β in . By hypothesis, β is mono, so g is also mono. But then α, h and g form a triangle as in Definition 4.3 so g ∈ and we have a diagram such that gh ∈ . Lemma 2.2 implies that f id = β g , so Lemma 4.2 implies that is semisaturated.
The semi-saturated subcategories we are interested in come from bases. Definition 4.5 A basis K on a small category C is called a basis of dense monos if: 1. every K-cover is proper and 2. for every C in C, for any F ∈ K(C) and any proper family P of maps with codomain C, if every map in F factors through some map in P then P ∈ K(C).
Recall (Lemma 3.6) that any basis K on C such that all K-covers are proper determines a subcategory K → FamC admitting a calculus of fractions such that every map in K is mono.

Corollary 4.6
If K is a basis of dense monos on C then K → FamC is semi-saturated.
Proof By Proposition 4.4 it is enough to show that K → FamC admits a calculus of dense monos. So consider a diagram of monos in FamC as below and assume that [u i | i ∈ I ] is in the subcategory K → FamC. We need to show that [v j | j ∈ J ] is in K . It is enough to concentrate on the case when Z is in C. By Lemma 3.4 the families (u i | i ∈ I ) and (v j | j ∈ J ) are proper and, by hypothesis, the former is in KZ. Since K is assumed to be a basis of dense monos, the latter family is also in KZ. So [v j | j ∈ J ] is also in K → C.

Compatibility and Weights
Let X be a distributive category and → X a bijective-on-objects subcategory admitting a calculus of fractions. The only assumption that we make on → X is that X [ −1 ] is distributive and that the universal functor X → X [ −1 ] preserves finite coproducts. (Recall that this functor preserves finite products automatically. In fact, all finite limits by the 'rightfractions' analogue of Proposition I.3.1 in [7].) For example, this is what happens in the case of Proposition 3.7. See also remark after Definition 1.1.
Fix also a rig R.
Compatibility interacts well with semi-saturation.

Proposition 5.2 If is semi-saturated and
Because is semisaturated, we can assume that f (as well as s) is in . Our hypothesis on γ implies that Let C be a small category with finite products and denote by BC the multiplicative monoid of iso-clases of objects. Proof The morphism γ sends coproducts in FamC to sums in R. We leave the details for the reader.
Let us fix a map γ : BC → R of multiplicative monoids. We now explain how the compatibility of the extension B(FamC) → R with a subcategory (admitting a calculus of fractions) induced by a basis on C can be reduced to a condition in terms of the basis.

Definition 5.4 For any finite family
This notion of weight is analogous to the one introduced in p. 11 of Blass' paper. It plays essentially the same role in the proof.
Assume now that C is equipped with a basis K such that every K-cover is proper. (So that FamC is equipped with the subcategory K → FamC, admitting a calculus of fractions, as explained in Lemma 3.6.) This is justified by the following.
Proof Assume first that γ : BC → R is compatible with K. A morphism m in the subcategory K → FamC (admitting a calculus of fractions) is given by a coproduct as we needed to prove. The converse holds because if γ is compatible then, for every so the proof is complete.
Let us summarize what we have achieved so far. To prove the main result we will take C to be the free theory L determined by a nonconstant polynomial L(X) ∈ N[X]. The hard part is to find a basis of dense monos K such that the canonical L → N[X]/(X = L(X)) is compatible with K in the sense of Definition 5.5. For this purpose the following will be useful eventually.

Lemma 5.8 (Weights of composite families)
using the definition of weight.

Amplitude
The Lemma in p. 16 of [3] deals with a Grothendieck topology on a free (algebraic) theory and, among other things, identifies the sieves in J (C) as those that contain a finite family such that every constant 1 → C in the theory factors through a map in that finite family. I claim that such a characterization is possible because the polynomial L(X) = 1 + X 2 is such that L0 = 0. Part of our proof was influenced by the wish to generalize this lemma to theories that may lack constants. This is the origin of the notion of 'ample family' that we introduce in this section.
We first recall the notion of normed category suggested in pages 139-140 of [12] and, after that, we define ample families in suitable normed categories.
in V for every pair of maps f : X → Y and g : Y → Z in C, and the assignment of a morphism k → d X,X id X in V for every object X in C, subject to the evident associativity and unit conditions (that we need not emphasize because they automatically hold in our main example of base monoidal category).
(Although I have not been able to obtain a copy of [2], it appears that normed categories have been also considered there. Indeed, it seems clear from the Zentralblatt Autorreferat and the AMS review by Linton that it is proved loc. cit. that normed categories may be seen as categories enriched in suitable monoidal categories; solving, in a general way, an exercise suggested in page 140 of [12].) Let (N, +, 0) be the usual commutative monoid of natural numbers under addition and consider its extension N ∞ = (N + {∞}, +, 0) with an element ∞ such that, for every n ∈ N + {∞}, ∞ + n = ∞ = n + ∞. The monoid structure induces a total order (N ∞ , ≤) with ∞ as terminal object. Moreover, addition extends to a symmetric monoidal structure on the category (N ∞ , ≤). The resulting monoidal category ((N ∞ , ≤), +, 0) will be denoted simply by (N ∞ , ≤).
In concrete terms, an holds for every f : X → Y and g : Y → Z. (Notice that it automatically holds that, for every X in C, 0 ≤ d X,X id X .) We will drop the subscripts and write d instead of d X,Y ; so the key condition may be expressed as dg + df ≤ d(gf ).
Assume from now on that C is an (N ∞ , ≤)-normed category with 'norm' d.

Lemma 6.2 For any
Proof Simply observe that d(id Y ) + df ≤ df and similarly for the other equality.
On the other hand, if we let df = ∞ for all f then we obtain a somewhat extreme sort of N ∞ -normed category.

Definition 6.3 Let F be a family of maps in
For brevity let us say that F is a family on Y if it is a family of maps with codomain Y . Also, if F is a family on Y then we will say that f : X → Y factors through F if it factors through some map in F .
The family of all maps f with codomain Y and df ≥ n is clearly n-ample. We are interested in categories that contain less trivial examples. In fact, we are going to be mainly interested in ample families that are also proper in the sense of Definition 3.5. Definition 6.5 A family F on Y is strictly n-ample if it is n-ample and for every f ∈ F , df ≥ n. A family will be called strictly ample if it is strictly n-ample for some n.
Proper strictly ample families are unique up to iso in the following sense.
are both proper and strictly n-ample families then there exists a unique bijection φ : I → J such that for every i ∈ I , h φi is iso to f i over Y .
Proof Let i ∈ I . Since df i ≥ n there exists a j ∈ J and a t : Since H is monic, this j is unique so we may call it φi and in this way we obtain a function φ : I → J . Moreover, this t is unique (because h φi is mono) and mono (because f i is mono). We claim that t is an iso. Indeed, since dh φi ≥ n, there exists a unique k ∈ I and mono u : A φi → X k such that f k u = h φi . Then f k ut = h φi t = f i and, since F is monic, i = k and ut = id X i . So u is a monic with section t and hence t is an iso.
It remains to show that φ : I → J is bijective. To prove surjectivity let j ∈ J . As before, there exists an i ∈ I such that h j factors through f i which, in turn, factors through h φi . So h j factors through h φi and, since H is monic, j = φi.
Finally, for i 0 , i 1 ∈ I assume that φi 0 = j = φi 1 for some j ∈ J . Once again, h j factors through f k for some k ∈ I and then both f i 0 and f i 1 factor through f k . Since F is monic, Assume for the rest of the section that K is a function that assigns to each X in C a collection of ample and proper families on X. Definition 6. 7 We say K has long covers if for every Y in C and k ∈ N there exists an We are interested in cases where K is the basis of a Grothendieck topology. For the moment we just show that having long covers is sufficient to ensure the key Coverage condition for bases. Lemma 6.8 The following hold: 1. If f : X → Y is an iso then the family (f | 1) on Y is 0-ample and in KY . 2. If K has long covers then for every G ∈ KZ and g : Y → Z there exists an F ∈ KY such that for every f ∈ F , gf factors through G.
Proof The first item is easy because every map with codomain Y factors through f . For the second item assume that G is k-ample. Since K has long covers there exists an F ∈ KY such that for every In a category with terminal object 1, a map X → Y is constant if it factors through the terminal object.
Sieves on an object containing all the points of that object play an important role in [3]. For this reason we highlight the following.

Lemma 6.10 If C is a fort and F is an ample family on
Part of the remaining work involves showing that free theories with some non-constant operation are strong forts in a canonical way. This will follow from the acquisition of more subtle information present in free theories.

Rigged Theories
Fix a rig R = (R, ·, 1, +, 0) in Set. For any A in fSet, R A will denote the exponential in Set.
such that the following hold: for every a : J → I in fSet and, We are going to use the formulas as displayed above but it seems also useful to express them in more general terms. For any A in fSet we have a function A : transposes to a map that, with little risk of confusion, we may call B : R B×A → R A . It sends ψ ∈ R B×A to j ∈B ψ(j, ) in R A . So far this has nothing to do with any algebraic theory, but if we are given the family then there is a function such that the second equation of Definition 7.1 may be formulated as (gf ) = B (g * f ) or by requiring that the diagram commutes. Fix a theory T rigged in R.

Lemma 7.2 For any
Proof Calculate (id B f ) using the second item in Definition 7.1.
That is, the function f : A → R is determined by the projections π j f : T A → T .

Definition 7.3 For any
The operation d can also be calculated in terms of the projections, in the following sense.
using Definition 7.3 and Lemma 7.2.
The next result explains how d behaves with respect to composition.
for some n ∈ R.
Proof The calculation below shows that the left equality in the statement holds. Using Lemma 7.4 we may calculate as follows which proves that the right equality in the statement holds because we can take n to be the right summand in the last binary addition.
The following will also be relevant.
An algebraic theory rigged in N ∞ is then an algebraic theory T equipped with a family hold. Readers familiar with [3] may recognize the idea of 'depth of a node in a tree'. If we picture a map f : T A → T as a tree with leaves in A then f (i ∈ A) may be thought of as measure of how far is the leaf i from the root of f . We will give a precise meaning to this analogy later. For the moment, consider the following alternative formulation of the first item above.
For example, for k : 1 → I in fSet and the induced projection π k = T k : T I → T , for every i ∈ I . As another example of how the general facts manifest in the case of N ∞ , and also for future reference, we state the following particular case of Lemma 7.2.
and the other statements follow from Lemma 7.5 which, in particular, implies the existence of an n ∈ N ∞ such that d(gf ) ∧ n = (dg) + (df ).
We can now relate theories rigged in N ∞ with the material of Section 6.
Proposition 7.10 With the notation above, d makes T into a fort.
Proof Lemma 7.9 shows that d makes T into an (N ∞ , ≤)-normed category and dp = i∈0 p i = ∞ holds for any point p : In particular, ample families make sense inside theories rigged in N ∞ . We now embark on the construction of rigged theories.

Extensivity of Fibered Categories
In this section we make explicit a particular case of the Grothendieck semi-direct product construction. Let  (gf, (g f ) · f ) where the map (g f ) · f : X → M is the composite on the right above.
(If we force the notation a little bit then we could write (gf ) = (g f ) · f , which is reminiscent of the chain rule; but we stress: in the notation above, f is not determined by f .) There is an obvious functor C M → C which is the identity on objects and sends (f, f ) to f . This functor has a section C → C M that sends f : It is relevant to observe that maps in the image of C → C M remain in C after pulling back along any map in C M. More precisely: Proof The diagram on the right of the statement commutes because the left one does and because (g π 0 ) · 1 = g π 0 = 1 · (g π 0 ) = (1π 1 ) · (g π 0 ). Assume now that the following square commutes in C M, which means that gf = uh and (g f ) Consider now a map (t, t ) : X → P in C M such that the equations hold. This means that the equations below hold. Equivalently, the following hold and it is clear that the first three equations uniquely determine t and t . The last equation holds automatically, given the first one, because (g π 0 t) · t = (g f ) · f = h . Altogether, the map (t, t ) exists and is unique.

M. Menni
If C has finite coproducts then so does C M. More explicitly:

Lemma 8.2 If 0 is initial in C then it is initial as an object in C M. If the diagram on the left below is a coproduct in
then the diagram on the right above is a coproduct in C M.
Proof We concentrate on binary coproducts. g ). That is, the equations below hold or, equivalently, the equations below In the examples we are interested in C is extensive.

Proposition 8.3 If C is extensive then so is C M.
Proof Recall that a category with finite coproducts is extensive if and only if coproducts are stable and disjoint [5]. Stability and disjointness are easy to prove using the description of coproducts in Lemma 8.2 and the description of the relevant pullbacks in Lemma 8.1.
We say that M is conical if the following square is a pullback. f ) is a section then f is a section and f = 1. It is easy to show that if (f, 1) is a retraction then f is a retraction.

The Category of Restricted Spans
In this section we let C be an extensive category with finite limits and equipped with a monoid M = (M, ·, 1). We can then consider the (non full) coproduct-preserving inclusion C → C M.
More concretely, a restricted span is a span as on the left below and, for example, for each object X in C, we have the distinguished span on the right above from X to X. Given consecutive spans as on the left below and a pullback square as on the right above (recall Lemma 8.1) then we define the induced composite as the span which is clearly a restricted span from X to Z. Notice that this is just the usual composition of spans, relying on Lemma 8.1, to conclude that restricted spans are closed under composition. We will not need general spans so, from now on, we use the word 'span' to mean 'restricted span'.
Given spans from X to Y as on the left below then a map from the first to the second is a morphism (s, s ) : A → B in C M such that the triangles in the right square above commute. This is the standard definition of morphism of spans, but notice that in this case, the lower triangle means that the composite (b, 1)(s, s ) = (bs, (1s) · s ) = (bs, s ) equals (a, 1). That is, bs = a and s = 1. Then, the upper triangle means that (g, g )(s, 1) = (gs, (g s) · 1) = (gs, g s) equals (f, f ). That is, gs = f : A → Y and g s = f : A → M. So it is convenient to simplify the notation as follows.
From now on, spans from X to Y will be denoted as on the left below and a map from the first to the second will be just a map s : A → B in C such that bs = a, gs = f and g s = f . Given a choice of pullbacks in C (which gives a choice of the relevant pullbacks in C M) we define 'the' bicategory Sp(C, C M) as in [1]. The objects of Sp(C, C M) are the objects of C. For any pair of objects X, Y , the category Sp(C, C M)(X, Y ) is the category of spans from X to Y and morphisms between them. Composition is defined using pullbacks as above and the rest of the structure is determined by the universal property of pullbacks as in the standard case.
(Surely, the above discussion generalizes to a setting starting with a category D with a suitable bijective on objects subcategory C that is closed under pullbacks in D; but we have not found the details spelled out in the literature.) The associated 'classifying category' (7.2 in [1]) will be denoted by Span(C, C M). Concretely, the objects of Span(C, C M) are the objects of C and a map from X to Y is an equivalence class of spans from X to Y . Two spans being equivalent if they are iso in Sp(C, C M)(X, Y ). More explicitly: Proof It is easy to check that the assignments in the statement are functorial. It is also easy to check that C op → Span(C, C M) is faithful. To prove that it reflects isos assume first that the map [a; id, 1] has a retraction [b; (g, g )] with b : B → Y and g : B → X. That is, [b; g, g ][a; id, 1] = [ba; g, g ] = [id X , id X , 1]. So, without loss of generality, we may assume that B = X and conclude that a is a section of b, g = id X and g = 1. Now, assume further that [b; g, g ] = [b; id, 1] is a section of [a; id, 1]. Then it is easy to check that b is a section of a, so a is an iso.
To prove that C M → Span(C, C M) is faithful let (f, f ), (g, g ) : X → Y in the category C M and assume that [id X ; f, f ] = [id X ; g, g ]. Then there exists an iso σ : X → X in C such that id X σ = id X , gσ = f and g σ = f in C. So σ = id X , f = g and f = g . To prove that the functor C M → Span(C, C M) reflects isos assume first that [id X ; f, f ] : X → Y has a section. This means that there is a span In general, though, the diagonal fill-in need not be unique. To see this it seems clearer to make explicit the phenomenon in the usual category of spans of sets. Take the commutative square so that the composite is just the map determined by the span 1 ← 2 → 1. Consider now the two maps [id; id], [τ ; id] : 2 → 2 where τ : 2 → 2 is the only non-identity bijection. It is clear that the two maps are different and it is easy to check that they both make the relevant triangles commute.
To prove uniqueness first observe that a map A → X + Y in Span(C, C M) is determined by some span given by a map d : The object 0 is terminal because every restricted span X ← A → 0 is forced to be the unique X ← 0 → 0. Finally, for every X, points 0 → X are determined by spans 0 ← A → X; that is, by the unique span 0 ← 0 → X.

Rigged Theories of Finite Spans
Fix a conical monoid M = (M, ·, 1) in Set so that, by Proposition 9.5, we can consider the category Span(Set, Set M) with finite products and the product preserving inclusion Set op → Span(Set, Set M).
Let fSet → Set the fixed small version of the category of finite sets and functions that we used to introduce algebraic theories. We may consider the full subcategory of Set M determined by fSet. We will denote this subcategory by fSet M → Set M, even if M is not in fSet.

Definition 10.1 A finite (restricted) span in Set M is a span as on the left below
with A, X and Y in fSet. As before, we will denote the span as on the right above. Now fix a rig R = (R, +, 0, ·, 1) together with a morphism M → R of multiplicative monoids that, for simplicity, we assume that it is an inclusion. We will show that the theory E : fSet op → T M can be rigged in R. This process sending spans to R-valued functions is well behaved with respect to the equivalence of spans.
The hypothesis implies the existence of an iso σ in fSet such that bσ = a, gσ = f and g σ = f in Set. The calculation f j so that we may conclude the following. It is fair to picture a map f : X → 1 in fSet N as the record of certain information related to a tree with leaves in X. In practice f will record the distances from the leaves to the roots. If we embed fSet N into the algebraic theory T N then we need the rig N ∞ in order to extend the information recorded by maps in fSet N to arbitrary maps in the theory T N .
We are going to use T N to record relevant information of maps in free theories. On the other hand, it should be possible to generalize our construction of T M in order to give an alternative construction of free theories; a construction that is intermediate between their conceptual construction as 'coproducts of operations' as in [13] and their construction in terms of strings of symbols.

Ranked Theories
If T is an algebraic theory such that T : fSet op → T is conservative then the maps in this subcategory will be called bureaucratic. The intuition is that bureaucratic maps just forget things, repeat things or permute things. If T is liberal then the maps in (fSet op ) ↓ → T will be called efficient. It follows from the results in [19] that free theories are liberal. In relation to the work just cited some remarks on terminology are in order. Notice that if the subcategory fSet op → T is part of a factorization system then it must contain all the isos. So, in this case, the bureaucratic maps coincide with the class of structural maps as defined loc. cit. For this reason the maps in the subcategory (fSet op ) ↓ → T are exactly the analytic maps defined there. In other words, if structural maps are bureaucratic then efficient and analytic maps coincide. We keep the bureaucratic/efficient terminology in order to emphasize the condition that bureaucratic maps are part of a factorization system. (A theory is called analytic if structural and analytic maps form a factorization system. Marek Zawadowski observes that it follows from [19] that liberal theories are exactly the analytic theories with no non-trivial unary invertible operations. For example, any free theory. He also produced a characterization of the analytic theories which satisfy that analytic maps are mono. It follows easily from this characterization that efficient maps in free theories are mono.) Lemma 11.2 If T : fSet op → T is liberal then, e : T X → T Y is efficient if and only if there is a family (e i : T X i → T | i ∈ Y ) of efficient maps and an iso b : i∈Y X i → X in fSet such that the following diagram commutes.
Proof First notice that, for general reasons about factorization systems, if the family (e i | i ∈ Y ) of efficient maps exists then the product e i is also efficient. Let f i : T X → T be the composite of e : T X → T Y followed by the projection T i : : i∈Y X i → X determined by the universal property of the coproduct. Then the diagram in the statement commutes and since the product e i is efficient then the top map T b is an iso and hence b is an iso.
In other words, efficient maps are products of efficient maps with codomain T . Before the next definition recall the algebraic theory T N rigged in N ∞ and exalted in Corollary 10.7. Proof We have already observed that free theories are liberal. The monoid (N, +, 0) has a distinguished element 1 ∈ N and, for any X in fSet, we have the associated constant function 1 : X → N. (To avoid a possible confusion we stress that 1 ∈ N is not the multiplicative unit of the rig N ∞ .) The map [id X ; !, 1] : E X → E 1 in the subcategory fSet N → T N will be denoted by f X : E X → E. Notice that f 0 : 1 → E 1 is the unique point of T N .
Recall the adjunction F U : Th → Set N . For any 'signature' P ∈ Set N we have the constant function P n → (U T N )n = T N (E n , E) that sends everything in P n to f n . These functions underlie a natural transformation P → U T N and hence a morphism of theories F P → T N . It remains to show that this morphism sends efficient maps to maps in the subcategory fSet N → T N . For this, notice that every efficient map in F P is a composite of products of maps coming from P . Since the elements in P n are sent to f n : E n → E in the subcategory fSet N → T N , the morphism of theories F P → T N sends efficient maps to maps in the same subcategory.
The other basic fact we need about ranked theories is the following. Proposition 11.5 If ρ : T → T N is a ranked theory then the assignment that sends Proof The theory T is rigged in N ∞ by Lemma 7.6 and Corollary 10.7. Now let us assume that β : T A → T B is efficient in T . By hypothesis ρβ = [id; g, g ] for some (g, g ) : A → B in fSet N and so, by Corollary 10.7 again, We invite the reader to think of an efficient map f : T X → T 1 in T as a tree with leaves in X; and the function (ρf ) as assigning to each x ∈ X its distance from the root. The fact that (ρf ) lands in N means that all these distances are finite. If X is non-empty then df ∈ N. If X = 0 then (ρf ) : 0 → N → N ∞ and df = ∞.
Fix a ranked category ρ : T → T N . For any f : T X → T Y in T we write f instead of (ρf ) . Similarly, we write df ∈ N ∞ instead of d(ρf ). As we have already mentioned, we can consider ample families in T .

Lemma 11.6 If G is a strictly ample and proper family on T Z then every map in G is efficient.
Proof Assume that G is strictly n-ample. Let g : T Y → T Z be a map in G and let g = h(T b ) with b : A → Y in fSet and h : T A → T Z efficient. Before the next calculation notice that, for any j : 1 → A, d(T bj ) = 0 by Lemma 7.9. Since G is strictly n-ample we have that: so h factors through some map g in G. In this case g also factors through g but, since G is monic, g = g . In other words, h factors through g, say, as h = gr with r : Since g is mono, gr(T b ) = h(T b ) = g implies that r is a retraction of T b . We also have h(T b )r = gr = h so the following diagram commutes and since T b is bureaucratic and h is efficient (T b )r = id so T b is an iso. (The last part is, of course, an instance of a more general fact about factorization systems.) Another feature of ranked theories is that they are naturally equipped with a notion of covering family that is closed under composition. In the cases we are interested in, these families will form the basis of a Grothendieck topology.
Definition 11.7 (The natural potential basis of a ranked theory) Let K be the function that assigns to each T X in T , the collection of ample and proper families of efficient maps.
Before the next result recall that we could not prove that ample families compose in the general context of a fort.
Proof It is clear that H is a proper family of efficient maps. Assume that G is n-ample. Since all the maps in G are efficient then {(g i ) y | i ∈ I, y ∈ Y i } is a finite set of natural numbers by Proposition 11.5. So we can choose an N ∈ N that is above all these and also above n. Similarly, for each i ∈ I , if F i is m i -ample then we can choose an M ∈ N that is above every m i . We claim that H is (N + M)-ample. Let h : T X → T Z be such that dh ≥ N + M. Since M + N ≥ N ≥ n there exists an i ∈ I and an f :

Developments
Let T be a ranked theory with associated natural potential basis K (Definition 11.7). In this section we assume that T is equipped with a distinguished family B ∈ K(T ) of 'basic operations'. Notice that for any j : 1 → Y , the pullback π * j B exists.

Lemma 12.1 For any
Proof The family π * j B is proper by Lemma 3.8 and ample by Lemma 6.4. Finally, the maps in π * j B are efficient by the basic properties of factorization systems.
The next concept is analogous to the notion of development introduced in p. 10 of [3].

Definition 12.2 (Development of a cover) For
l ∈ K and j ∈ Y l , the development of G at g l and j is the family obtained by composing G with the families Roughly speaking the development of G at g l and j is the result of replacing g l with the family g l (π * j B). Lemmas 11.8 and 12.1 imply that any such development is in K(T Z ).
If B is 1-ample and (g l ) j < n then the development H of G at g l and j is also n-ample.
Proof Let h : T X → T Z be such that dh ≥ n. By hypothesis, h factors through G. That is, there exists k ∈ K and f : X → Y k such that h = g k f . If k = l then f factors through the trivial family and so h factors through H . If k = l then it is enough to show that f factors through π * j B. In turn, it is enough to prove that π j f : T X → T factors through B. Since B is 1-ample we are left to show that d(π j f ) ≥ 1. Calculate: Let us say that the family G can be developed to H if there exists a sequence of families G = G 0 , G 1 , G 2 , . . . , G n = H such that for each m < n, G m+1 is the development of G m at some g : T Y → T Z in G m and j ∈ Y .

Proposition 12.4
If B is strictly 1-ample then any n-ample family in K(T Z ) can be developed to a strictly n-ample family.
Proof Let G ∈ K(T Z ) be n-ample. If it is not strictly n-ample there exists a g : T Y → T Z in G such that dg = k∈Y g k < n. Then there exists j ∈ Y such that g j < n. Let G be the result of developing G at g and j . If every map h in G is such that dh ≥ n then we are done. If not, repeat the process. If the process terminates then the resulting family H is such that for every h in H , dh ≥ n. Moreover, Lemma 12.3 implies that, at each stage of the development, n-amplitude is preserved; so H is strictly n-ample. Now, why does the process terminate? Loosely speaking the reason is that since B is strictly 1-ample then, at each stage, the chosen 'tree' g is replaced by a family of trees whose 'leaves' are further from the root. Since this 'distance' can only grow up to n, and all the families involved are finite, then the process must terminate.
To be more precise let g : T Y → T Z in G and j ∈ Y as above. Each map in g(π * j B) is of the form g(π * j f ) for some f : T X → T in B. Since B is strictly 1-ample, df ≥ 1. That is, for every x ∈ X, f x ≥ 1. If we let Y j → Y be the complement of j : 1 → Y then we can picture g(π * j f ) as follows where the square is a pullback. For each z ∈ Y j + X, and and so (g(π * j f )) z = y∈Y j g y + (π y ) z = g z which is reasonable because we have developed at j ∈ Y j . On the other hand, if z ∈ X then π y z = ∞ for every y ∈ Y j so (g(π * j f )) z = (g j) + x∈X (f x) + (π x ) z = (g j) + (f z) > g j because f z ≥ 1. Roughly speaking, the new labels that have appeared in the tree g(π * j f ), i.e. those in X, are further from the root than the label j in the tree g. (Of course, if f is a constant the X is empty.) The proof of Proposition 12.4 should be compared with the discussion in p. 10 of [3]. Also on the relation with [3], notice that one can develop the trivial family in K(T X ) to a strictly n-ample one. These are analogous to the families denoted by S n in [3].
Corollary 12.5 (Potential is actual) If B is strictly 1-ample then K is the basis for a Grothendieck topology on T .
Proof Lemma 11.8 implies that K-coverings compose. So, by Lemma 6.8, it is enough to check that K has long covers in the sense of Definition 6.7. That is, that for every T Z in T and n ∈ N there is a H ∈ K(T Z ) such that for all h ∈ H , dh ≥ n. The trivial family (id | 1) on T Z is 0-ample so it is also n-ample and, by Proposition 12.4, can be developed to a strictly n-ample family.
The next result is analogous to the lemma in p. 16 of [3]. Notice that the structure of the proof is essentially the same. Proof By our current hypotheses, B is in K so the topology in the first item is included in that of the fourth.
To prove that the topology of the second item is included in that of the first we must show that, for every A in fSet and n ∈ N, the essentially unique proper and strictly n-ample family on T A is a cover with respect to the topology of the first item. This follows from Proposition 12.4 because it shows that using only B one can develop the trivial family (id T A | 1) to 'the' strictly n-ample family (Lemma 6.6).
It is trivial that all the sieves described in the third item are in the topology of the second. (The fact that the sieves in the third item form a topology will follow once we show that it includes K, for then all four items are equal.) So, to complete the proof, we consider an arbitrary sieve R on T A containing a K-cover and show that this sieve is among the sieves described in the third item. Assume that R contains an n-ample family F in K(T A ). It is clear that the maps in any development of F must be in R but then, by Proposition 12.4, the strictly n-ample family on T Y must be in R.
It must be stressed that Corollary 12.6 above is, in a sense, weaker than the lemma in p. 16 of [3] because the latter does not resort to efficient maps; something we used in the proof that K is a basis (see Lemma 11.8). So it is natural to search for a result showing that, under reasonable hypotheses, the efficiency requirement in the definition of K is superfluous. We do this in the next section.

Inconstant Maps
Let T be a ranked theory with associated natural potential basis K (Definition 11.7). In this section we assume that T is equipped with a distinguished strictly 1-ample family B ∈ K(T ) so that K is the basis for a Grothendieck topology by Corollary 12.5. We will show that in certain cases the efficiency requirement in the definition of K is unnecessary. First, a piece of very basic category theory. Lemma 13.1 Let the following diagram be a pullback. If π 1 is an iso, r is split epi and π 0 is epi then r is an iso.
Proof Without loss of generality we can assume that the following diagram is a pullback. Let s : B → C be a section of r. Since rsa = a the pullback property implies that sa = π 0 . Since, π 0 is epi, so is s. Proof Let f : T A → T B be efficient with a : A → B as above. Now let r : C → B be any map in fSet and let the diagram on the left below be a pullback in fSet. It follows that the rectangle on right above is also a pullback, and it induces a canonical q j : A rj → P as displayed there, which is mono because in rj is. Let h : T P → T C be the unique map such that the square on the left below commutes for every j ∈ C. Notice that h is a product of f i 's so it is efficient. We claim that (T r )f = h(T p 1 ) : T A → T C . In other words, that h, T p 1 form the efficient/bureaucratic factorization of (T r )f . To prove the claim calculate as on the right above and, together with (T j )(T r ) = T rj : T B → T 1 , we may conclude that for every point j : 1 → C, (T j )(T r )f = (T j )h(T p 1 ). That is, (T r )f = h(T p 1 ), so the claim is proved.
Consider now g as in the statement. Since g is bureaucratic it is of the form T r for some r : C → B in fSet and, as g is mono, r must be epi. Since f is inconstant, there is an a : A → B as above which is also surjective. By the argument above, gf = (T r )f = h(T p 1 ) so, if gf is efficient, then T p 1 is an iso; so p 1 is an iso and, by Lemma 13.1, r is an iso. Efficiency is 'reflected by developments' in the following sense.

Lemma 13.4 Let G be a proper family on T Z and let H be a development of G such that every map in H is efficient. If B contains an inconstant map then every map in G is efficient.
Proof Let H be the development of G at g : T Y → T Z and i ∈ Y . Then, to prove the result, we need only check that g is efficient. So let g = h(T b ) with h efficient and b : B → Y . Since g is mono, so is (T b ). We need to prove that T b is actually an iso.
By hypothesis there is an inconstant map in B and, since inconstant maps are closed under pullback, there is an inconstant map in π * i B. Let us call this inconstant map f : T X → T Y and consider the composite gf = h(T b )f . Since h is efficient, and gf is efficient by hypothesis, (T b )f is also efficient (by general considerations about factorizations systems). Then T b is an iso by Lemma 13.3.
We believe that Corollary 12.6, together with the next result, form a fair generalization of the lemma in p. 16 of [3] to our context. Proposition 13.5 If B has an inconstant map then the basis K may be described as assigning, to each T X in T , the collection of ample and proper families on T X . Proof Definition 11.7 introduces K as the function that assigns, to each T X in T , the collection of ample and proper families of efficient maps on T X . So it is enough to show that the requirement that the maps are efficient is superfluous. Let G be a proper and n-ample family on T X . Let H be the result of developing G to a strictly n-ample family. By Lemma 11.6, every map in H is efficient. By Lemma 13.4, every map in G is efficient.
We will use the following.

Corollary 13.6 If B has an inconstant map then K is a basis of dense monos in the sense of Definition 4.5.
Proof Every K-cover is proper by definition of K so to prove that K is a basis of dense monos consider an F ∈ K(T Y ) and a proper family P on T Y such that every map in F factors through P . Since F is ample then so is P . By Proposition 13.5, P is in K.

Weights for a Ranked Theory
Let T be a ranked theory with natural potential basis K. Assume that T is equipped with a distinguished strictly 1-ample family B ∈ K(T ) so that K is the basis for a Grothendieck topology. Moreover, let R be a rig and let γ : B(T ) → R be a multiplicative-monoid morphism so that we can consider weights as in Section 5.
to complete the proof.
The following is also easy.

Lemma 14.2 (Development does not change weight) Assume that wB
and H is the development of G at g l : T Y l → T Z and j ∈ Y l then wH = wG.
Proof By definition H is the family obtained by composing G with the families F k on T Y k , where F k is the trivial family (id Y k | 1) if k = l and F l = π * j B. Clearly, for any k = l, wF k = γ [T ] Y k and, by Lemma 14.1, wF l = γ [T ] Y l . Then, by Lemma 5.8, It is time to summarize. We are assuming that T is a ranked theory equipped a strictly 1ample family B in K(T ) so that K is the basis for a Grothendieck topology by Corollary 12.5. We are also assuming a rig R and a morphism γ : BT → R of multiplicative monoids. Proof Corollary 13.6 implies that K is a basis of dense monos and Lemma 14.3 implies that γ is compatible with K so Proposition 5.7 is applicable.

The Main Result
It may be convenient to start by recalling a basic fact. A morphism in a category is iso if and only if it is epi and has a retraction. In particular, if R is a rig and C is a distributive category then, a surjective R → BC is an iso if and only if it has a retraction. Most of the work in the paper was done to construct such a retraction. Lemma 15.1 below completes the construction. Theorem 15.2 makes explicit the simple surjection R → BC and proves that it is a section of the previous morphism.
Fix a polynomial L(X) ∈ N[X]. As discussed in Section 1, L(X) determines a 'signature' ∈ Set N and its associated free theory that we denote by L.
We let L be ranked as in Lemma 11.4 and we consider its natural potential basis K (Definition 11.7). For every n ≥ 0, each f ∈ n induces an efficient monic map in L that we denote by f : T n → T . (In free theories, efficient maps are mono. See discussion after Definition 11.1.) Let B be the (proper) family (f : T n → T | n ∈ N, f ∈ n). The proof of Lemma 11.4 implies that df ≥ 1 for each f . The family B is also 1-ample because any map h : T X → T with dh ≥ 1 is of the form fg for some f in B. Altogether, B in K(T ) is strictly 1-ample so K is the basis for a Grothendieck topology by Corollary 12.5 and we can consider the distributive category Fam(L, K). is the identity. It follows that N[X]/(X = L(X)) → B(Fam(L, K)) is an iso.

A Presentation of the Theory Classified by Sh(L, K)
In this section we give a presentation of the theory classified by the topos Sh(L, K). The argument is a straightforward generalization of the argument used in the proof of Theorem 4 in [3], but it seems useful to split it in a way that gives a solution to a particular case of a more general problem Let T be an algebraic theory and let T ‡ be the opposite of the category of finitely presented T -models. It is well-known that the classifying topos for T may be identified with the presheaf topos T ‡ . See, e.g., Corollary D3.1.1 in [10]. The full subcategory T → T ‡ induces a subtopos T → T ‡ . What does T classify? Diaconescu's Theorem gives one answer, but it is natural to strive for a more specific one when starting with an algebraic theory instead of an arbitrary (internal) category. We don't know of a general satisfactory answer to this question but we observe that a simple variation of Theorem 4 in [3] provides an answer for the case of free algebraic theories. At each set we have either proved that A | E is cocovered by the empty family or obtained a smaller presentation of the same model. In the second case we can repeat the steps. We can only do this a finite number of times because the size of A cannot decrease infinitely often and, after it stops decreasing, the 'total length' of E cannot decrease infinitely often. So the only way that the process can stop is if A | E is cocovered by the empty family or E has become empty.
Notice also that, at each step, the conclusion that A | E is cocovered by the empty family is the result of exhibiting a map from the model presented by the antecedent of one of the sequents in the second or third items in the statement. But free models cannot receive maps from these, so the topos Sh(C, J ) is equivalent to the topos Sh(T , J ) where T → C is the full subcategory determined by the free models and J is the trivial topology. In other words, the presheaf topos T classifies the theory presented in the statement.
Notice that the models of the theory classified by T are very similar to the Peano algebras discussed in [6]. See also Section 8 in [15].
We now give the analogue of Theorem 4 in [3] in the more general context of the present paper. Let L(X) ∈ N[X], let ∈ Set N be the induced signature, and L be the resulting free algebraic theory. (we stress that this is a finite disjunction because n = ∅ if n > deg L(X)).
Proof The method of D3.1.10 in [10], together with the calculations performed in Theorem 16.1, show that the classifying topos for the 'extended' presentation of the present result may be described as the topos of sheaves on the site (L, J ) where J is the Grothendieck topology generated by the new axiom; but this axiom simply says that the family (f : T n → T | n ∈ N, f ∈ n) covers T . So J is the smallest topology generated by this family. In other words, J is the topology generated by K (Corollary 12.6).
Much of the above seems to work without the one-sorted restriction. In particular, the notion of ample family. So it might be possible to extend the results in this paper to fixedpoint quotients of N[X 1 , . . . , X n ].