Monads of regular theories

We characterize the category of monads on $Set$ and the category of Lawvere theories that are equivalent to the category of regular equational theories.


Introduction
The category of algebras of a (finitary) equational theory can be equivalently described as either a category of models of a Lawvere theory or as a category of algebras of a finitary monad on the category Set or a category of algebras of a (generalized) operad, in which not only permutations but all functions between finite sets act on operations. In fact, the four categories of (finitary) equational theories, Lawvere theories, finitary monads on Set and (generalized) operads are equivalent. These equivalences induce a correspondence between various subcategories. In [SZ] we have given an intrinsic characterizations of equational theories and Lawvere theories that correspond to the analytic and polynomial monads on Set. This was achieved via correspondence with symmetric and rigid operads.
Recall that an equational theory is regular if it has a set of axioms in which each equation contains the same variable on both sides. Thus the theories of monoids and of sup-lattices are regular but the theory of groups is not. The category of regular equational theories and regular morphisms RegET was defined in [SZ] but the notion of a regular theory was discovered and studied in universal algebra (cf. [Pl]) and will turn fifty soon.
The main objective of this paper is to describe the categories of regular Lawvere theories RegLT, of semi-analytic monads SanMnd, and regular operads RegOp that correspond to the category of regular theories.
In [SZ] symmetric operads proved to be very useful in describing correspondences of this kind. Full and regular operads (cf. [Tr], Section 2) play the same role here. The category of full operads can be thought of as the category of monoids in the monoidal category Set F , where F is the skeleton of the category of finite sets, with the substitution tensor. Thus a monoid in Set F consists of operations that are equipped with not only a multiplication operator but also an action of the whole category F. As the left Kan extension Set F → End along the inclusion F → Set is an equivalence of monoidal categories (End is the category of finitary endofunctors on Set) the category of full operads is equivalent to the category of finitary monads on Set. Regular operads can be thought of as monoids in the monoidal category Set S with substitution tensor, where S is the skeleton of the category of finite sets and surjections. Thus in regular operads only surjections act on operations.
We identify the essential image of the left Kan extension Set S → End along inclusion S → Set, as the category of semi-analytic functors and semi-analytic natural transformations San. Semi-analytic functors have similar characterizations as analytic ones; cf. Section 2. A finitary endofunctor on Set is semi-analytic iff it preserves pullbacks along monomorphisms. Semi-analytic functors also have presentations via series similar to the series that represent analytic functors. A natural transformation is semi-analytic iff the naturality squares for monomorphism are pullbacks. The category of semi-analytic functors San is a monoidal subcategory of End and the monoids there form the category of semi-analytic monads equivalent to the category of regular theories; cf. Section 5.
The category of regular Lawvere theories is defined very much in the spirit of the definition of the category of analytic Lawvere theories, cf. Section 2. A regular Lawvere theory is a Lawvere theory with nicely behaving isomorphisms and a factorization system consisting of the class of projections and the class of regular morphisms. Regular morphisms in a Lawvere theory are by definitions those morphisms that are right orthogonal to all projections. A regular interpretation of Lawvere theories is an interpretation of Lawvere theories that preserves regular morphisms. We show that the category of regular Lawvere theories is equivalent to the category of regular operads and hence also to the category of semi-analytic monads and to the category of regular equational theories; cf. Section 6.
The category of semi-analytic monads SanMnd contains subcategories of cartesian CartMnd and weakly cartesian wCartMnd monads (i.e. those that preserve and weakly preserve pullbacks, respectively). One may think that such categories of monads can correspond to some natural subcategories of the category of Lawvere theories and of the category of equational theories. In Section 7 we characterize the subcategories of Set S that have as their essential images wCartMnd and CartMnd. As these characterizations are a bit technical we do not try to rephrase those conditions in terms of either Lawvere theories or equational theories.
In this way we shall describe the second level (level r) of the following diagram The vertical lines denote adjoint equivalences. Thus up to equivalence there are only four categories in it, one on each level. One equivalent to the category of all finitary monads on Set, second equivalent to the category of all semi-analitic monads on Set, third equivalent to the category of all analytic monads on Set, and forth equivalent to the category of all polynomial monads on Set. These levels are denoted by letters f , r, a, and p, respectively. Thus all four columns of equational theories, Lawvere theories, monads and operads are equivalent. These columns are denoted by letters e, l, m and o, respectively. The vertical functors heading up, in all columns but the column of operads, are inclusions of subcategories. In the column of operads the functors heading up are more like free extensions of the actions. The lower functors are full embeddings and the upper are embeddings that are full on isomorphisms. The vertical functors heading down, the right adjoints to those heading up are monadic. All the squares in the diagram commute up to canonical isomorphisms. The notation concerning categories involved is displayed in the above diagram.
The notation concerning functors is not on the diagram but it is meant to be systematic referring to levels and columns they 'connect'. The horizontal functors are denoted using letters from both columns they connect; the codomain by the script letter, the domain by its subscript, and the level is denoted by superscript. Thus the functor AnMnd → AnLT will be denoted by L a m . We usually drop superscripts when it does not lead to confusion. Thus we can write, for example, The vertical functors heading up are denoted by the script letter P with superscript indicating the column and subscript indicating the level of the codomain. The vertical functors heading down are denoted by the script letter Q with subscript and superscript as with those heading up. Thus we have, for example, functors P = P o = P o a : RiOp → SOp and Q = Q a = Q m a : Mnd → AnMnd. We will also refer to various diagonal morphisms and then we need to extend the notation concerning vertical functors by specifying both the columns of the domain and the codomain. For example, we write P ol f : SOp → LT to denote one such functor and its right adjoint will be denoted by Q lo a : LT → SOp. In principle this notation will leave the codomain not always uniquely specified but in practice it it sufficient, and in fact, usually much less is needed.

Notation
. . , n}, (n] = {1, . . . , n}, ω -denotes the set of natural numbers. The set X n is interpreted as X (n] when convenient. The skeletal category equivalent to the category of finite sets Set f in will be denoted by F. We will be assuming that the objects of F are sets (n] for n ∈ ω. The subcategories of F with the same objects as F but having as objects bijection, surjections and injections will be denoted by B, S, I, respectively. When S n acts on a set A n on the right and on the set B n on the left, the set A ⊗ n B is the usual tensor product of S n -sets.

Presentations of categories of algebras Equational theories
The category of regular theories, denoted by RegET, was introduced in [SZ]. It is a subcategory of all equational theories, denoted ET.

Operads
The symmetric operads provide yet another way of presenting models of an equational theory. This kind of presentation is usually very convenient, however the models defined by such operads are more specific than models of arbitrary equational theories. For example, if O is a symmetric operad then the free algebra functor Set → Alg(O) preserves weak wide pullbacks. Below we extend the definition of an operad so that it captures all the equational theories but still keeps the operadic flavor. The main difference is that instead of having just symmetric groups acting on sets of operations we have actions of the morphisms of the whole skeleton of the category of finite sets F. Symmetric operads can be thought of as monoids for the substitution tensor on the category Set B . Similarly F-operads can be thought of as monoids for the substitution tensor on the category Set F . By End we denote the category of finitary endofunctors of Set. It is a strict monoidal category with the tensor being composition. The substitution tensor on Set F makes the equivalence of categories Set F → End, given by left Kan extension, a strong monoidal equivalence. This immediately shows that the category of F-operads is equivalent to the category of finitary monads on Set. The definition below was first explicitly spelled out in [Tr].
A full operad (or F-operad) O consists a family of sets O n , for n ∈ ω, a unit element ι ∈ O 1 for any k, n, n 1 , . . . , n k ∈ ω with n = k i=1 n i a multiplication operation a left action of the morphisms on operations for n, m ∈ ω, such that the multiplication is associative with unit ι and compatible with the category action, i.e. for a ∈ O n a * ι = a = ι, . . . , ι * a; 1 n · a = a for a ∈ O n , b i ∈ O ki , c ij ∈ O mij , for i ∈ (n], j ∈ (k i ] we have c 1,1 , . . . , c 1,k1 , . . . , c n,1 , . . . , c n,kn * ( b 1 , . . . , b n * a) = = c 1,1 , . . . , c 1,k1 * b 1 , . . . , c n,1 , . . . , c n,kn * b n * a is a function such that j-th summand of the domain (k φ(j) ] is sent to the φ(j)-th summand (l φ(j) ] of the codomain by the function ψ φ(j) , i.e. j, r → φ(j), ψ φ(j) (r) for j ∈ (n] and r ∈ (k φ(j) ]. This definition refers to the obvious lexicographic order on both It is an extension of the multiplication operation in the operad of symmetries.
Remark The operation ⋆ is defined on a family of function indexed by k, l ∈ ω between sets: n,m∈ω, φ∈F(n,m) ki,li∈ω, k= A morphism of full operads f : O → O ′ is a function that respects arities of operations, unit, compositions, and the actions of functions from F.
The operation ⋆, when applied to morphisms in S, returns a morphism in S. Therefore these definitions make sense if we restrict morphisms to surjections i.e. morphisms in S 1 . In this way, we obtain the notion of a regular operad, a morphism of regular operads and the whole category of regular operads denoted RegOp.
We have functors Clearly, these definitions make also sense if we restrict morphisms in F to injections i.e. morphisms in I. But we will study such operads elsewhere.
The action of the category S in P o r (O) where φ : (n] → (m], f : (k] → (n] are surjection and a ∈ O k . The composition in P o where φ : (n] → (m] is a function, f : (k] → (n] is an injection, a ∈ O k and φ ′ , f ′ is the epi-mono factorization of φ • f : The rest of the definition of P o f (O) is similar to the remaining part of the definition of P o r (O) given above.

Lawvere theories
The category of Lawvere theories will be denoted by LT, see [Lw], [KR], [SZ] for details. F op is the initial Lawvere theory with the obvious projections. Let T be any Lawvere theory. By π : F op → T we denote the unique morphism from F op to T. The class of projections in T is the closure under isomorphisms of the image of the injections in F under π. A morphism r in T is regular iff r is right orthogonal to all projection morphisms in T. By a factorization system we mean a factorization system in the sense of [FK], see [CJKP] sec 2.8.
Aut(n) is the set of automorphisms of n in T. Recall from [SZ], that a Lawvere theory has simple automorphisms if the canonical function ρ n : S n × Aut(1) n −→ Aut(n) such that (σ, a 1 , . . . , a n ) → a 1 × . . . × a n • π σ is a bijection, for n ∈ ω. A Lawvere theory T is a regular Lawvere theory iff the projection morphisms and the regular morphisms form a factorization system and T has simple automorphisms. A regular interpretation of Lawvere theories is a morphism of Lawvere theories that preserves regular morphisms. Thus we have a non-full subcategory of LT of regular Lawvere theories with regular interpretations RegLT. The theory F op is regular.
We have inclusion functors

Monads
We introduce here a notion of a semi-analytic monad that is broader then that of an analytic monad but it still retains some combinatorial flavor.
Recall that an analytic (endo)functor on Set can be defined by any of the following conditions 1. finitary functor preserving weak wild pullbacks; 2. Kan extension of a functor from B to Set; 3. functor (isomorphic to one) having an analytic presentation n∈ω X n ⊗ n A n , where the n-coefficient A n is a (left) S n -set for n ∈ ω.
Similarly we shall define a semi-analytic (endo)functor on Set as a functor satisfying any of three equivalent conditions (see Proposition 2.1, Theorem 2.2) 1. finitary functor preserving pullbacks along monomorphisms; 2. Kan extension of a functor from S to Set along the inclusion functor i S : S → Set; 3. functor (isomorphic to one) having a semi-analytic presentation n∈ω X n ⊗ n A n , where the category S acts on coefficients A n on the left (see below).
The first two characterizations of the semi-analytic functors are clear. We shall describe the third one and show that it is equivalent to the other two. For the time being, the first definition of a semi-analytic functor is the official one. A natural transformation φ : F → G is semi-cartesian iff the naturality squares for monomorphisms are pullbacks. The category of semi-analytic functors with the semi-cartesian natural transformations will be denoted by San.

Examples
1. The functor P ≤n : Set −→ Set associating to a set X the set of subsets of X with at most n-elements is not analytic, if n > 2, as it can be easily seen that it does not preserve weak pullbacks. However, it preserves pullbacks along monos and hence it is semi-analytic.
2. If U is a set, n ∈ ω then the functor (−) U ≤n : Set → Set, associating to a set X the set of functions from U to X with an at most n-element image, is not analytic, if |U | > n > 2. Again it can be easily seen that it does not preserve weak pullbacks. However, it is semi-analytic.
3. We will see later that the functor part of any monad on Set that comes from a regular equational theory is semi-analytic.
For a set X and n ∈ ω the set X n denotes the set of monomorphisms from (n] to X. If X has less than n elements, this set is empty and if X is a finite set, then it has |X| n · k! elements. The notation is in analogy with the notation X n denoting the set of n-element subsets of X.
Clearly, X n is not functorial in X on its own but if we build a series with such sets and coefficients that are related by surjections we do get a functor. To see this let A : S → Set be a functor, or equivalently a sequence of coefficient sets {A n } n∈ω on which the category S acts on the left. As S n acts on X n on the right, by composition it makes sense to form a set X n ⊗ n A n and a whole coproductÂ The above formula is functorial in X, i.e.Â can be defined on morphisms as follows. Let f : X → Y be a function and [ x, a] an element of X n ⊗ n A n . We take the epi-mono factorization α, y of Proposition 2.1. The functors(−) : Set S −→ End is well defined and it is isomorphic to the left Kan extension along the inclusion functor i S : S → Set.
Proof. It is well known (see [CWM]) that the Kan extensions can be calculated with coends. Thus for A : S → Set and a function f : X → Y we have where x, a is the equivalence class of the equivalence relation ≈ on n∈ω X n × A n generated by the relation ∼ such that y • φ, a ∼ y, φ · a for y : (m] → X, φ : (n] → (m] ∈ S and a ∈ A n . Let us call an representant x, a of a class x, a minimal iff x is injective. Any element x, a is ∼-related to one of form y, f · a where f , y is the surjective-injective factorization of x, i.e. any element is ∼-related to a minimal one. It is easy to see that any two minimal representatives of ≈-equivalence class are ∼-related.
We define an isomorphism where x : (n] → X and a ∈ A n . Taking the minimal elements in the equivalence class is the inverse function. Thus (κ A ) X is a bijection. It is easy to see that κ A defined this way is natural in X, i.e. it is a natural isomorphism. For a natural transformation τ : obviously commutes, as both compositions of an element [ x, a] ∈Â(X) are equal to x, τ (a) .
is also a natural isomorphism, as required.
Theorem 2.2. The functor(−) : Set S −→ End is faithful, full on isomorphisms, and its essential image is the category of semi-analytic functors San.
The theorem will be proved via a series of lemmas.
Lemma 2.3. The essential image of the functor(−) : Set S −→ End is contained in the category San.
Proof. First we check that for A ∈ Set S ,Â preserves pullbacks along monos. Let be a pullback in Set. We need to show that the square Let α : (m] → (n ′ ] and z ′ : (n] → X be a surjection-injection factorization of the function f • z : (m] → X. Then there is a σ ∈ S n such that Hence there is a function p : (m] → P as in the following diagram Moreover, on representatives we have ThusÂ preserves pullbacks along monos indeed. Now let τ : A → B be a morphism in Set S . We shall show thatτ :Â →B is semi-cartesian. Let f : X → Y be an injection in Set. We need to show that the squarê where a ∈ A n . Note that as f is an injection, so is f • x and hence f • x, b represents an element inB(Y ). The above equality means that b ∈ B n and there is a σ ∈ S n such that On representatives we have and hence we haveτ Lemma 2.4. The functor(−) : Set S −→ End is faithful and full on semi-cartesian morphisms. In particular it is full on isomorphisms.
Proof. One can easily verify that if two natural transformationsτ ,τ ′ :Â →B agree on elements of the form [1 (m] , a] for m ∈ ω and a ∈Â(m] then natural transformation τ, τ ′ : A → B are equal. Thus(−) is faithful.
To show that(−) is full, let us fix two functors A, B ∈ Set S and a semi-cartesian natural transformation ψ :Â →B. Let m ∈ ω and a ∈ A m . We shall define τ m (a). We have for some injection f : (k] → (m] and b ∈ B k . We claim that k = m and f is a bijection. Suppose to the contrary, that k < m. We have thatB(f ) ([1 (k] Since f is an injection and ψ is semi-cartesian, the squarê is a pullback and hence there is an element [g, c] ∈Â(k] such that The first equality implies that the proper mono f • g is epi (as its codomain is (m] and its mono part is 1 (m] ). This is a contradiction. Hence f is a bijection and we can apply the functor B to it. We put To show that τ is natural, let us fix a surjection β : (m] → (n] and a ∈ A m . Then using definitions ofÂ,B and the naturality of ψ we have Since a and f were arbitrary, τ is natural. Finally, we show thatτ = ψ. Let us fix a set X and [ x : Using the naturality of ψ andτ on x and the definition of τ we have Since X and [ x, a] were arbitraryτ = ψ. Lemma 2.5. Each semi-cartesian functor is in the essential image of(−) : Set S −→ End.
Proof. Let us fix a semi-cartesian functor. We define a functor A : S → Set and a natural isomorphism τ :Â → F . Put Note that the sum over the empty index set is empty.
We form a pullback of a surjection α and a proper mono f Thus f ′ is again a proper mono. F preserves this pullback, i.e. the square The later equality means that a ∈ A(n] contrary to the supposition. Thus A : S → Set is a well defined functor. For a set X we define the function where n ∈ ω, x : (n] → X is an injection and a ∈ A(n].
First we show that the transformation τ :Â → F is natural. Let f : X → Y be a function.
i.e. τ is natural. It remains to show that τ is an isomorphism. Fix a set X. Let x ∈ F (x). As F is finitary, there is an n ∈ ω, f : (n] → X and y ∈ F (n] such that F (f )(y) = x. Let α, g be an epi/mono factorization of f Thus τ X is onto.
We form a pullback of monos As a ∈ A(n] and b ∈ B(m] we must have that n = k = m and both f and g are bijections. Put This τ X is an injection, as required. A monad (M, η, µ) on Set is a semi-analytic monad iff M is a semi-analytic functor and both η and µ are semi-cartesian natural transformations. The category of semi-analytic monads with the semi-cartesian morphisms will be denoted by SanMnd.
We have inclusion functors In this section we describe the equivalence of the category of full operads FOp to categories of equational theories ET, Lawvere theories LT and monads Mnd. As we said, the categories FOp and Mnd are categories of monoids of two equivalent monoidal categories thus they are obviously equivalent. Therefore we will just define the equivalences of categories with the domain FOp and their essential inverses leaving all the verifications to the reader. In [Tr] it was shown that the category of full operads is equivalent to the category of abstract clones. The latter category in known to be equivalent to ET. Let us fix a morphism h : O → O ′ of F-operads. We denote by κ n i : (1] → (n] the function that sends 1 to i, where n ∈ ω and we denote π n i the action of κ n i on the unit ι ∈ O 1 , i.e. π n i = κ n i · ι.

The functor L o
The identity on n is π n 1 , . . . , π n n : n −→ n and the i-th projection from n is π n i : n −→ 1 Recall that π n i is the value of the action of the function κ n i : (1] → (n] that picks i on the unit ι. The composition of morphisms is Q n = T(n, 1). The action in Q is given by where f ∈ Q n , φ as above, and π : F op → T is the unique morphism from the initial Lawvere theory. The composition in Q is given by products and composition in T Thus an element of this set is an equivalence class of pairs x, t ∈ X n × O n . We put The unit of the monad wherex : (1] → X sends 1 to x. The multiplication 'picking' s and the k terms t 1 , . . . , t k . The action (t 1 , . . . , t k ) ⋆ s is the element 'picked' by the morphism where functions α i are as above.

Regular Lawvere theories vs regular operads
In this section we study the relations between Lawvere theories and regular operads. We shall describe the adjunction Q ol r ⊣ P lo f and the properties of the embedding P lo f .

RegOp RegLT
The functor P f = P ol f : RegOp → LT Let O be a regular operad; ι, ·, * denote the unit, action (of S), and composition in O, respectively. We define a Lawvere theory P f (O) as follows. The set of objects of P f (O) is the set of natural numbers ω. A morphism from n to m is an equivalence class of spans such that φ : (r] → (n] is a function called amalgamation, f : (r] → (m] is a monotone function called the arity function (as it determines the arities of the operations g i ), r i = |f −1 (i)| and we have g i ∈ O ri for i ∈ (m] and r = m i=1 r i , moreover φ and f are jointly mono (or what comes out the same, φ is mono on the fibers of f ). Two spans φ, f, g i i∈m and φ ′ , f ′ , g ′ j j∈m ′ are equivalent whenever f = f ′ and there are permutations σ i : In particular |f −1 (i)| = r i = |f ′−1 (i)|, for i ∈ (m]. By i σ i : r → r we mean the permutation that is formed by placing permutations σ i 'one after another'. Thus, it respects the fibers of f , i.e. f • i σ i = f ′ . A morphism in the category P f (O) is a class of spans modulo the above equivalence relation. We could represent morphisms in P f (O) as spans without the requirement that φ and f are jointly mono. But then the relation identifying the spans would be more complicated. Instead of permutations σ i we would need to consider surjections. But this relation is not an equivalence relation and we would need to work with the equivalence relation generated by such a relation. However, it might happen, as with the compositions defined below, that an operation on spans naturally leads to a span whose amalgamation φ is not injective on fibers of f . In such a case we can regularize the span as follows. Let φ, f, g i i∈m be a span as above but with φ not necessarily injective on the fibers of f . Let φ i be the restriction of φ to the fiber f −1 (i) for i ∈ (m]. Let Then the regularization of the span φ, f, g i i∈m is the span f ′ is the monotone map sending the elements in the domain of φ ′ i to i, for i ∈ (m]. The composition of morphisms φ, f, g i i∈(m] : n → m and φ ′ , f ′ , g ′ j j∈(k] : m → k to φ ′′ , f ′′ , g ′′ j i∈(k] : n → k is defined in two steps as follows. In the diagram the square is a pullback of f along φ ′ . The functionf is so chosen that it is monotone. We define a span by f ′′ = f ′ •f , φ ′′ = φ ′ •φ, and g ′′ j = g ′ j * g φ ′ (l) l∈f ′−1 (j) . Finally, we take a regularization of this span to get a regular span that represents the composition. We leave for the reader the verification that the composition is a congruence with respect to the equivalence relation on spans.
The identity on n is the span n n n 1 n ✠ 1 n , ι i ❅ ❅ ❅ | The i-projection π n i : n → 1 on i-th coordinate is the span For a morphism of regular operads h : O → O ′ , we define a functor so that for a morphism φ, f, g i i∈(m] : n → m in P f (O), we have . This ends the definition of the functor P f .
The functor Q r = Q lo r : LT −→ RegOp Let T be a Lawvere theory. Recall that π : F op → T is the morphism from the initial Lawvere theory. The operad Q r (T) consists of operations of T, i.e. morphisms to 1. In detail it can be described as follows. The set of n-ary operations Q r (T) n is the set of n-ary operations T(n, 1) of T, for n ∈ ω. The action · : S(n, m) × Q r (T) n −→ Q r (T) m is given, for f ∈ T(n, 1) and φ ∈ S(n, m), by The identity of Q r (T) is ι = 1 1 ∈ T(1, 1). The composition * : Q r (T) n1 × Q r (T) n k × Q r (T) k −→ Q r (T) n is given, for f ∈ Q r (T) k and f i ∈ Q r (T) ni , where i ∈ (k], n = i∈k n i , by where f 1 × . . . , ×f k is defined using the chosen projections in T and • is the composition in T.
If F : T → T ′ is a morphism of Lawvere theories then the map of regular operads This ends the definition of the functor Q r .
The adjunction P f ⊣ Q r and the properties of the functor P f We note for the record We have an easy Lemma 4.2. Let O be a regular operad and n ∈ ω. An isomorphism on n in P r (O) has a representation by a span of the following form n n n φ ✠ 1 n , a i i ❅ ❅ ❅ | where φ : (n] → (n] is a bijection, a i ∈ O 1 is an invertible operation, i.e. there is b i ∈ O 1 such that a i * b i = ι = b i * a i for i ∈ (n]. It is the unique span in its equivalence class.
Proof. Suppose that we have two spans (2) representing two morphisms that compose to identity 1 n both ways. Then, since the displayed composition is 1 n , the functions φ, f ′ are surjections. Hence the functions φ ′ , f are surjections, as well. Having this it is easy to notice that φ sends elements in different fibers of f to different elements. Thus φ (and φ ′ ) must be in fact a bijection. If we take a composition g i i∈f ′−1 (j) * h j and regularize it multiplying by a surjection, we will get ι. This implies that h j cannot be a nullary operation and that the arity of g i 's is at most one. Now, as the above spans represent morphisms that compose both ways to identity, it is easy to see that we get the required description.
Proposition 4.3. We have an adjunction P f ⊣ Q r . The functor P f is faithful.
Proof. We shall show that P f ⊣ Q r . For a regular operad O the unit is We verify the triangular equalities. For g ∈ Q r (T) n = T(n, 1) we have As the unit η O is mono, P f is faithful. LT commutes up to an isomorphism.

RegOp
FOp commutes up to an isomorphism, where the functor Op f l is the left adjoint to L f o and together they form an adjoint equivalence. The functor Op f l is defined as the functor Q r except the action involved is the action of the whole category F rather than its subcategory S. As the functor Q o r forgets this additional part of the action the above diagram clearly commutes.
Proposition 4.5. The functor P f : RegOp → LT is full on isomorphisms and its essential image is RegLT. In particular RegOp is equivalent to RegLT.
Proof. Recall that we have a unique morphism of Lawvere theories from the initial theory π : F op → P f (O). For a function φ : m → n, π φ the morphism π φ is represented by the span of form Clearly, both classes contain isomorphisms and are closed under composition.
Any morphism φ, f, g i i∈(m] : n → m in P f (O) has a projection-regular factorization as follows Thus to show that projections and (what we have described as) regular morphisms form a factorization system it remains to show that projection morphisms are left orthogonal to the regular morphisms. Let with the left vertical morphism φ, 1 r , a i j∈(k] a structural map and right vertical morphism 1 m , !, g a regular map. We have chosen the right bottom to be 1 to simplify notation but the general case is similar. The commutativity means that r = r ′ and there is a permutation σ ∈ S r such that we see that the permutations 1 r and σ show that both triangles commute. Thus regular morphisms are indeed right orthogonal to the structural ones and P f (O) is a regular Lawvere theory.
From the description of the functor P f (h) : P f (O) → P f (O ′ ) and the description of the structure of P f (O), it is clear that P f (h) sends the regular (projection) morphisms to the regular (projection) morphisms. Thus P f (h) is a regular morphism of Lawvere theories. Now let T be any Lawvere theory. As the class of regular morphisms in T is right orthogonal to a class of morphisms and it is closed under composition, finite products and isomorphisms. Moreover, for any surjection in ψ : (n] → (m] in F the image π ψ : m → n in T is regular as it is orthogonal to all projection morphisms in T. Thus surjections in S act on all regular morphisms f : n → 1 in T on the left · : S(n, m) × T(n, 1) −→ T(m, 1) Hence the regular operations of any Lawvere theory T form a regular operad. The unit is the identity morphism on 1. The composition f 1 , . . . , f n * f is defined to be f • (f 1 × . . . × f n ). The action of S is defined as above. Let us denote this operad by T r . We have an inclusion morphism of regular operads T r → Q r (T). By adjunction we get a morphism of Lawvere theories Clearly, ξ T is bijective on objects. If T is regular then ξ T is full (faithful) since the projectionregular factorization exists (is unique and π : F op → T is faithful).
If I : T → T ′ is a regular interpretation between any Lawvere theories, then the diagram where I r is the obvious restriction of I to T r . Thus the essential image of P f is indeed the category of regular Lawvere theories and regular interpretations. An isomorphic interpretation of Lawvere theories is always regular. Therefore P f is full on isomorphisms.
We have Proposition 4.6. The functor Q r : LT → RegOp is monadic.
Proof. We shall verify the assumption of the Beck monadicity theorem. By Proposition 4.3 Q r has a left adjoint. It is easy to see that Q r reflects isomorphisms. We shall verify that LT has and Q r preserves Q r -contractible coequalizers.
Let I, I ′ : T ′ → T be a pair of interpretations between Lawvere theories so that is a split coequalizer in RegOp. We define a Lawvere theory T O so that a morphism from n to m in T O is an m-tuple g 1 , . . . , g m with g i ∈ O n , for i = 1, . . . , m. The compositions and the identities in T O are defined in the obvious way from the compositions and the unit in O. The projectionsπ n i in T O are the images of the projections π n i in T, i.e.π n i = q(π n i ). The functorq : First we verify, that T O has finite products. For this, it is enough to verify that f 1 , . . . , f n * π n i = f i , where * is the multiplication in the operad O. The uniqueness of the morphism into the product is obvious from the construction. We have routine calculations It is obvious thatq is a morphism of Lawvere theories and that Q r (q) = q. It remains to verify thatq is a coequalizer in LT. Let p : T → S be a morphism in LT coequalizing I and I ′ .
The morphism Q r (p) coequalizes Q r (I) and Q r (I ′ ) in RegOp. Thus there is a unique morphism k in RegOp making the triangle on the right commute. We define the functork so that for any morphism f 1 , . . . , f n in T O . The verification thatk is the required unique functor is left for the reader.

Semi-analytic monads vs regular operads
The main objective of this section is to show that the square In particular for a set X we have In the set X n ⊗ n O n we identify x • σ, a with x, σ · a for a ∈ O n , x : (n] → X and σ ∈ S n .
Let γ : S → F be the inclusion functor. It induces the following diagram of categories and functors that we describe below γ * is the functor of composing with γ. It has a left adjoint Lan γ , the left Kan extension along γ. For C ∈ Set S it is given by the coend formula The functor i sa : San → End is just an inclusion. The equivalence i S : Set S −→ San is defined by left Kan extension that may be given by the coend formula and a coproduct where C ∈ Set S . Similarly, the equivalence is defined by left Kan extension that is given by the coend formula where B ∈ Set F . The following calculation shows that the right hand square in the above diagram commutes: The functor (−) sa , right adjoint to i sa , is given by the formula for F ∈ End. (−) sa associates to functors and natural transformations their 'semi-analytic parts'. Note that both San and End are strict monoidal categories with tensor given by composition, and i sa is a strict monoidal functor. Thus its right adjoint (−) sa has a unique lax monoidal structure making the adjunction i sa ⊣ (−) sa a monoidal adjunction. This in turn gives us a monoidal monad (W, η, µ) on San.

MonCat
Cat where MonCat is the 2-category of monoidal categories, lax monoidal functors, and monoidal transformations; Mon is the 2-functor associating the monoids objects to monoidal categories, | − | is the forgetful functor forgetting the monoidal structure, and U is a 2-natural transformation whose component at a monoidal category M is the forgetful functor from monoids in M to the underlying category of M : U M : Mon(M ) → |M |.
Applying U to the monoidal adjunction and i sa ⊣ (−) sa and monoidal monad W we get an adjunction between categories of monoids and a monad on Mon(San). The unnamed arrow in the above diagram is Mon(i sa ). But the monoids in End and San are exactly monads and hence we get the left most adjunction P m f ⊣ Q m r together with the monad (W,η,μ) on the category of semi-analytic monads.
On the other hand, on the categories Set F and Set S there are substitution tensors making i F and ı S monoidal equivalences and γ * and Lan γ monoidal adjunctions. Thus we can apply the 2-functor Mon to this adjunction and obtain an adjunction Mon(Lan γ ) ⊣ Mon(γ * ) as in the diagram The unnamed functor is Mon((−) sa ). But monoids in Set F and Set S are (equivalent to) the categories of full and regular operads, respectively. The verification that the right most square commutes serially is left for the reader. We obtain Proposition 5.1. The square (3) of categories and functors commutes up to an isomorphism.
Proof. Both horizontal adjunctions in the square (3) are obtained from equivalent monoidal adjunctions. It remains to show that the identifications we obtained above are isomorphic to the functors M f o and M r o , respectively. This is left for the reader. There are free monads on finitary functors (cf. [Barr]) and free semi-analytic monads on semianalytic functors. The adjunctions F ⊣ U and F ⊣ U induce monads R and R, respectively. R is the finitary version of what is called 'the monad for all monads' in [Barr]. Putting this additional data to the above diagram and simplifying it at the same time we get a diagram

SanMnd San
Mnd In the above diagram the square of the left adjoints commutes. Thus, the square of the right adjoint commutes as well. This shows in particular that the free monad on a semi-analytic functor is semi-analytic.
The monadW is a lift of a monad W to the category of R-algebras SanMnd and, by [Be] we obtain Theorem 5.2. The monad R for regular monads distributes over the monad W for finitary functors, i.e. we have a distributive law λ : RW −→ WR The category of algebras of the composed monad WR on SanMnd is equivalent to the category Mnd of all finitary monads on Set.
Remark. We arrived at the above theorem with essentially no calculations. We give below explicit formulas how to calculate the values of some functors mentioned above and we shall also describe the coherence morphism on the monoidal monad W. The coherence morphism ϕ : First we describe the adjunction i sa ⊣ (−) sa . We shall drop the inclusion i sa when possible. Let A ∈ San and G ∈ End and X be a set. The regular functor A is given by the (functor of) its coefficients. Its value at X is The unit of the adjunction i sa ⊣ (−) sa at X where x : (n] → X and a ∈ A n . The counit of the adjunction at X where x : (n] → X is an injection and t ∈ G(n].
The multiplication in the monad W where x : (n] → X, g : (m] → (n], f : (k] → (m], and a ∈ A k . This ends the definition of the monad W. Now we shall describe the monoidal structure on W. If B is another analytic functor, the n-coefficient of the composition A • B is given by where the equivalence relation ∼ n is such that for σ ∈ S n , σ i ∈ S ni , τ ∈ S m , b i ∈ B i , for i ∈ (m] and a ∈ A m we have The coherence morphism ϕ for W at the n-th coefficient of the functor A is Note that this map is well defined at the level of equivalence classes.

Equational theories vs regular operads
In this section we study the relations between regular equational theories and regular operads. We shall show that the square (4) commutes up to an isomorphism, with P e f being an inclusion and both horizontal functors being equivalences of categories. P o f ' was defined in Section 2 and E f o was defined in Section 3. We shall define E r o .
The functor E r o : RegOp → ET Let O be a regular operad. We define an equational theory E r o (O) = (L, A). As the set of n-ary function symbols we put L n = O n for n ∈ ω. The set of axioms A contains the following three kinds of equations in context: For a ∈ O n we put E r o (h)(a) = h(a)(x 1 , . . . , x n ) : x n for n ∈ ω. Clearly, E r o (h) is a regular interpretation.
Proposition 6.1. The square (4) commutes up to a natural isomorphism.
Proof. Let O be a regular operad. We define an interpretation of equational theories where a ∈ O n and [1 n , a] is an n-ary operation symbol of the theory E f o P o f (O). We need to verify that I is a well defined natural isomorphism. First we need to verify that I O preserves axioms. The unit axiom is obvious. To prove the action axioms, we fix a ∈ O m and τ : (m] → (n] and we calculate using the theory E The calculations for the composition axioms are similar. The naturality of I O is left for the reader. We shall show that I O is an isomorphism of theories. To this end we define an inverse interpretation given by Finally, we need to verify that I O and J O are mutually inverse one to the other. The composition J O I O sends operation a ∈ O n to the term a(x 1 , . . . , x n ) : x n so it is the identity. For an operation It is left for the reader to verify that h has the required property.
To see that E r o is essentially surjective let us fix a regular theory T = (L, A). Then the regular terms in T form a regular operad called T ro . The unit is the term x 1 : x 1 . The composition is defined by the substitution (making sure that we make the variable disjoint in different substituted term, via α-conversion). The action of a surjection φ : (n] → (m] on a regular term t(x 1 , . . . , x n ) : x n is again a regular term t(x φ(1) , . . . , x φ(n) : x n . Again it is a matter of a routine verification that E r o (T ro ) ∼ = T .

Examples
1. Let 1 be the terminal equational theory. It has one constant, say e, and can be axiomatized by a single axiom: v 1 = e : v 1 . As a Lawvere theory it is the category that has exactly one morphism between any two objects. It is best seen at the level of Lawvere theories. Both theories 1 a and the theory of commutative monoids are linear-regular and have exactly one analytic morphism a : n → 1, for any n.
2. The functor Q e r : ET → RegET is a right adjoint, it preserves the terminal object. Hence 1 r , the regular part 1, is the terminal regular theory. It is the theory of suplattices.
The embedding of the regular theories into all equational theories has a right adjoint (−) sr , as well. The value of the functor (−) sr on the terminal equational theory 1 is the terminal regular theory, i.e. the theory of suplattices: 3. The theory 1 has a proper subcategory, in which 0 ∼ = 1. It has no function symbols, and can be axiomatized by a single axiom: v 1 = v 2 : v 2 . The regular part of this theory is the theory of suplattices without a bottom element.

Cartesian and weakly cartesian monads
In this section we shall investigate two (strict monoidal) subcategories of San and their categories of monoids. The category of (weakly) cartesian functors and (weakly) cartesian natural transformations will be denoted by Cart (wCart). The corresponding categories of monoids: the category of (weakly) cartesian monads will be denoted by CartMnd (wCartMnd). Thus we have embeddings full on isomorphisms Cart −→ wCart −→ San which are strict monoidal and induce embeddings of categories of monoids The characterizations of the subcategories of equational theories ET and of Lawvere theories LT corresponding to CartMnd and wCartMnd are a bit technical and we are not going to describe it in detail here. Clearly, the objects are some regular theories satisfying additional conditions and similarly for morphisms. We shall content ourselves with a description of subcategories of Set S whose essential images are wCart and Cart, respectively. Note, however, that if (T, η, µ) is a semianalytic monad such that the functor part T is the left Kan extension of a functor R : S → Set then R is the functor of all regular operations in the equational theory corresponding to the monad T . Thus our description will in fact provide a description of the equational theories corresponding to monads in wCart and Cart.
Remarks. If in a weak pullback square the morphism m is mono then the square is a pullback. Since m is mono iff the square is a pullback, it follows that if a functor weakly preserves pullbacks it does preserve monos as well.
Recall the description of functor(−) : Set S −→ End from Section 2, the Kan extension along i S : S → Set. We begin with the following observation then there are m ∈ ω, surjections g : n → m, g ′ : n ′ → m, and an injection y : m → Y as in the diagram Proof. Exercise. The following two Propositions identify the subcategory of Set S whose essential image in End is wCart.  Thus ( x, τ m (c)) ∼ (1 (m] , b) and we have that k = m, x is a bijection, ( x, c) ∼ (1 (m] , A( x(c)). Hence we also have τ m (A( x)(c)) = b. Moreover as the square (n] (n] ✲ 1 (n] (m] Thus A( x)(c) is the element sought for a and b. Since f , a and b were arbitrary, τ is weakly cartesian. The final two Propositions identify the subcategory of Set S whose essential image in End is Cart.
Proposition 7.4. Let A : S → Set be a functor. The functorÂ : Set → Set preserves pullbacks iff the functor A satisfies the condition (WPB) from the Proposition 7.2, and additionally satisfies the following condition (PB): suppose that the square is a pullback of surjections in F, x : (q] → (p], x ′ : (q ′ ] → (p] are two injections, c ∈ A(q], c ′ ∈ A(q ′ ] are two elements so that the functions

are surjections and
Then q = q ′ and there is σ ∈ S q such that x ′ • σ = c, A(σ)(c) = c ′ .
Proof. Assume that A satisfies (WPB) and (PB). Thus, by Proposition 7.3,Â weakly preserves pullbacks. Let be a pullback in Set. We shall show that the squarê is a pullback, i.e. it satisfies the uniqueness condition. So let [h, a] ∈ P q ⊗ q A(q), and [h ′ , a ′ ] ∈ P q ′ ⊗ q ′ A(q ′ ) be such thatÂ The first equality means that [ x, τ n (a)] = [ x ′ , τ n (a ′ )]. Hence n = n ′ and there is a σ ∈ S n such that x ′ • σ = x, and τ n (a ′ ) = B(σ)(τ n (a)) = τ n (A(σ)(a)) By Lemma 7.1 and the second equality in (7) there are surjections g and g ′ and an injection y as in the diagram Then we have As y is mono g = g ′ • σ. Thus A(g ′ )(a ′ ) = A(g)(a) = A(g ′ )(A(σ)(a)) As τ is a cartesian natural transformation we get, from the fact that the naturality square for g' is a pullback, that a ′ = A(σ)(a). But this means that