Theory and Applications of Categories, Vol. 11, No. 11, 2003, pp. 252–282.

CONTINUOUS CATEGORIES REVISITED

J. ADÁMEK^{∗)}, F. W. LAWVERE, J. ROSICKÝ^{∗)}

ABSTRACT. Generalizing the fact that Scott’s continuous lattices form the equational hull of the class of all algebraic lattices, we describe an equational hull of LFP, the category of locally finitely presentable categories, over CAT. Up to a set-theoretical hypothesis this hull is formed by the category of all precontinuous categories, i.e., categories in which limits and filtered colimits distribute. This concept is closely related to the continuous categories of P. T. Johnstone and A. Joyal.

1. Introduction

1.1. Operations on the Category LFP. In our previous work we have shown that algebra is not algebraic in the sense that the 2-category VAR of varieties is not equational over CAT. And we have described an equational hull of VAR as the 2-category of all algebraically exact categories, see [ALR]. The present paper solves the same question for the 2-category LFP of all locally finitely presentable categories of Gabriel and Ulmer. We prove that, analogously to Scott’s continuous lattices forming the equational hull of all algebraic lattices, an equational hull of LFP is formed by all “precontinuous” categories, a concept closely related to continuous categories of [JJ]. Actually, our present work is stronger than that concerning VAR: we present a description of the equational hull with respect to all operations of small arities (under the set-theoretic assumption (R) explained below), whereas for varieties we had to restrict ourselves to the accessible operations, which do not include all operation of small arities.

Recall that the morphisms considered by Gabriel and Ulmer between locally finitely presentable categories are the functors preserving limits and filtered colimits – or, equivalently, the finitary right adjoints. With natural transformations as 2-cells, this yields a 2-category LFP considered as a concrete category over CAT via the (non-full) embedding

U : LFP ⟶ CAT .

The question of equationality of LFP can now be formulated precisely: is the functor U pseudomonadic? The answer, as we prove below, is negative.

Since we speak about equational hull, we must first explain what operations on the 2-category LFP are and what are equations. The concept of operation follows the idea of Lawvere’s dissertation where, given a category 𝒞 concrete over Set, i.e., with a forgetful functor U : 𝒞 → Set, a k-ary operation ω is a natural transformation ω : U^{k} → U (assigning, thus, to every object C of 𝒞 a k-ary operation map ω_{C} on the set UC so that all morphisms f of 𝒞 fulfil: Uf preserves the k-ary operations map in the usual sense). Here k can be any cardinal number, but we often want to restrict ourselves to k finite. The first idea of a generalization to LFP would be: a k-ary operation is a natural transformation ω : U^{k} → U. Here k can be any category, but we often want to restrict ourselves to k small. Example: the formation of limits of type k should be a k-ary operation ω. Indeed, all morphisms of LFP preserve limits, thus, they preserve ω. Alas, this is only true non-strictly, i.e., the operation “limits of type k” is not a natural transformation (= preserved by functors strictly) but a pseudonatural transformation. This leads us to the following

∗) Supported by the Grant Agency of the Czech Republic under the grant No. 201/99/0310.

Received by the editors 2002-02-28 and, in revised form, 2003-05-21.

Transmitted by R.J. Wood. Published on 2003-05-28.

2000 Mathematics Subject Classification: 18A35, 06B35.

Key words and phrases: locally finitely presentable category, precontinuous category, continuous lattice, pseudomonad.

\begin{matrix}c \\ ◯ \end{matrix} J. Adámek, F. W. Lawvere, J. Rosický, 2003. Permission to copy for private use granted.

Definition. By a k-ary operation on LFP is meant a pseudonatural transformation ω : U^{k} → U.

Thus, ω assigns to every locally finitely presentable category A an “operation functor” ω^{A} : A^{k} → A and to every finitary right adjoint F : A → B a natural isomorphism ω̂^{F} as follows:

\xymatrix{A^{k} \ar[d]^{\underrightarrow{ω̂_{F}}}_{F^{k}} \ar[r]^{ω^{A}} & A \ar[d]^{F} \\ B^{k} \ar[r]_{ω^{B}} & B}

with natural coherence conditions.

Examples. (1) “Limits of type k” is a k-ary operation k-lim on LFP for every small category k.

(2) “Colimits of type k” is a k-ary operation k-colim_{→} on LFP for every small filtered category k.

(3) The trivial operation (corresponding to variables, considered as the appropriate projection operations, in universal algebra) is ev_{x}, evaluation at x, for every object x of the category k:

ev_{x}^{A} : A^{k} → A , F ⟼ F x for all F ∈ A^{k} .

One of the main results of our paper is that the above three types of operations are exhaustive: all operations of small arities on LFP are generated (in a sense made precise below) by the trivial operations, the limit-operations and the filtered-colimit-operations. This result requires a certain restriction on the set theory we work with. It holds whenever the following axiom

(R) every uniform ultrafilter is regular is assumed – we discuss this axiom in Section 5 below. And this result fails to hold assuming that arbitrarily large measurable cardinals exist.

We will also consider operations of large arities. However, we do not have any nice generating families for them.

1.2. Equational Laws for LFP-Operations. We not only prove that limits and filtered colimits generate all operations of small arities on LFP, we also describe all equational properties of those operations. They, of course, include

(i) Equational properties of limits (in general categories). Recall that completeness is equational over CAT in the sense that for the pseudomonad

Lim

of free limit-completion on CAT the 2-category of algebras is the 2-category of complete categories. Since locally finitely presentable categories are complete, they satisfy (i). Analogously for

(ii) Equational properties of filtered colimits (in general categories) expressing algebras over Grothendieck’s pseudomonad

Ind

of free completion with respect to filtered colimits on CAT. And the third obvious equational property holding in every locally finitely presentable category is

(iii) Finite limits commute with filtered colimits. Is this all? No, there is one item missing: all locally finitely presentable categories have the following property

(iv) Products distribute over filtered colimits. This means, for a given category 𝒞 with products and filtered colimits, that for any set D_{i} (i ∈ I) of filtered diagrams in 𝒞 the canonical morphism

colim_{→} Π_{i∈I}D_{i}d_{i} ⟶ Π_{i∈I}colim_{→}D_{i}d_{i}

is an isomorphism. (This is less “popular” than commutation of finite limits with filtered colimits. But e.g. Grothendieck and Verdier work with the property (iv) explicitly in [AGV].)

Now what we prove assuming (R) above, is that the equational properties of (i)–(iv) generate all equations between operations of small arities on LFP. Let us stress here that equations

ω_{1} ≈ ω_{2} (for ω_{1}, ω_{2} : U^{k} → U)

are, once again, meant non-strictly: the meaning is not that the components ω_{1}^{A} and ω_{2}^{A} are equal for every locally finitely presentable category A, but that a coherent collection of natural isomorphisms i^{A} : ω_{1}^{A} ⟶^{≅} ω_{2}^{A} is given.

How is the above result proved? We introduce first an equational hull of LFP with respect to operations of small and large arities. This is performed by analogy to the case of algebraic lattices (a posetal shadow of LFP) which we explain in the next subsection: we find a pseudomonad 𝔻^{∗} on CAT whose category of algebras is an equational hull of LFP. We then turn to the small-core 𝔻_{small}^{∗} of that pseudomonad 𝔻^{∗} and argue that algebras for that pseudomonad form an equational hull of LFP with respect to operations of small arities. By a recent result of Adámek, Koubek and Trnková [AKT] we derive the formula

𝔻_{small}^{∗}K = Ind (Lim K) for all K small,  (∗)

assuming the set-theoretical axiom (R). It turns out that the two pseudomonads Lim (free completion under limits) and Ind (free completion under filtered colimits) have a distributive law – here we were surprised that Grothendieck and Verdier have not come to that conclusion in [GV]: they prove that if a functor F preserves finite limits then so does Ind F, but actually, this is true (and needed) for all limits, see Lemma 5.5 below. This distributive law turns Ind ∘ Lim into a pseudomonad biequivalent to 𝔻_{small}^{∗}. Thus, algebras of Ind ∘ Lim form an equational hull of LFP with respect to operations of small arities. But those algebras are just algebras of the lifting of Ind to the category of all Lim -algebras (i.e., to the category of all complete categories). We call them precontinuous categories. Explicitly:

Definition. A category A with limits and filtered colimits is called precontinuous provided that the functor

colim_{→} : Ind A ⟶ A

(replacing formal filtered colimits with actual filtered colimits in A) preserves limits.

This concept is closely related to continuous categories of Johnstone and Joyal [JJ] defined as categories A with filtered colimits such that colim_{→} is a right adjoint. The precise relationship of the concepts is discussed in Section 5.

The last step we use is the following result of Adámek, Rosický and Vitale [ARV]: precontinuity is equivalent to the above conditions (iii) and (iv). Thus, categories satisfying (i)–(iv) are precisely the precontinuous categories, and assuming (R), they form an equational hull of LFP with respect to operations of small arities.

1.3. From Continuous Lattices to Precontinuous Categories. Recall that the fundamental concept of algebraic lattice in universal algebra is nothing else than a poset that is, as a category, locally finitely presentable. That is, a complete lattice in which every element is a (directed) join of compact (= finitely presentable) elements. It has been known since the 1970’s that algebraic lattices do not form a variety, and that Scott’s continuous lattices are an equational hull. Recall that a lattice is continuous iff (infinite) meets distribute over directed joins – which is precisely the conditions (iv) above.

Now algebraic lattices are precisely the lattices

Ind L L a (meet-)semilattice

where Ind L, a free completion under directed joins, is the well known lattice of ideals of L. Homomorphisms for algebraic lattices are the maps preserving all meets and all directed joins. Consider now the category Slat of semilattices and homomorphisms as a monadic category over the category Pos of partially ordered sets

Slat = Pos^{𝔻}

for the monad 𝔻 of free semilattices (= semilattices of all finitely generated down sets in a given poset). We obtain an adjoint situation using the two-element chain 2 as a dualizing object:

[figure: Slat(−,2); _{Slat_{op}}; Pos; Pos(−,2)]

This adjoint situation defines a monad 𝔻^{∗} over Pos (the filter monad) whose algebras are precisely the continuous lattices. This has been proved by A. Day [D] and O. Wyler [W]. We present a different proof here: the monad 𝔻^{∗} of the above adjunction is a composite of the monad Ind (of free directed joins) with the monad Meet (of free meets) in the sense of Beck’s distributive laws [B]. This implies immediately that algebras for this monad 𝔻^{∗} are Ind -liftings to complete semilattices (= Meet-algebras), and this is precisely the category of continuous lattices. Observe that the comparison functor

K : Slat^{op} ⟶ Pos^{𝔻^{∗}}

is full and faithful and maps Slat^{op} to the category of algebraic lattices. Observe further that K is a right adjoint to Pos^{𝔻^{∗}}(−, 2) : Pos^{𝔻^{∗}} ⟶ Slat^{op}; thus, algebraic lattices are reflective in their equational hull.

We now generalize this from posets to categories by replacing

(i) the category of algebraic lattices with LFP,

(ii) Pos with CAT,

(iii) Slat with CAT_{lex}, the 2-category of lex (= finitely complete) categories, lex (= finite-limit preserving) functors and natural transformations and

(iv) Meet with Lim and the ideal lattice with Grothendieck’s completion. Let us recall that for small categories A we have

Lim A ≅ (Set^{A})^{op}

and for all categories A we can describe Ind A as the category of all small lex functors from A^{op} to Set. Small means: a small colimit of hom-functors.

Using the category Set as the dualizing object, we obtain a 2-adjoint situation

[figure: CAT_{lex}(−,Set); _{CAT{op \atop lex}}; CAT; CAT(−,Set)]

defining a 2-monad 𝔻^{∗} over CAT. A characterization of 𝔻^{∗}-algebras is an open problem. For small categories A we have

𝔻^{∗}A = CAT_{lex}(Set^{A}, Set) ,

and assuming (R) every lex functor from Set^{A} to Set is small, as proved in [AKT]. Thus 𝔻^{∗}A is a free completion of Lim A = (Set^{A})^{op} under filtered colimits and we get (∗).

Set-theoretical foundations. In the present paper we are not concerned with set theory (except for the discussion of the axioms (R) and non(M) restricted to section Section 5 below), but here we want to make clear what (very meager) requirements are needed below. A chosen universe of small sets is assumed satisfying the Axiom of Choice (AC). In a higher universe categories (not assumed locally small in general) live. And when “categories” outside of this universe are considered, they are called quasicategories; example: CAT. Analogously with 2-categories and 2-quasicategories. But where a 2quasicategory is biequivalent to a 2-category (such as LFP, due to Gabriel-Ulmer duality), we call it a 2-category. And we assume that every category is an ∞-filtered union of its small subcategories (i.e., a large union which is λ-filtered for every cardinal λ). All limits and colimits are meant to be small (unless explicitly stated large).

2. Continuous Lattices Revisited

2.1. The Monad 𝔻 of Semilattices. We recall shortly that the forgetful functor

U : Slat → Pos

of the category Slat of (meet-) semilattices and their homomorphisms into the category Pos of poset has a left adjoint: For each poset X a free semilattice over X,

η_{X}^{𝔻} : X → 𝔻(X) ,

is given by the poset 𝔻(X) of all finitely generated up sets (i.e., all sets ↑M, where M ⊆ X is finite), ordered by dual inclusion, and η_{X}^{𝔻} is the principal-filter embedding x ↦↑{x}.

This gives rise to the free-semilattice monad 𝔻 = (𝔻, μ^{𝔻}, η^{𝔻}) over Pos whose category of algebras is (isomorphic to) Slat. Moreover, 𝔻 is an enriched monad over the (cartesian closed) category Pos, i.e., the maps

hom(X, Y) → hom(𝔻(X), 𝔻(Y))

are order-preserving. Consequently, every 𝔻-algebra S yields an enriched contravariant hom-functor

Slat(−, S) : Slat^{op} ⟶ Pos .

For S = 2, the chain 0 < 1, this assigns to each semilattice A the poset

Slat(A, 2) = all filters on A

ordered by inclusion; here we identify a homomorphism h : A → 2 with the set h^{−1}(1) which is a filter (i.e., up set closed under finite meets).

2.2. The Dual Monad 𝔻^{∗}. Since the monad 𝔻 is enriched over Pos, exponentiation can be transferred from Pos to 𝔻-algebras. In particular, we get the canonical functor

2^{(−)} : Pos → Slat^{op}

assigning to every poset X the semilattice 2^{X} of all up sets ordered by inclusion. This functor is left adjoint to Slat(−, 2): for posets X and semilattices A we have the following natural bijections:

[figure: X; Slat(A, 2); [in Pos]; X × A; 2; [∧-preserving in the 2^{nd} coordinate]; A; 2^{X}; [in Slat]; 2^{X}; A; [in Slat^{op}]]

This adjoint situation

2^{(−)} ⊣ Slat(−, 2)

leads to a new monad

𝔻^{∗} = (𝔻^{∗}, η^{∗}, μ^{∗})

called dual monad of 𝔻 (with respect to the dualizer 2). Here 𝔻^{∗} assigns to each poset X the poset

𝔻^{∗}X = Slat(2^{X}, 2) = all filters in 2^{X}

and η_{X}^{∗} : X → 𝔻^{∗}X is the principal-filter embedding: η_{X}^{∗}(x) is the filter of all up sets containing x.

2.3. 𝔻^{∗}-Algebras. They are easy to describe when we observe that (a) the well-known monads Ind (of free directed joins) and Meet (of free meets) distribute in Beck’s sense and (b) the monad 𝔻^{∗} is the corresponding composite

𝔻^{∗} ≅ Ind ∘ Meet .

In more detail, for each poset X we denote by

η_{X}^{Meet} : X → Meet(X)

the principal-filter embedding into the poset Meet(X) of all up sets of X ordered by dual inclusion (i.e., dual ordering of 2^{X}). This is well-known to be a free completion of X under meets. And by

η_{X}^{Ind} : X → Ind(X)

we denote the principal-ideal embedding into the poset Ind(X) of all ideals (i.e., directed down sets), ordered by inclusion. This is well-known to be a free completion of X under directed joins.

2.4. Beck’s Distributive Laws. We recall here briefly the concept of distributive law of a monad 𝕊 over monad 𝕋 (on the same category K) as presented by J. Beck [B]. Let (𝕊, η^{𝕊}, μ^{𝕊}) and (𝕋, η^{𝕋}, μ^{𝕋}) be monads on the category K.

A distributive law of 𝕊 over 𝕋 is a natural transformation

ℓ : 𝕊𝕋 → 𝕋𝕊

such that (i) ℓ · η^{𝕊}𝕋 = 𝕋η^{𝕊}, (ii) ℓ · 𝕊η^{𝕋} = η^{𝕋}𝕊, (iii) ℓ · μ^{𝕊}𝕋 = 𝕋μ^{𝕊} · ℓ𝕊 · 𝕊ℓ,

and

(iv) ℓ · 𝕊μ^{𝕋} = μ^{𝕋}𝕊 · 𝕋ℓ · ℓ𝕋 This defines a composite monad

𝕋𝕊 = (𝕋𝕊, η^{𝕋}𝕊 · η^{𝕊}, μ^{𝕋}𝕊 · 𝕋𝕋μ^{𝕊} · 𝕋ℓ𝕊) .

(Conversely, every composite monad gives rise to a unique distributive law.) By a lifting of the monad 𝕋 into the category of 𝕊-algebras is meant a monad

𝕋̃ = (𝕋̃, η̃, μ̃) over K^{𝕊}

such that (α) U^{𝕊}𝕋̃ = 𝕋U^{𝕊}, (β) U^{𝕊}η̃ = ηU^{𝕊}

and

(γ) U^{𝕊}μ̃ = μU^{𝕊}. Every distributive law ℓ defines a lifting by the rule

𝕋̃(A, α) = (𝕋A, 𝕋α · ℓ_{A}) ,
η̃(A, α) = η_{A}

and

μ̃(A, α) = μ_{A} .

Conversely, every lifting gives rise to a unique distributive law.

𝕊𝕋 ⟶^{𝕊𝕋η^{𝕊}} 𝕊𝕋𝕊 = 𝕊𝕋U^{𝕊}F^{𝕊} = 𝕊U^{𝕊}𝕋̃F^{𝕊} =

U^{𝕊}F^{𝕊}U^{𝕊}𝕋̃F^{𝕊} ⟶^{U^{𝕊}ε^{𝕊}𝕋̃F^{𝕊}} U^{𝕊}𝕋̃F^{𝕊} = 𝕋U^{𝕊}F^{𝕊} = 𝕋𝕊

where F^{𝕊} is left adjoint to U^{𝕊} and ε^{𝕊} : F^{𝕊}U^{𝕊} → Ind_{K^{𝕊}} is a counit.

There is an isomorphism

K^{𝕋𝕊} ≅ (K^{𝕊})𝕋̃

commuting with the natural forgetful functors into K.

2.5. Distributivity of Meet over Ind. Consider K = Pos, 𝕋 = Ind, and 𝕊 = Meet. The following lemma gives a lifting of Ind over complete meet-semilattices, i.e., over Pos^{𝕊}, and therefore shows a distributive law of Meet over Ind:

Lemma.

(i) Ind(A) is a complete (semi)lattice for each poset A;

(ii) Ind(f) is a complete-semilattice homomorphism whenever f is such;

(iii) η_{A}^{Ind} and μ_{A}^{Ind} are complete-semilattice homomorphisms for each complete lattice A.

Proof. (i) Ind(A) is closed under meets (=intersections) in 2^{A}.

(ii) Ind(f) : Ind(A) → Ind(B) assigns to every ideal Z the ideal ↓ f(Z). Given Z = ⋂_{t∈T} Z_{t} in Ind(A) we prove ⋂_{t∈T} ↓f(Z_{t}) ⊆↓f(Z)– thus, Ind(f) preserves meets. For

every element z ∈ ⋂_{t∈T} ↓f(Z_{t}) we choose z_{t} ∈ Z_{t} with z ≤ f(z_{t}) for each t ∈ T to conclude

that

z ≤ ⋀_{t∈T} f(z_{t}) = f(⋀_{t∈T} z_{t}) ∈ f(Z) ,

thus, z ∈↓f(Z).

(iii) The principal-ideal embedding η_{A}^{Ind} obviously preserves meets. And μ_{A}^{Ind} preserves meets because it is right adjoint to Ind(η_{A}^{Ind}): given ideals Z ∈ Ind(A) and Y = {Y_{t}} ∈ Ind(Ind(A)) then

Z ≤ μ_{A}^{Ind}(Y)

means Z ⊆ ⋃_{t∈T} Y_{t}, and since Y is a down set, this is equivalent to ↓ {z} ∈ Y for each

z ∈ Z, i.e., to

↓η_{A}^{Ind}(Z) ≤ Y .

Remark. This corollary yields a distributive law

ℓ : Meet ∘ Ind → Ind ∘ Meet ;

since Ind(Meet(A)) is a complete lattice, Ind(η_{A}^{Meet}) extends uniquely to

ℓ_{A} : Meet(Ind(A)) → Ind(Meet(A))

preserving joins. This resembles the distributive law studied by Marmolejo, Rosebrugh and Wood in [MRW].

Corollary. Ind∘Meet is a monad over Pos whose algebras are Ind-liftings to complete semilattices, i.e., complete (semi)lattices L with directed joins such that the directed-join map

Ind(L) → L

preserves meets. (These are precisely continuous lattices L.) And Ind ∘ Meet-homomor-phisms are the maps preserving meets and directed joins. (These are the homomorphisms of continuous lattices.)

Notation. Cont denotes the category of continuous lattices and their homomorphisms, that is, the category of Ind ∘ Meet-algebras.

2.6. Corollary. The monads 𝔻^{∗} and Ind ∘ Meet are isomorphic.

In fact, their Kleisli categories are concretely isomorphic since a free 𝔻^{∗}-algebra on any poset X is just the free Ind ∘ Meet-algebra on X:

Slat(Pos(X, 2), 2) = Slat(Meet(X), 2) = Ind(Meet(X)).

3. Equational Doctrines

3.1. Enriched Monads in CCC. Here we just briefly recall some concepts (introduced for the case of the category C of all categories in [L_{2}]) in the more general setting of an arbitrary cartesian closed category C. By an enriched monad 𝔻 over C we understand an enriched functor 𝔻 : C → C together with enriched natural transformations η : 1_{C} → 𝔻 and μ : 𝔻𝔻 → 𝔻 satisfying the usual monad axioms. The category

C^{𝔻}

of (strict) algebras and homomorphisms is then also enriched over C: the internal homfunctor

Hom_{𝔻} : (C^{𝔻})^{op} × C^{𝔻} → C

is defined by means of the following equalizer for 𝔻-algebras (A, α) and (B, β):

\xymatrix{Hom_{𝔻}((B, β), (A, α)) \ar[r]^{e} & A^{B} \ar[rr]^{A^{β}} \ar[dr]^{𝔻} &  & A^{(𝔻B)} \\  &  & (𝔻A)^{(𝔻B)} \ar[ur]_{α^{(𝔻B)}}}

Example. C = Pos, 𝔻 the monad of finitely generated up sets (see 1.1): this is an enriched monad with

Hom_{𝔻}((B, ≤), (A, ≤)) = all order-preserving maps in B^{A} ordered pointwise.

3.2. The Dual Monad 𝔻^{∗}. We now fix a 𝔻-algebra

𝔻S ⟶^{σ} S

called dualizer, and observe that the enriched hom-functor

Hom_{𝔻}(−, (S, σ)) : (C^{𝔻})^{op} → C

has a left adjoint which can be constructed as follows (see [L_{2}]): for every object C ∈ C we denote by

eval_{C}^{∗} : C → (𝔻S)^{𝔻(S^{C})}

the composite of the evaluation map C → S^{(S^{C})} and the 𝔻-map S^{(S^{C})} → 𝔻S^{𝔻(S^{C})}, and we consider the following adjunctions:

[figure: C; ^{eval∗ C}; (𝔻S)^{𝔻(S^{C})}; C × 𝔻(S^{C}); 𝔻S; 𝔻(S^{C}); ϱ_{C}; (𝔻S)^{C}]

We now define a functor

S^{(−)} : C → (C^{𝔻})^{op}

on objects by assigning to C ∈ C the 𝔻-algebra

𝔻(S^{C}) →^{ϱ_{C}} (𝔻S)^{C} →^{σ^{C}} S^{C}

and to every morphism f : C → C̄ in C the 𝔻-homomorphism

S^{f} : (SC̄, σC̄ · ϱC̄) → (S^{C}, σ^{C} · ϱ_{C}) .

Then we get an enriched adjoint situation

S^{(−)} ⊣ Hom_{𝔻}(−, S)

(the argument in this general case is analogous to that in 2.2 above).

Notation. The enriched monad generated by the last adjoint situation is denoted by 𝔻^{∗} = (𝔻^{∗}, η^{∗}, μ^{∗}) and is called the dual monad of 𝔻 with respect to the dualizer (S, σ). Thus,

𝔻^{∗}C = Hom_{𝔻}(S^{C}, S)

3.3. Equational Hull. An equational hull of the category dual to C^{𝔻}, the category of 𝔻-algebras with respect to the “forgetful” functor Hom_{𝔻}(−, (S, σ)) : (C^{𝔻})^{op} → C is given by the comparison functor

K : (C^{𝔻})^{op} → C^{𝔻^{∗}} .

Obviously, this is a right adjoint to an attempted inverse

K^{∗} = Hom_{𝔻^{∗}}(−, S).

Of course, in general neither composite is the identity, but if one of them is, then the other is a closure operation or “hull”. This further left adjoint was mentioned in 1. for the case of posets.

Examples. (i) In the above-mentioned example we have C = Pos, 𝔻 = the semilattice monad (thus, (C^{𝔻})^{op} ≅ the category of algebraic lattices) and K the full embedding into the category of continuous lattices.

(ii) Put C = Set and 𝔻 the semilattice monad, again, i.e., 𝔻X is the set of all finite subsets of X with η_{X}^{𝔻} the singleton map and μ_{X}^{𝔻} the union map. Using the dualizing object 2 again, we obtain an adjoint situation

[figure: Slat(−,2); _{Slat_{op}}; ⊤; Set; Set(−,2)]

This generates as 𝔻^{∗} the filter monad on Set, see [D]: 𝔻^{∗}X is the set of all filters on the set X (= on the semilattice 2^{X}) whose Eilenberg-Moore category is Cont, and the comparison functor K : Slat^{op} → Cont has, again, as image the category of algebraic lattices.

(iii) For C = Set and 𝔻 = Ind we obtain an adjoint situation

[figure: Set(−,2); _{Set_{op}}; ⊤; Set; Set(−,2)]

whose dual monad 𝔻^{∗} (given by X ↦ 2^{2^{X}}) is the monad of complete atomic Boolean algebras. Here the comparison functor K : Set^{op} → Set^{𝔻^{∗}} is an equivalence.

3.4. Pseudomonads and Algebras. (1) Recall the concept of a pseudomonad on a 2-category K, as introduced by B. Day and R. Street [DS]; we use the notation of F. Marmolejo [M_{2}]. A pseudomonad 𝔻 = (𝔻, d, m, η, β, μ) consists of a bifunctor

𝔻 : K → K

with 1-cells

d : id_{K} → 𝔻 and m : 𝔻 · 𝔻 → 𝔻

together with invertible 2-cells

\xymatrix{𝔻 \ar[r]^{d𝔻} \ar[dr]_{Id_{𝔻}} & 𝔻𝔻 \ar[d]_{⇐_{β}}_{m} & 𝔻 \ar[l]^{⇐_{η}}_{𝔻d} \ar[dl]^{Id_{𝔻}} & 𝔻𝔻𝔻 \ar[r]_{⇐_{η}}^{𝔻m} \ar[d]_{m𝔻} & 𝔻𝔻 \ar[d]^{m} \\  & 𝔻 &  & 𝔻𝔻 \ar[r]_{m} & 𝔻 ,}

such that two equations of the expected type must be satisfied.

(2) Recall further the concept of an algebra for a pseudomonad 𝔻: it is a quadruple

(X, x, ψ, χ)

consisting of an object (0-cell) X, a morphism (1-cell) x : 𝔻X → X

and invertible 2-cells

[figure: X; dX; Id_{X}; 𝔻X; ⇐^{ψ}; x; X; 𝔻𝔻X; ⇐^{χ}; 𝔻x; mX; 𝔻X; x; 𝔻X; x; X]

This data must satisfy two expected equations.

We denote by

K^{𝔻}

the 2-category of 𝔻-algebras. Its 1-cells, called homomorphisms, from (X, x, ψ, χ) to (Z, z, ξ, θ) are pairs (h, ϱ) where

h : X → Z

is a 1-cell of K and

\xymatrix{𝔻X \ar[r]_{⇐^{ϱ}}^{𝔻h} \ar[d]_{x} & 𝔻Z \ar[d]^{x} \\ X \ar[r]_{h} & Z}

is an invertible 2-cell, such that two expected equations are satisfied.

The 2-cells of K^{𝔻} are defined as follows:

Given homomorphisms (h, ϱ), (h′, ϱ′) : (X, x, ψ, χ) ⟶ (Z, z, ζ, θ), a 2-cell

ξ : (h, ϱ) ⟶ (h′, ϱ′)

is a 2-cell ξ : h ⟶ h′ such that (ξ ∘ x) · ϱ = ϱ′ · (z ∘ 𝔻ξ). Vertical composition is the obvious one.

Horizontal composition: for (h, ϱ) : (ψ, χ) ⟶ (ζ, θ) and (k, π) : (ζ, θ) ⟶ (τ, σ) we define (k, π) ∘ (h, ϱ) = (k ∘ h, (k ∘ ϱ) · (π ∘ 𝔻h)).

4. Gabriel-Ulmer Duality Revisited

4.1. The Pseudomonad 𝔻 of Lex Categories. Generalizing the situation of U : Slat → Pos in 2.1, we consider here the forgetful functor (non-full inclusion)

U : CAT_{lex} → CAT

where CAT is the 2-quasicategory of all categories (and functors, and natural transformations), and CAT_{lex} is the sub-2-quasicategory of all categories with finite limits (and functors preserving finite limits, and natural transformations).

This functor U has a left adjoint: for every category X we consider a free completion of X under finite limits. We obtain a 2-adjoint situation and denote by 𝔻 = (𝔻, η^{𝔻}, μ^{𝔻}) the pseudomonad on CAT it generates.

The category CAT^{𝔻} of 𝔻-algebras (see 3.4) is biequivalent to CAT_{lex} – this easily follows from the fact that 𝔻 is a (typical example of a) co-KZ-doctrine, see [K], [M_{1}].

4.2. The Dual Pseudomonad 𝔻^{∗}. We use the category Set of (small) sets as the dualizing objects. We thus consider the enriched hom-functor

CAT_{lex}(−, Set) : CAT_{lex}^{op} → CAT

assigning to every lex category A the category

LEX A = CAT_{lex}(A, Set)

of all lex functors in Set^{A}. Since 𝔻 is an enriched monad over CAT, exponentiation is transferred from CAT to CAT^{𝔻}, and we obtain the canonical 2-functor

Set^{(−)} = CAT(−, Set) : CAT → CAT_{lex}^{op} .

Analogously to 2.2, the 2-functors above form a 2-adjoint situation: for a category X and a lex category A we have the following natural bijections:

[figure: X; F; LEX A; [in CAT]; X × A; Set; [lex in the 2^{nd} coordinate]; A; F′; _{Set_{X}}; [in CAT_{lex}]; _{Set_{X}}; A; [in CAT_{lex}^{op}]]

This adjoint situation

Set^{(−)} ⊣ LEX(−)

generates the dual pseudomonad

𝔻^{∗} = (𝔻^{∗}, η^{∗}, μ^{∗})

over CAT. Thus, 𝔻^{∗} is the 2-functor defined on objects (categories) X by

𝔻^{∗}X = LEX(Set^{X})

on 1-cells (functors) F : X → Y by

𝔻^{∗}F : H ⟼ H · Set^{F} for each lex H : Set^{X} → Set ,

and on 2-cells (natural transformations) f : F → F′ by

(𝔻^{∗}f)_{H} = H ∗ Set^{f} .

The unit

η_{X}^{∗} : X → LEX(Set^{X})

is given by X ↦ ev_{X}, where

ev_{X} = Set^{X} (X (X, −), −)

is the evaluation-at-X functor. And the multiplication

μ_{X}^{∗} : LEX(Set^{𝔻^{∗}X}) → LEX(Set^{X})

is defined by means of the evaluation functor

ev : Set^{X} → Set^{LEX(Set^{X})}

as follows

μ_{X}^{∗} (G) = G · ev for each lex G : Set^{𝔻^{∗}X} → Set .

Notation. For every functor

F : X → LEX A (X ∈ CAT, A ∈ CAT_{lex})

we denote by

F̂ : 𝔻^{∗}X → LEX A

the functor which to every H ∈ 𝔻^{∗}X = LEX(Set^{X}) assigns the composite of the lex functor F′ : A → Set^{X} (see 4.2) with H:

F̂(H) = H · F′ for all H ∈ 𝔻^{∗}X .

Observation. (1) For each functor F : X → Y, the functor

𝔻^{∗}F : H ⟼ H · Set^{F}

preserves limits and filtered colimits (even those large ones that exist in 𝔻^{∗}X). In fact, 𝔻^{∗}F preserves all object-wise (large or small) limits, and 𝔻^{∗}X is closed under all existing limits in Set^{Set^{X}}; analogously with filtered colimits.

(2) Also η_{X}^{∗} and μ_{X}^{∗} preserve limits and filtered colimits (including all existing large ones). For η_{X}^{∗} this follows from the formula η_{X}^{∗} X = Set^{X} (X (X, −), −) and the fact that LEX Set^{X} is closed under limits and filtered colimits in Set^{Set^{X}}. For μ_{X}^{∗} : G ↦ G · ev this is also evident.

(3) Finally, each F̂ : 𝔻^{∗}X → LEX A preserves (all existing) limits and filtered colimits.

4.3. Open Problem. Characterize the category CAT^{𝔻^{∗}} of 𝔻^{∗}-algebras.

Remark. The answer to the above problem could be called a full equational hull of LFP. Here “full” refers to the possibility of forming k-ary operations where k is an arbitrary (possibly large) category.

In the next section we will aim at describing an equational hull of LFP with respect to small-ary operations.

4.4. Comparison Functor.

K : CAT_{lex}^{op} → CAT^{𝔻^{∗}} .

This functor assigns to every lex category A the 𝔻^{∗}-algebra whose underlying category is LEX A, and whose operation-map is the functor

Îd : 𝔻^{∗}(LEX A) → LEX A

(see Notation 4.2), thus,

KA = (LEX A, Îd) .

For every lex functor F : A → B we have

K(F) : (LEX B, Îd) → (LEX A, Îd)

defined by

K(F) : H ⟼ H · F for all lex H : B → Set

For reasons analogous to the above (4.2), K(F) preserves limits and filtered colimits for each F.

4.5. Gabriel-Ulmer Duality. The image of the comparison functor

K : CAT_{lex}^{op} → CAT^{𝔻^{∗}}

restricted to small categories is precisely LFP. That is, denote by

Cat_{lex}

the 2-category of all small lex categories (and functors preserving finite limits and natural transformations). Then K[Cat_{lex}^{op}] is biequivalent to LFP. In fact:

(a) for every small lex category A the algebra KA = (LEX A, Îd) is determined by the (locally finitely presentable) category LEX A. And conversely, every object of LFP is isomorphic to one of the form LEX A, A small and lex;

(b) for every lex functor F : A → B the homomorphism K(F) : LEX B → LEX A preserves limits and filtered colimits (see 4.4). The main ingredient of the Gabriel-Ulmer duality is, then, the following converse: every homomorphism from LEX B to LEX A has the form K(F) for some lex functor F : A → B.

Lemma. Let A and B be small lex categories. For every functor H : LEX B → LEX A equivalent are:

(i) H : (LEX B, Îd) → (LEX A, Îd) is a 𝔻^{∗}-homomorphism;

(ii) H preserves limits and filtered colimits, and

(iii) H ≅ K(F) for some lex functor F : A → B.

Proof. i→ii. Let J : J → LEX B be a diagram. Since Îd preserves limits, we have

lim J ≅ Îd(lim η_{LEX B} · J) .

Hence

H(lim J) ≅ H(Îd(lim η_{LEX B} · J))
≅ Îd · 𝔻^{∗}(H)(lim η_{LEX B} · J)
≅ lim(Îd · 𝔻^{∗}(H) · η_{LEX B} · J)
≅ lim(Îd · η_{LEX A} · H) ≅ lim(H · J)

because 𝔻^{∗}(H) preserves limits. Therefore H preserves limits. Analogously we prove that H preserves filtered colimits.

ii→iii. By the Adjoint Functor Theorem 1.66 in [AR], H has a left adjoint G : LEX A → LEX B. Since H preserves filtered colimits, the functor G preserves finite presentability of objects. Since A is lex, finitely presentable objects of LEX A are precisely the representable functors, analogously for B. Consequently, there is a functor F : A → B with G(A(A, −)) ≅ B(F A, −) for all A ∈ A. The functor K(F) is easily seen to be a right adjoint to G, thus, K(F) ≅ H.

iii→i. Trivial.

4.6. Corollary. The quasicategory

CAT^{𝔻^{∗}}

of 𝔻^{∗}-algebras is an equational hull of LFP.

That is, given a pseudomonad 𝕋 over CAT and a functor

E : LFP → CAT^{𝕋}

such that the functor U_{𝕋}·E : LFP → CAT is naturally isomorphic to the forgetful functor U : LFP → CAT, then there exists a pseudofunctor

E^{∗} : CAT^{𝔻^{∗}} → CAT^{𝕋}

with

E ≅ KE^{∗} and U_{𝔻^{∗}} ≅ U_{𝕋} · E^{∗}

\xymatrix{LFP \ar[rrrr]^{K} \ar[dddd]_{U} \ar[ddrr]^{E} &  &  &  & CAT^{𝔻^{∗}} \ar[ddll]^{_{E_{∗}}} \ar[dddd]^{U_{𝔻^{∗}}} \\  &  & ≅ \\  & ≅ & _{CAT_{𝕋}} \ar[ddll]^{U_{𝕋}} \ar[ddrr]_{U_{𝕋}} & ≅ \\  &  & = \\ CAT \ar[rrrr]_{Id} &  &  &  & CAT}

Proof. It is sufficient to show that

(a) every 𝔻^{∗}-algebra is an absolute pseudocolimit of limits of locally finitely presentable categories

and

(b) the forgetful functors U_{𝕋} (in particular, U_{𝔻^{∗}}) create absolute pseudocolimits and limits.

In fact, we then define E^{∗} using E^{∗}(K(A)) = A and extending this to CAT^{𝔻^{∗}} by means of absolute pseudocolimits and limits.

Concerning (a), every 𝔻^{∗}-algebra is an absolute pseudocolimit of free 𝔻^{∗}-algebras, as proved in [MLV], and every free 𝔻^{∗}-algebra

(𝔻^{∗})^{2}A ⟶^{μ_{A}} 𝔻^{∗}A = LEX(Set^{A})

is a limit of the (large) ∞-cofiltered diagram of all LEX(𝒞) where 𝒞 is a small subcategory of Set^{A} closed under finite limits (and where for 𝒞 ⊆ 𝒞 ′ we have an embedding I : 𝒞 → 𝒞 ′ giving a connecting morphism LEX(I) : LEX(𝒞 ′) → LEX(𝒞)). The limit cone is then, analogously, given by LEX-images of the embeddings 𝒞 ↪ Set^{A}. Each LEX(𝒞) is a locally finitely presentable category.

Concerning (b), this has been proved by [MLV].

4.7. Corollary. LFP is not pseudomonadic over CAT.

This follows from the case of continuous lattices and the fact that LFP contains all algebraic lattices (which, being posets, are strict 𝔻^{∗}-algebras). In fact, let L be a continuous lattice which is not algebraic. Being an Ind ∘ Meet-algebra, L is a split coequalizer of homomorphisms of algebraic lattices (recall that Ind(−) is always algebraic). This gives, then, a split coequalizer of two strict homomorphisms of strict 𝔻^{∗}-algebras such that the codomain of the coequalizer (necessarily, a strict 𝔻^{∗}-algebra) L lies outside LFP: in fact, lattices which are in LFP are precisely the algebraic ones.

4.8. Remark. As stated in 4.3, we do not know how to describe the equational hull of LFP, i.e., the category CAT^{𝔻^{∗}} – we do not even know whether this is a “property” (i.e., a subcategory of CAT) or a “structure”.

It is natural to restrict our attention to operations of small arities, more precisely, to work instead of the pseudomonad 𝔻^{∗} with its “small core” 𝔻_{small}^{∗} defined on objects by forming the ∞-filtered colimit

𝔻_{small}^{∗}A = colim 𝔻^{∗}B

of the free algebras generated by all small subcategories B of A. Can we, then, describe the “small-equational hull”

CAT^{𝔻_{small}^{∗}}

of LFP?

5. Small-Equational Hull of LFP

5.1. Remark. We have seen in Section 4 that the pseudomonad 𝔻^{∗} whose algebras form an equational hull of the category LFP is given by

𝔻^{∗}K = LEX(Set^{K}) .

In the present section we describe algebras of the pseudomonad

𝔻_{small}^{∗}

which is the small core of 𝔻^{∗}; see 4.8. We prove that, under a certain set-theoretical axiom (R), we have the formula

𝔻_{small}^{∗}k = Ind (Lim k) for all k small,

analogous to the formula for algebraic lattices in 2.6 above. And we start this section by observing a distributive law of the pseudomonad Lim on Ind , from which it follows that Ind ∘ Lim is a pseudomonad. The above formula yields a biequivalence

𝔻_{small}^{∗} ≈ Ind ∘ Lim .

Therefore, an equational hull of LFP with respect to small-ary operations is the category of algebras of Ind ∘ Lim . We call these algebras precontinuous categories.

5.2. Pseudomonad Ind . The role that Ind, the free completion under directed joins, played for continuous lattices in 2.3 above is now played by Grothendieck’s completion Ind , i.e., free completion under filtered colimits. For every category X we describe η_{X}^{Ind} : X → Ind X as the codomain restriction of the Yoneda embedding X → Set^{X^{op}} into the full subcategory Ind X of all flat functors (i.e., filtered colimits of hom-functors) in Set^{X^{op}}. This leads to a pseudomonad

(Ind , η^{Ind}, μ^{Ind})

over CAT.

An alternative description of Ind X can be presented for all categories with finite coproducts:

5.3. Notation. For every category ℒ with finite products denote by

Lexℒ

the full subcategory of Set^{ℒ} formed by all lex functors which are small, i.e., small colimits of hom-functors.

5.4. Remark. If X is a category with finite coproducts, then

Ind X = LexX^{op} .

More precisely, the Yoneda embedding

X ⟶ LexX^{op}

is a free completion of X under filtered colimits.

Now we prove that Ind distributes over the pseudomonad of free completion under limits:

5.5. Lemma.

(i) Ind X is complete for every complete category X.

(ii) Ind F preserves limits whenever F : X → Y preserves limits (X and Y complete)

(iii) η_{X}^{Ind} and μ_{X}^{Ind} preserve limits for each complete category X.

Proof. (i) See [AGV].

(ii) We use the following description of Ind K (see e.g. [JJ]): objects are all filtered diagrams in K. Morphisms from D : D → K to D′ : D′ → K are compatible families of equivalence classes [f_{d}] (d ∈ objD) of morphisms f_{d} : Dd → D′d′ in K under the smallest equivalence ∼ with f_{d} = D′δ · f_{d} for every δ ∈ morD′ with domain d′; compatibility means [f_{d}] = [fd̄ · Dδ̄] for all δ̄ : d → d̄ in D. The embedding η_{K}^{Ind} sends an object X to the corresponding single-morphism diagram (X). For finite limits (ii) is also proved in [AGV]. Thus, it remains to prove that Ind F preserves products. The functor Ind F takes a diagram X = (X_{i})_{i∈I}, an object of Ind X, to the diagram F X = (F X_{i})_{i∈I}, an object of Ind Y. Products of objects X^{t}, t ∈ T, of Ind X can be described as follows: let I = Π_{t∈T}I^{t}

be the usual product category of the filtered categories I^{t} which are the domains of X^{t}, then

Π_{t∈T}X^{t} = (X_{i})_{i∈I}

where for each i = (i^{t}) ∈ I the object X_{i} is a product of X_{i^{t}}^{t}, t ∈ T, and the equivalence classes of the projection π_{i}^{t} : X_{i} → X_{i^{t}}^{t} form then the projection morphisms of the product ΠX^{t} in Ind X. The claim

Ind F ( Π_{t∈T}X^{t}) = Π_{t∈T}Ind F (X^{t})

now easily follows from the fact that F preserves products, thus, from X_{i} = Π_{t∈T}X_{i^{t}}^{t} we

conclude F X_{i} = Π_{t∈T}F X_{i^{t}}^{t}.

(iii) The proof that η_{X}^{Ind} preserves limits can be found in [AGV]; it is based on the description of Ind via flat functors above: Ind X is obviously closed under limits in Set^{X^{op}}. Finally, μ_{X}^{Ind} preserves limits because Ind is a KZ-monad (and therefore μ_{X}^{Ind} has Ind(η_{X}^{Ind}) as a left adjoint), see [M_{1}]).

5.6. Pseudomonad Lim . We next consider a free completion of categories under limits. This yields a pseudomonad

(Lim , η^{Lim}, μ^{Lim})

on CAT.

Corollary. Ind ∘ Lim is a pseudomonad over CAT whose quasicategory of algebras is equivalent to the quasicategory of algebras of Ĩnd. The latter is the lifting of Ind to the quasicategory of complete categories.

Proof. In fact, given pseudomonads 𝕊 and 𝕋 on a 2-category K, then a lifting 𝕊̃ of 𝕊 to the category K^{𝕋} of algebras for 𝕋 is equivalent to providing a pseudomonad structure 𝕊 ∘ 𝕋 of the corresponding composite endofunctor of K. And the category of algebras for 𝕊 ∘ 𝕋 is biequivalent to that of 𝕊̃ (see [M_{2}] and [M_{3}]). Thus the result follows from Lemma 5.5 which precisely yields a lifting of Ind to CAT^{Lim}.

5.7. Remark. We are prepared to describe a small equational hull of LFP. As seen in Section 4, for all small categories k we have (due to Lim k = (Set^{k})^{op})

𝔻_{small}^{∗}k = 𝔻^{∗}k = LEX(Set^{k}) = LEX(Lim k)^{op} .

If LEX could be substituted by Lex (i.e., if every lex functor from Set^{k} to Set, k a small category, were small), we would have the desired formula

𝔻_{small}^{∗}k = Lex(Lim k)^{op} = Ind (Lim k) .

The problem of

LEX(Set^{k}) ? = Lex(Set^{k})

has been proved to depend on set theory in the recent paper [AKT], inspired by an earlier version of the present work.

Recall that an ultrafilter U on a set of cardinality k is called uniform if its members all have cardinality k. And U is called regular provided a subset V of U of cardinality k exists such that every member of U intersects only finitely many of members of V. The following statement

(R) every uniform ultrafilter is regular is consistent with set theory, as proved by H.-D. Donder, see [Do]. Now the following has been proved in [AKT]:

5.8. Theorem. For every small category k, (R) implies

LEX(Set^{k}) = Lex(Set^{k}).

5.9. Corollary. Assuming (R), the pseudomonads 𝔻_{small}^{∗} and Ind ∘ Lim are biequivalent.

Proof. Since both pseudomonads 𝔻_{small}^{∗} and Ind ∘ Lim preserve ∞-filtered colimits, it suffices to show that they are biequivalent on small categories. Following Theorem 5.8, they are biequivalent as bifunctors. It is easy to see that they have the same unit and multiplication.

5.10. Corollary. Assuming (R), the category of algebras of the pseudomonad Ind ∘ Lim is a small equational hull of LFP.

That is, given a pseudomonad 𝕋 over CAT preserving ∞-filtered colimits and a functor

E : LFP → CAT^{𝕋}

as in 4.6, then there exists a pseudofunctor

E^{∗} : CAT^{𝔻_{small}^{∗}} → CAT^{𝕋}

with the properties from 4.6. The proof is analogous to that in 4.6.

Now, we are going to spell out the property of being an algebra of Ind ∘Lim explicitly.

5.11. Definition. A category A is called precontinuous provided that it has limits and filtered colimits, and they distribute in the following sense: the functor

colim_{→} : Ind A → A

(replacing formal filtered colimits with actual colimits in A) preserves limits.

5.12. Examples. (i) Every complete, continuous category in the sense of [JJ] is precontinuous. We recall that a category A is called continuous if it has filtered colimits and colim_{→} : Ind A → A is a right adjoint.

(ii) Every locally finitely presentable category is precontinuous (in fact, continuous). (iii) For every complete category A the category Ind A is precontinuous because it is

complete by 5.5 and continuous by [JJ].

(iv) The following category is precontinuous but not continuous: we start with a large class X and form the ordered class P of all pairs

(V, i) with V ⊆ X and i ≤ ω

(i.e., V is a subclass of X and i is a natural number or ω), subject to the condition

i = ω ⇒ V is small,

and ordered as follows:

(V, i) ≤ (W, j) iff W ⊆ V and i ≤ j .

We claim that the ordered class P_{⊥} obtained by adding the least element ⊥ to P is precontinuous but not continuous.

(a) P_{⊥} is precontinuous. In fact, P is a subposet of the ordered class Q = (exp X)^{op} × (ω + 1) which is easily seen to be a continuous category. The following maps

f : Q → P_{⊥} , f(V, i) = (V, i) if (v, i) ∈ P, =⊥ else

and

g : P_{⊥} → Q , g(V, i) = (V, i) and g(⊥) = (X, ω)

preserve small meets, thus, Ind g does (see Lemma 5.1). And colim_{→} : Ind P_{⊥} → P_{⊥} is the composite of Ind g, colim_{→} : Ind Q → Q, and f.

(b) P_{⊥} is not continuous because colim_{→} : Ind P_{⊥} → P_{⊥} does not preserve large meets. In fact, for each v ∈ X we have an ideal I_{v} = {x} × ω ∪ {⊥} in P_{⊥} and ∩_{v∈X}I_{v} = {⊥}. However, colim I_{v} = ∨I_{v} = ({v}, ω) and the meet of all the latter elements is (V, ω).

5.13. Theorem [ARV]. A category with limits and filtered colimits is precontinuous iff

(i) finite limits commute with filtered colimits

and

(ii) products distribute over filtered colimits (see Introduction).

5.14. Lemma. A category with a dense set of finitely presentable objects is precontinuous iff it is locally finitely presentable.

Proof. Let A be such a dense set in a precontinuous category K. The functor E : K → Set^{A^{op}} assigning to every object K the domain-restriction of hom(−, K) to A^{op} is fully faithful (since A is dense) and preserves limits and filtered colimits (since objects of A are finitely presentable). Consequently, it has a left adjoint (see Theorem 2.48 in [AR]). Thus, K is equivalent to a full, reflective subcategory of a locally finitely presentable category L = Set^{A^{op}}, and being closed in L under filtered colimits, is itself locally finitely presentable.

5.15. Proposition. For every small category K, the embedding K ↪ Ind (Lim K) which is a codomain restriction of η_{K}^{𝔻^{∗}} is a free precontinuous category on K. That is,

(a) Ind (Lim K) is precontinuous

and

(b) every functor F : K → L, where L is precontinuous, has an essentially unique extension to a functor F^{∗} : Ind (Lim K) → L preserving limits and filtered colimits.

Proof. (a) follows from Example 5.12 (iii).

To prove (b), denote by F′ : Lim K → L the essentially unique limit-preserving extension of F, and by F^{∗} : Ind (Lim K) → L the essentially unique extension of F′ preserving filtered colimits. It is sufficient to prove that F^{∗} preserves limits. By Lemma 5.5, Ind F′ preserves limits, and since the functor colim_{→}L : Ind L → L preserves filtered colimits, we have F^{∗} ≅ colim_{→} · Ind F′. Since L is precontinuous, colim_{→} preserves limits, therefore F^{∗} preserves limits.

5.16. Notation. We denote by PCONT

the 2-quasicategory of all (0-cells) precontinuous categories (1-cells) functors preserving limits and filtered colimits (2-cells) natural transformations.

5.17. Corollary. Assuming (R), the 2-quasicategory PCONT of precontinuous categories is a small equational hull of LFP.

In other words, as claimed in the Introduction, the operations “limits of type k” and “filtered colimits of type k” generate all operations of small arity on LFP. And the only equational laws, besides the general laws of limits in categories and filtered colimits in categories, are (by 5.13) the commutation of finite limits with filtered colimits, and the distributivity of products over filtered colimits.

5.18. Remark. The above Corollary does not hold absolutely, i.e., without any restriction on set theory. In fact, Jan Reiterman [R] has proved that under the following assumption

non(M): there exist arbitrarily large measurable cardinals a large lex functor K : Set → Set exists. That is, K lies in LEX(Set) − Lex(Set). We will see in the next section that this implies that the small equational hull of LFP is strictly smaller than PCONT .

6. Equational Description

6.1. Remark. We have described in the Introduction what operations and equations of LFP are. Let us add that every object

K ∈ 𝔻^{∗}K

defines a K-ary operation ω_{K} on LFP by the (standard) rule:

ω_{K}^{A} : A^{K} → A , F ⟼ F̂(K) for all F : K → A

(see Notation 4.2; recall that each locally finitely presentable category A is equivalent to LEX B following the Gabriel-Ulmer duality). This generalizes all examples from 1.1: let k be a small category and η_{k} : k → 𝔻^{∗}k be the universal arrow, then we have

k- lim = ω_{K} where K is the limit of
η_{k} in 𝔻^{∗}k;
k-colim_{→} = ω_{K} where K is the filtered colimit
of η_{k} in 𝔻^{∗}k;
ev_{x} = ω_{η_{k}(x)} .

6.2. Observation. (1) Every operation of LFP has the form ω_{K} for some K ∈ 𝔻^{∗}K. In fact, let ω be a K-ary operation, and let

K = ω^{𝔻K}(η_{K}) .

For every functor F : K → A, where A ∈ LFP, since F̂ preserves ω we have

ω^{A}(F) = ω^{A}(F̂ · η_{K}) = F̂(ω^{𝔻K}(η_{K})) = ω_{K}^{A}(F) .

(2) Every operation ω_{K} is a (large) filtered colimit of (large) limits of evaluations. In fact, K ∈ 𝔻^{∗}K is a lex functor form Set^{K} to Set, thus, K is a large filtered colimit of homfunctors, K = colim_{i∈I}_{→}Set^{K}(X_{i}, −). Express each X_{i} ∈ Set^{K} as a colimit of hom-functors, X_{i} = colim_{j∈J_{i}} K(x_{ij}, −), to obtain Set^{K}(X_{i}, −) = lim_{j∈J_{i}} Set^{K}(K(x_{ij}, −), −) = η_{K}(x_{ij}). Thus,

K = colim_{i∈I}_{→} lim_{j∈J_{i}} η_{K}(x_{ij}) .

Now, as observed in 4.2, the functors F̂ preserve large filtered colimits and large limits. Consequently,

ω_{K} = colim_{i∈I}_{→} lim_{j∈J_{i}} ev_{x_{ij}} .

(3) We do not know how to work with all of these (“large” or “illegitimate”) operations: as we have made clear in 4.3, we do not know how to describe the resulting category of algebras using all of those operations. Even when the arity K is small (and thus the limits in the above formula for ω_{K} are small), there may exist “large” operations, i.e., those which are not small filtered colimits of those limits of evaluations. But, as we make clear below, what we did was to describe the equational hull with respect to all small operations, i.e., all ω_{K} which are small filtered colimits of small limits of evaluations. And assuming the axiom (R), these are in fact all operations of small arities.

6.3. Composed operations. Besides operations as described above we want to work with composed operations, e.g., a limit of filtered colimits. Recall that in the classical case mentioned above the discovery of [L_{1}] was that one can work with natural transformations

U^{k} → U^{n}

(which, since U^{n} = U × U × ⋯ × U, are just n-tuples of k-ary operations). This makes the task of composing operations very easy: they were just composites of natural transformations between powers of U.

Analogously in our case of U : LFP → CAT: we work with n-tuples of k-ary operations (where n and k are categories) which we define as pseudonatural transformations

ω : U^{k} → U^{n}.

6.4. Equations in LFP. Given operations

ω_{1}, ω_{2} : U^{k} ⟶ U

of the same arity on LFP, we form an equation

ω_{1} ≈ ω_{2}

whose interpretation is not strict, of course. That is, we do not request that ω_{1}^{A} = ω_{2}^{A} for all locally finitely presentable categories A. Our interpretation is the following non-strict one:

Definition. For k-ary operations ω_{1} and ω_{2} on LFP we write

ω_{1} ≈ ω_{2}

provided that there is a collection

i^{A} : ω_{1}^{A} → ω_{2}^{A} (A ∈ LFP)

of natural isomorphisms satisfying the following coherence condition: for every morphism F : A → B of LFP the following square

\xymatrix{F ω_{1}^{A} \ar[r]^{ω_{1}^{F}} \ar[d]_{F_{i}^{A}} & ω_{1}^{B}F^{k} \ar[d]^{i^{B}F^{k}} \\ F ω_{2}^{A} \ar[r]_{ω_{2}^{F}} & ω_{2}^{B}F^{k}}

commutes.

6.5. Examples of equations. (1) Finite limits commute with filtered colimits. This is a property that all locally finitely presentable categories have. The commutation of k-lim (k finite) with f-colim_{→} (f small, filtered) can be expressed by equations as follows: we can form an operation of arity f × k by composing the k-tuple

(f-colim_{→})^{k} : (A^{f})^{k} → A^{k}

with k-lim. And another operation of the same arity f × k by composing the f-tuple

(k- lim)^{f} : (A^{k})^{f} → A^{f}

with f-lim. The fact that k-limits commute with f-colimits is expressed by the following equation

(k- lim) · (f-colim_{→})^{k} ≈ (f-colim_{→}) · (k- lim)^{f}  (f filtered, k finite).  (E1)

In fact, the canonical isomorphisms

i_{D}^{A} : k- lim(f-colim_{→})^{k}D → f-colim_{→}(k- lim)^{f}D

where A ranges through locally finitely presentable categories and D through diagrams of the scheme f × k in A, form natural isomorphisms

i^{A} : k- lim(f-colim_{→})^{k} → f-colim_{→}(k- lim)^{f}

which are obviously coherent.

(2) (Infinite) products distribute over filtered colimits. That is, given a set D_{i} : f_{i} → K (i ∈ k) of filtered diagrams in K, denote by D : Π_{i∈k}f_{i} → K the diagram obtained by forming products in K:

(d_{i})_{i∈k} ⟼ Dd_{i} .

then the canonical morphism colim_{→}D → Π_{i∈k}colim_{→}D_{i} is invertible.

To express this as an equation, denote by f the product of the categories f_{i} (for i ∈ k) and by g their coproduct; observe that f is filtered (due to the Axiom of Choice).

The passage from (D_{i})_{i∈k} to Π colim D_{i} is the composite of the product operation Π_{k} : A^{k} → A (k considered as a discrete category) with the product of the operations

f_{i}-colim_{→}:

A^{g} ≅ Π_{i∈k}A^{f_{i}} ⟶^{Π(f_{i}-colim_{→})} A^{k} ⟶^{Π_{k}} A

And the passage to colim D is expressed by using the trivial operations o_{i} : A^{f_{i}} → A^{f} corresponding to the i-th projection of f, and composing them with f-colim_{→} : A^{f} → A:

A^{g} ≅ Π_{i∈k}A^{f_{i}} ⟶^{(o_{i})_{i∈k}} A^{f} ⟶^{f-colim_{→}} A

This gives an equation expressing distributivity of filtered colimits over products as follows

Π_{k} · Π_{i∈k}(f_{i}-colim_{→}) ≈ ( Π_{i∈k}f_{i}-colim_{→}) · (o_{i})_{i∈k}  (f_{i} filtered, i ∈ k) .  (E2)

Again, the naturality conditions for the above canonical morphisms are easy to verify.

6.6. Conclusion. Following Theorem 5.13 and Examples 6.5, precontinuous categories are equational over CAT and, under (R), they form a small equational hull of LFP (cf. Corollary 5.17). And the equational laws (E1) and (E2) generate all equational laws between operations of small arity over LFP.

On the other hand, assuming non(M), PCONT is not a small equational hull of LFP: for the functor K of 5.18, the unary operation ω_{K} (see 6.1) cannot be extended to the category Lex(Set), thus, Lex(Set) is not included in the equational hull of LFP, although it is precontinuous. In fact, express K in Set^{Set} as a large chain colimit of small lex functors K_{i}, i ∈ Ord, then each of the operations ω_{K_{i}} (being small) defines a unary operation on Lex(Set), and if ω_{K} were extendible to Lex(Set), the chain ω_{K_{i}}^{Lex(Set)} would have a colimit, which is not the case.

References

[AGV] M. Artin, A. Grothendieck and J. L. Verdier, Théorie des topos et cohomologie étale des schémas, Lect. Notes in Math. 269, Springer-Verlag, Berlin 1972.

CONTINUOUS CATEGORIES REVISITED  281

[AKT] J. Adámek, V. Koubek and V. Trnková, How large are left exact functors, Theory and Applications of Categories 8(13), 2001, 377–390.

[ALR] J. Adámek, F. W. Lawvere and J. Rosický, How algebraic is algebra, Theory and Applications of Categories 8(9), 2001, 253–283.

[AR] J. Adámek and J. Rosický, Locally presentable and accessible categories, Cambridge Univ. Press 1994.

[ARV] J. Adámek, J. Rosický and E. M. Vitale, On algebraically exact categories and essential localizations of varieties, J. Alg. 244 (2001), 450–477.

[B] J. Beck, Distributive laws, Lect. Notes in Math. 80 (1969), 119–140.

[D] A. Day, Filter monads, continuous lattices and closure systems, Canad. J. Math. 27 (1975), 50–59.

[Do] H.-D. Donder, Regularity of ultrafilters and the core model, Israel J. Math. 63 (1988), 289–322.

[DS] B. Day and R. Street, Monoidal bicategories and Hopf algebroids, Adv. Math. 129 (1997), 99–157.

[JJ] P. Johnstone and A. Joyal, Continuous categories and exponentiable toposes, J. Pure Appl. Algebra 25 (1982), 255–296.

[K] A. Kock, Monads for which structures are adjoint to units, J. Pure Appl. Alg. 104 (1995), 41–59. [L_{1}] F. W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Soc. 50 (1963),

865–872. [L_{2}] F. W. Lawvere, Ordinal sums and equational doctrines, Lect. Notes in Math. 80 (1969), 141–155. [M_{1}] F. Marmolejo, Doctrines whose structure forms a fully faithful adjoint string, Theory and Applica-

tions of Categories 3(2) 1997, 24–44. [M_{2}] F. Marmolejo, Distributive laws for pseudomonads, Theory and Applications of Categories 5(5)

1999, 91–147. [M_{3}] F. Marmolejo, Distributive laws for pseudomonads II, to appear.

[MLV] F. Marmolejo, I. J. Le Creurer and E. M. Vitale, Beck’s theorem for pseudo-monads, to appear.

[MRW] F. Marmolejo, R. D. Rosebrugh and R. J. Wood, A basic distributive law, J. Pure Appl. Alg. 168 (2002), 209–226.

[R] J. Reiterman, An example concerning set functors, Comm. Math. Univ. Carol. 12 (1971), 227–233.

[T] V. Trnková, On descriptive classification of set functors I, Comm. Math. Univ. Carolinae 12 (1971), 143–173.

[W] O. Wyler, Algebraic theories of continuous lattices, In: Continuous lattices (ed. B. Banaschewski and R.-E. Hoffmann), Lecture Notes in Math. 871, 390–413, Springer Verlag, Berlin 1981.

Technical University Braunschweig Postfach 3329 38023 Braunschweig, Germany

State University of New York 244 Mathematics Building Buffalo, N.Y. 14260-2900, U.S.A.

Masaryk University, Janáčkovo nám. 2a 662 95 Brno, Czech Republic

Email: adamek@iti.cs.tu-bs.de
      wlawvere@math.buffalo.edu
      rosicky@math.muni.cz

This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anony-

mous ftp at ftp://ftp.tac.mta.ca/pub/tac/html/volumes/11/11/11-11.{dvi,ps}

THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods.

Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication.

The method of distribution of the journal is via the Internet tools WWW/ftp. The journal is archived electronically and in printed paper format.

Subscription information. Individual subscribers receive (by e-mail) abstracts of articles as they are published. Full text of published articles is available in .dvi, Postscript and PDF. Details will be e-mailed to new subscribers. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh,

rrosebrugh@mta.ca.

Information for authors. The typesetting language of the journal is TEX, and L^{A}TEX is the preferred flavour. TEX source of articles for publication should be submitted by e-mail directly to an appropriate Editor. They are listed below. Please obtain detailed information on submission format and style files from the journal’s WWW server at http://www.tac.mta.ca/tac/. You may also write to tac@mta.ca to receive details by e-mail.

Editorial board.

John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@barrs.org, Associate Managing Editor Lawrence Breen, Université Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of Wales Bangor: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, Palo Alto Research Center: paiva@parc.xerox.com Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, University of Western Sydney: s.lack@uws.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca, Managing Editor Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca