Reiterman's Theorem on Finite Algebras for a Monad
aa r X i v : . [ m a t h . C T ] J a n Reiterman’s Theorem on Finite Algebras for a Monad
JIŘÍ ADÁMEK ∗ , Czech Technical University in Prague, Czech Republic
LIANG-TING CHEN,
Academia Sinica, Taiwan
STEFAN MILIUS † , Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
HENNING URBAT ‡ , Friedrich-Alexander-Universität Erlangen-Nürnberg, GermanyProfinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman’stheorem states that they precisely specify pseudovarieties, i.e. classes of finite algebras closed under finiteproducts, subalgebras and quotients. In this paper, Reiterman’s theorem is generalized to finite Eilenberg-Moore algebras for a monad T on a category D : we prove that a class of finite T -algebras is a pseudovarietyiff it is presentable by profinite equations. As a key technical tool, we introduce the concept of a profinitemonad b T associated to the monad T , which gives a categorical view of the construction of the space of profiniteterms. ACM Reference Format:
Jiří Adámek, Liang-Ting Chen, Stefan Milius, and Henning Urbat. 2021. Reiterman’s Theorem on Finite Al-gebras for a Monad. 1, 1 (January 2021), 48 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn
One of the main principles of both mathematics and computer science is the specification of struc-tures in terms of equational properties. The first systematic study of equations as mathematicalobjects was pursued by Birkhoff [6] who proved that a class of algebraic structures over a finitarysignature Σ can be specified by equations between Σ -terms if and only if it is closed under quo-tient algebras (a.k.a. homomorphic images), subalgebras, and products. This fundamental result,known as the HSP theorem , lays the ground for universal algebra and has been extended and gen-eralized in many directions over the past 80 years, including categorical approaches via Lawveretheories [4, 12] and monads [15].While Birkhoff’s seminal work and its categorifications are concerned with general algebraicstructures, in many computer science applications the focus is on finite algebras. For instance, inautomata theory, regular languages (i.e. the behaviors of classical finite automata) can be character-ized as precisely the languages recognizable by finite monoids. This algebraic point of view leadsto important insights, including decidability results. As a prime example, Schützenberger’s the-orem [21] asserts that star-free regular languages correspond to aperiodic finite monoids, i.e. mon-oids where the unique idempotent power 𝑥 𝜔 of any element 𝑥 satisfies 𝑥 𝜔 = 𝑥 · 𝑥 𝜔 . As an immediateapplication, one obtains the decidability of star-freeness. However, the identity 𝑥 𝜔 = 𝑥 · 𝑥 𝜔 is notan equation in Birkhoff’s sense since the operation (−) 𝜔 is not a part of the signature of monoids.Instead, it is an instance of a profinite equation , a topological generalization of Birkhoff’s concept ∗ Jiří Adámek acknowledges support by the Grant Agency of the Czech Republic under the grant 19-00902S. † Stefan Milius acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) under project MI 717/5-2 and aspart of the Research and Training Group 2475 “Cybercrime and Forensic Computing” (393541319/GRK2475/1-2019) ‡ Henning Urbat acknowledges support by Deutsche Forschungsgemeinschaft (DFG) under project SCHR 1118/8-2.Authors’ addresses: Jiří Adámek, Department of Mathematics, Faculty of Electrical Engineering, Czech TechnicalUniversity in Prague, Czech Republic, [email protected]; Liang-Ting Chen, Institute of Information Sci-ence, Academia Sinica, Taiwan, [email protected]; Stefan Milius, Friedrich-Alexander-Universität Erlangen-Nürnberg, Martensstr. 3, Erlangen, 91058, Germany, [email protected]; Henning Urbat, Friedrich-Alexander-Universität Erlangen-Nürnberg, Martensstr. 3, Erlangen, 91058, Germany, [email protected]. XXXX-XXXX/2021/1-ART $15.00https://doi.org/10.1145/nnnnnnn.nnnnnnn , Vol. 1, No. 1, Article . Publication date: January 2021.
J. Adámek, L.-T. Chen, S. Milius, and H. Urbat introduced by Reiterman [19]. (Originally, Reiterman worked with the equivalent concept of an implicit equation , cf. Section 5.) Given a set 𝑋 of variables and 𝑥 ∈ 𝑋 , the expression 𝑥 𝜔 can be in-terpreted as an element of the Stone space c 𝑋 ∗ of profinite words , constructed as the cofiltered limitof all finite quotient monoids of the free monoid 𝑋 ∗ . Analogously, over general signatures Σ onecan form the Stone space of profinite Σ -terms . Reiterman proved that a class of finite Σ -algebrascan be specified by profinite equations (i.e. pairs of profinite terms) if and only if it is closed un-der quotient algebras, subalgebras, and finite products. This result establishes a finite analogue ofBirkhoff’s HSP theorem.In this paper, we develop a categorical approach to Reiterman’s theorem and the theory of profin-ite equations. The idea is to replace monoids (or general algebras over a signature) by Eilenberg-Moore algebras for a monad T on an arbitrary base category D . As an important technical device,we introduce a categorical abstraction of the space of profinite words. To this end, we consider afull subcategory D f of D of “finite” objects and form the category Pro D f , the free completion of D f under cofiltered limits. We then show that the monad T naturally induces a monad b T on Pro D f ,called the profinite monad of T , whose free algebras b T 𝑋 serve as domains for profinite equations.For example, for D = Set and the full subcategory
Set f of finite sets, we get Pro
Set f = Stone ,the category of Stone spaces. Moreover, if T 𝑋 = 𝑋 ∗ is the finite-word monad (whose algebras areprecisely monoids), then b T is the monad of profinite words on Stone ; that is, b T associates to eachfinite Stone space (i.e. a finite set with the discrete topology) 𝑋 the space c 𝑋 ∗ of profinite wordson 𝑋 . Our overall approach can thus be summarized by the following diagram, where the skewedfunctors are inclusions and the horizontal ones are forgetful functors. Stone / / @GAFED (cid:15) (cid:15) š (−) ∗ Set
GFBECD (cid:15) (cid:15) (−) ∗ Set f b b b b ❊❊❊❊❊❊❊❊ @ @ @ @ ✂✂✂✂✂✂✂ Pro D f / / @GAFED (cid:15) (cid:15) b T D GFBECD (cid:15) (cid:15) T D f a a a a ❉❉❉❉❉❉❉❉ C C C C ✟✟✟✟✟✟✟ It turns out that many familiar properties of the space of profinite words can be developed at theabstract level of profinite monads and their algebras. Our main result is the
Generalized Reiterman Theorem.
A class of finite T -algebras is presentable by profinite equa-tions if and only if it is closed under quotient algebras, subalgebras, and finite products.Here, profinite equations are modelled categorically as finite quotients 𝑒 : b 𝑇 𝑋 ։ 𝐸 of the object b 𝑇 𝑋 of generalized profinite terms. If the category D is Set or, more generally, a category of first-order structures, we will see that this abstract concept of an equation is equivalent to the familiarone: b 𝑇 𝑋 is a topological space and quotients 𝑒 as above can be identified with sets of pairs ( 𝑠, 𝑡 ) of profinite terms 𝑠, 𝑡 ∈ b 𝑇 𝑋 . Thus, our categorical results instantiate to the original Reitermantheorem [19] ( D = Set ), but also to its versions for ordered algebras ( D = Pos ) and for first-orderstructures due to Pin and Weil [17].Our proof of the Generalized Reiterman Theorem is purely categorical and relies on generalproperties of (codensity) monads, free completions and locally finitely copresentable categories. Itdoes not employ any topological methods, as opposed to all known proofs of Reiterman’s theoremand its variants. The insight that topological reasoning can be completely avoided in the profiniteworld is quite surprising, and we consider it as one of the main contributions of our paper. , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 3
Related work . This paper is the full version of an extended abstract [8] presented at FoSSaCS2016. Besides providing complete proofs of all results, the presentation is significantly more generalthan in op. cit. : there we restricted ourselves to base categories D which are varieties of (possiblyordered) algebras, and the development of the profinite monad and its properties used results fromtopology. In contrast, the present paper works with general categories D and develops all requiredprofinite concepts in full categorical abstraction.An important application of the Generalized Reiterman Theorem and the profinite monad canbe found in algebraic language theory [24]: we showed that given a category C dually equivalentto Pro D f , the concept of a profinite equational class of finite T -algebras dualizes to the concept of a variety of T -recognizable languages in C . For instance, for D = Set and
Pro D f = Stone , the classicalStone duality yields the category C = BA of boolean algebras, and for the monad T 𝑋 = 𝑋 ∗ on Set the dual correspondence gives Eilenberg’s fundamental variety theorem for regular languages [9].Using our duality-theoretic approach we established a categorical generalization of Eilenberg’stheorem and showed that it instantiates to more than a dozen Eilenberg-type results known in theliterature, along with a number of new correspondence results.Recently, an abstract approach to HSP-type theorems [16] has been developed that not onlyprovides a common roof over Birkhoff’s and Reiterman’s theorem, but also applies to classes ofalgebras with additional underlying structure, such as ordered, quantitative, or nominal algebras.The characterization of pseudovarieties in terms of pseudoeuqations given in Proposition 3.8 is aspecial case of the HSP theorem in op. cit . In this preliminary section, we review the profinite completion (commonly known as pro-completion)of a category and describe it for the category Σ - Str of structures over a first-order signature Σ . Remark 2.1.
Recall that a category is cofiltered if every finite subcategory has a cone in it. For ex-ample, every cochain (i.e. a poset dual to an ordinal number) is cofiltered. A cofiltered limit is a limitof a diagram with a small cofiltered diagram scheme. A functor is cofinitary if it preserves cofilteredlimits. An object 𝐴 of a category C is called finitely copresentable if the functor C (− , 𝐴 ) : C → Set op is cofinitary. The latter means that for every limit cone 𝑐 𝑖 : 𝐶 → 𝐶 𝑖 ( 𝑖 ∈ I ) of a cofiltered diagram,(1) each morphism 𝑓 : 𝐶 → 𝐴 factorizes through some 𝑐 𝑖 : 𝐶 → 𝐶 𝑖 as 𝑓 = 𝑔 · 𝑐 𝑖 , and(2) the morphism 𝑔 : 𝐶 𝑖 → 𝐴 is essentially unique , i.e. given another factorization 𝑓 = ℎ · 𝑐 𝑖 , thereis a connecting morphism 𝑐 𝑗𝑖 : 𝐶 𝑗 → 𝐶 𝑖 with 𝑔 · 𝑐 𝑗𝑖 = ℎ · 𝑐 𝑗𝑖 : 𝐶 𝑓 / / 𝑐 𝑖 (cid:15) (cid:15) 𝑐 𝑗 ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ 𝐴𝐶 𝑗 𝑐 𝑗𝑖 / / ❴❴❴ 𝐶 𝑖 𝑔 ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ ℎ ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ The dual concept is that of a filtered colimit . Notation 2.2. (1) The free completion of a category C under cofiltered limits, i.e. the pro-completion ,is denoted by Pro C . This is a category with cofiltered limits together with a full embedding 𝐸 : C Pro C satisfyingthe following universal property: , Vol. 1, No. 1, Article . Publication date: January 2021. J. Adámek, L.-T. Chen, S. Milius, and H. Urbat (1a) Every functor 𝐹 : C → K into a category K with cofiltered limits admits a cofinitary exten-sion 𝐹 : Pro C → K , i.e. the triangle below commutes: C 𝐸 / / 𝐹 " " ❊❊❊❊❊❊❊❊ Pro C 𝐹 (cid:15) (cid:15) ✤✤✤ K (1b) The functor 𝐹 is essentially unique , i.e. for every cofinitary extension 𝐺 of 𝐹 there exists aunique natural isomorphism 𝑖 : 𝐹 (cid:27) −→ 𝐺 with 𝑖𝐸 = id 𝐹 .More precisely, the full embedding 𝐸 is the pro-completion, but we will often simply refer to Pro C as the pro-completion instead.(2) Dually, the free completion of C under filtered colimits, i.e. the ind-completion , is denoted by Ind C . Some standard results on ind- and pro-completions can be found in the Appendix.
Example 2.3. (1) Let
Set f be the category of finite sets and functions. Its pro-completion is thecategory Pro
Set f = Stone of Stone spaces , i.e. compact topological spaces in which distinct elements can be separated byclopen subsets. Morphisms are the continuous functions. The embedding
Set f Stone identifiesfinite sets with finite discrete spaces. This is a consequence of the Stone duality [10] between
Stone and the category BA of boolean algebras, and its restriction to finite sets and finite Booleanalgebras. In fact, since BA is a finitary variety, it is the ind-completion of its full subcategory BA f of finitely presentable objects, which are precisely the finite Boolean algebras. Therefore Pro
Set f = ( Ind
Set op f ) op (cid:27) ( Ind BA f ) op (cid:27) BA op (cid:27) Stone . (2) For the category of finite posets and monotone functions, denoted by Pos f , we obtain the cat-egory Pro
Pos f = Priest of Priestley spaces , i.e. ordered Stone spaces such that any two distinct elements can be separated byclopen upper sets. Morphisms in
Priest are continuous monotone functions. This follows from thePriestley duality [18] between
Priest and bounded distributive lattices. The argument is analogousto item (1): finite, equivalently finitely presentable, distributive lattices dualize to finite posets withdiscrete topology.
Notation 2.4 (First-order structures) . We will often work with the category Σ - Str of Σ -structures and Σ -homomorphisms for a first-order many-sorted signature Σ . Given a set S ofsorts, an S -sorted signature Σ consists of (1) operation symbols 𝜎 : 𝑠 , . . . , 𝑠 𝑛 → 𝑠 where 𝑛 ∈ N , thesorts 𝑠 𝑖 form the domain of 𝜎 and 𝑠 is its codomain, and (2) relation symbols 𝑟 : 𝑠 , . . . , 𝑠 𝑚 where 𝑚 ∈ N + = N \ { } . A Σ -structure is an S -sorted set 𝐴 = ( 𝐴 𝑠 ) 𝑠 ∈S in Set S with (1) an operation 𝜎 𝐴 : 𝐴 𝑠 × · · · × 𝐴 𝑠 𝑛 → 𝐴 𝑠 for every operation symbol 𝜎 : 𝑠 , . . . , 𝑠 𝑛 → 𝑠 ,and (2) a relation 𝑟 𝐴 ⊆ 𝐴 𝑠 × . . . 𝐴 𝑠 𝑛 for every relation symbol 𝑟 : 𝑠 , . . . , 𝑠 𝑛 . A Σ -homomorphism is , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 5 an S -sorted function 𝑓 : 𝐴 → 𝐵 which preserves operations and relations in the usual sense. Wedenote by Σ - Str f the full subcategory of Σ - Str given by all Σ -structures 𝐴 where each 𝐴 𝑠 is finite.When S is a singleton, the notion of Σ -structures boils down to a more common situation.Namely, the arity of an operation symbol is given solely by 𝑛 ∈ N and that of a relation symbolby 𝑚 ∈ N + . A Σ -structure is a set 𝐴 equipped with an operation 𝜎 𝐴 : 𝐴 𝑛 → 𝐴 for every 𝑛 -aryoperation symbol 𝜎 and with a relation 𝑟 𝐴 ⊆ 𝐴 𝑚 for every 𝑚 -ary relation symbol 𝑟 . Assumption 2.5.
Throughout the paper, we assume that every signature has a finite set of sortsand finitely many relation symbols. There is no restriction on the number of operation symbols.
Remark 2.6. (1) The category Σ - Str is complete with limits created at the level of
Set S . Moreprecisely, consider a diagram 𝐷 in Σ - Str indexed by I . Let 𝑈 𝑠 : Set S → Set be the projectionsending 𝐵 to 𝐵 𝑠 , and let 𝑏 𝑠𝑖 : 𝐵 𝑠 → 𝐷 𝑠𝑖 ( 𝑖 ∈ I) form limit cones of the diagrams 𝑈 𝑠 𝐷 in Set for every 𝑠 ∈ S . Then the limit of 𝐷 is the S -sortedset 𝐵 ≔ ( 𝐵 𝑠 ) , with operations 𝜎 𝐵 : 𝐵 𝑠 × · · · × 𝐵 𝑠 𝑛 → 𝐵 𝑠 uniquely determined by the requirementthat each 𝑏 𝑖 : 𝐵 → 𝐷 𝑖 preserves 𝜎 , and with relations 𝑟 𝐵 ⊆ 𝐵 𝑠 × · · · × 𝐵 𝑠 𝑛 consisting of all 𝑛 -tuples ( 𝑥 , . . . , 𝑥 𝑛 ) that each function 𝑏 𝑠 𝑖 × · · · × 𝑏 𝑠 𝑛 𝑖 maps into 𝑟 𝐷 𝑖 for all 𝑖 ∈ I . The limit cone is given by ( 𝑏 𝑠𝑖 ) 𝑠 ∈S : 𝐵 → 𝐷 𝑖 for 𝑖 ∈ I .(2) The category Σ - Str is also cocomplete. Indeed, let Σ op be the subsignature of all operationsymbols in Σ . Then Σ op - Str is a monadic category over
Set S . Since epimorphisms split in Set S ,all monadic categories are cocomplete, see e.g. [1]. The category Σ - Str has colimits obtained fromthe corresponding colimits in Σ op - Str by taking the smallest relations making each of the colimitinjections a Σ -homomorphism. Notation 2.7.
The category of Stone topological Σ -structures and continuous Σ -homomorphismsis denoted by Stone ( Σ - Str ) . A topological Σ -structure is an S -sorted topological space 𝐴 = ( 𝐴 𝑠 ) endowed with a Σ -structuresuch that every operation 𝜎 𝑠 : 𝐴 𝑠 × · · · × 𝐴 𝑠 𝑛 → 𝐴 is continuous and for every relation symbol 𝑟 the relation 𝑟 𝐴 ⊆ 𝐴 𝑠 × · · · × 𝐴 𝑠 𝑛 is a closed subset. Remark 2.8.
The category
Stone ( Σ - Str ) is complete with limits formed on the level of Set S . Thisfollows from the construction of limits in Stone S and in Σ - Str . Thus, the forgetful functor from
Stone ( Σ - Str ) to Σ - Str preserves limits.The following proposition describes the pro-completion of Σ - Str f . It is a categorical reformula-tion of results by Pin and Weil [17] on topological Σ -structures, and also appears in Johnstone’sbook [10, Prop. & Rem. VI.2.4] for the special case of single-sorted algebras. We provide a full prooffor the convenience of the reader. Definition 2.9.
A Stone topological Σ -structure is called profinite if it is a cofiltered limit in Stone ( Σ - Str ) of finite Σ -structures. Proposition 2.10.
The category
Pro ( Σ - Str f ) is the full subcategory of Stone ( Σ - Str ) on all profinite Σ -structures. Proof. (1) We first observe that cofiltered limits of finite sets in
Stone have the following prop-erty: If 𝑏 𝑖 : 𝐵 → 𝐵 𝑖 ( 𝑖 ∈ I) is a cofiltered limit cone such that all 𝐵 𝑖 are finite, then for every 𝑖 ∈ I there exists a connecting morphism of our diagram ℎ : 𝐵 𝑗 → 𝐵 𝑖 with the same image as 𝑏 𝑖 : 𝑏 𝑖 [ 𝐵 ] = ℎ [ 𝐵 𝑗 ] . (2.1) , Vol. 1, No. 1, Article . Publication date: January 2021. J. Adámek, L.-T. Chen, S. Milius, and H. Urbat
Since under Stone duality finite Stone spaces dualizes to finite boolean algebras, it suffices to verifythe dual statement about filtered colimits of finite Boolean algebras: if 𝑐 𝑖 : 𝐶 𝑖 → 𝐶 ( 𝑖 ∈ I ) is afiltered colimit cocone of finite Boolean algebras, then for every 𝑖 there exists a connecting morph-ism 𝑓 : 𝐶 𝑖 → 𝐶 𝑗 with the same kernel as 𝑐 𝑖 . But this is clear: given any pair 𝑥, 𝑦 ∈ 𝐶 𝑖 merged by 𝑐 𝑖 ,there exists a connecting morphism 𝑓 merging 𝑥 and 𝑦 , since filtered colimits are formed on thelevel of Set . Due to 𝐶 𝑖 × 𝐶 𝑖 being finite, we can choose one 𝑓 for all such pairs.(2) The argument is similar for cofiltered limits of finite Σ -structures in Stone ( Σ - Str ) : Consider alimit cone 𝑏 𝑖 : 𝐵 → 𝐵 𝑖 ( 𝑖 ∈ I) of a cofiltered diagram 𝐷 in Stone ( Σ - Str ) . For every 𝑖 ∈ I , we verify that there is a connectingmorphism ℎ : 𝐵 𝑗 → 𝐵 𝑖 with sorts ℎ 𝑠 for 𝑠 ∈ S such that 𝑏 𝑠𝑖 [ 𝐵 𝑠 ] = ℎ 𝑠 [ 𝐵 𝑠𝑗 ] for all 𝑠 ∈ S , (2.2)and 𝑏 𝑠 𝑖 × · · · × 𝑏 𝑠 𝑛 𝑖 [ 𝑟 𝐵 ] = ℎ 𝑠 × · · · × ℎ 𝑠 𝑛 [ 𝑟 𝐵 ] for all 𝑟 : 𝑠 , . . . , 𝑠 𝑛 in Σ . (2.3)Indeed, if we only consider (2.2) then the existence of such an ℎ follows from (1) by the assumptionthat S is finite and that I is cofiltered. For every sort 𝑠 , we have a cofiltered limit 𝑏 𝑠𝑗 : 𝐵 𝑠 → 𝐵 𝑠𝑗 in Stone , thus we can apply (1) and obtain a connecting morphism ℎ : 𝐵 𝑗 → 𝐵 𝑖 . Again, S is finite, sothe choice of ℎ can be made independent of 𝑠 ∈ S .Next consider (2.3) for a fixed relation symbol 𝑟 : 𝑠 , . . . , 𝑠 𝑛 . Form the diagram 𝐷 𝑟 in Stone withthe above diagram scheme I and with objects 𝐷 𝑟 𝑖 = 𝑟 𝐵 𝑖 (a finite discrete space) . Connecting morphisms are the domain-codomain restrictions of all connecting morphisms 𝐵 𝑗 ℎ −→ 𝐵 𝑘 : since ℎ preserves the relation 𝑟 , we have ℎ 𝑠 × · · · × ℎ 𝑠 𝑛 [ 𝑟 𝐵 𝑗 ] ⊆ 𝑟 𝐵 𝑘 , and we form the corresponding connecting morphism ℎ : 𝑟 𝐵 𝑗 → 𝑟 𝐵 𝑘 of 𝐷 𝑟 . From the description oflimits in Σ - Str in Remark 2.6 and the fact that limits in
Stone ( Σ - Str ) are preserved by the forgetfulfunctor into Σ - Str by Remark 2.8 we deduce that the limit of 𝐷 𝑟 in Stone is the space 𝑟 𝐵 ⊆ 𝐵 𝑠 ×· · ·× 𝐵 𝑠 𝑛 and the limit cone 𝑟 𝐵 → 𝑟 𝐵 𝑗 , 𝑗 ∈ I , is formed by domain-codomain restrictions of 𝑏 𝑠 𝑗 × · · ·× 𝑏 𝑠 𝑛 𝑗 for 𝑗 ∈ I . Apply (1) to this cofiltered limit to find a connecting morphism ℎ : 𝐵 𝑗 → 𝐵 𝑖 of 𝐷 satisfying (2.3) for any chosen relation symbol 𝑟 of Σ . Since we only have finitely many relationsymbols by Assumption 2.5, we conclude that ℎ can be chosen to satisfy (2.3).(3) Denote the full subcategory formed by profinite Σ -structures by L ⊆ Stone ( Σ - Str ) . In order to prove that L forms the pro-completion of Σ - Str f , we verify the conditions given inCorollary A.5. By construction, conditions (1) and (2) hold. It remains to prove condition (3): everyfinite Σ -structure 𝐴 is finitely copresentable in L . Hence, consider a limit cone 𝑏 𝑖 : 𝐵 → 𝐵 𝑖 ( 𝑖 ∈ I) of a cofiltered diagram 𝐷 in L . Due to the definition of L , we may assume that all 𝐵 𝑖 are finite.We need to show that for every homomorphism 𝑓 = ( 𝑓 𝑠 ) 𝑠 ∈S : 𝐵 → 𝐴 into a finite Σ -structure 𝐴 = ( 𝐴 𝑠 ) 𝑠 ∈S , there is an essentially unique factorization through some 𝑏 𝑖 . For every sort 𝑠 , wehave a projection 𝑉 𝑠 : L → Stone , and the cofiltered diagram 𝑉 𝑠 𝐷 has the limit cone 𝑓 𝑠𝑖 : 𝐵 𝑠 → , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 7 𝐵 𝑠𝑖 ( 𝑖 ∈ I) . Since each 𝐴 𝑠 is finite, the fact that Stone is the pro-completion of
Set f implies that forevery sort 𝑠 there is 𝑖 ∈ I and an essentially unique factorization of 𝑓 𝑠 as follows 𝐵 𝑠 𝑓 𝑠 / / 𝑏 𝑠𝑖 (cid:15) (cid:15) 𝐴 𝑠 𝐵 𝑠𝑖 𝑔 𝑠 > > ⑦⑦⑦⑦⑦⑦⑦⑦ By Assumption 2.5 the set S is finite, so we can choose 𝑖 independent of 𝑠 and thus obtain acontinuous S -sorted function 𝑔 = ( 𝑔 𝑠 ) : 𝐵 𝑖 → 𝐴 in Stone S which factorizes 𝑓 , i.e. 𝑓 = 𝑔 · 𝑏 𝑖 .All we still need to prove is that we can choose our 𝑖 and 𝑔 so that, moreover, 𝑔 is a Σ -homomorphism.The essential uniqueness of 𝑔 then follows from the corresponding property of 𝑔 in Stone .Let ℎ : 𝐵 𝑗 → 𝐵 𝑖 be a connecting map satisfying (2.2) and (2.3). Choose 𝑗 in lieu of 𝑖 and 𝑔 = 𝑔 · ℎ in lieu of 𝑔 . We conclude that 𝑔 is a morphism of Stone S factorizing 𝑓 through the limit map 𝑏 𝑗 : 𝐵 𝑏 𝑗 (cid:15) (cid:15) 𝑓 / / 𝑏 𝑖 ❆❆❆ 𝐴𝐵 𝑖 𝑔 ? ? ⑧⑧⑧⑧⑧⑧ 𝐵 𝑗 ℎ ? ? ⑧⑧⑧⑧⑧⑧ AB CD 𝑔 ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ _ _ Moreover, we prove that 𝑔 is a Σ -homomorphism:(3a) For every operation symbol 𝜎 : 𝑠 . . . 𝑠 𝑛 → 𝑠 in Σ and every 𝑛 -tuple ( 𝑥 , . . . , 𝑥 𝑛 ) ∈ 𝐵 𝑠 𝑗 ×· · ·× 𝐵 𝑠 𝑛 𝑗 we have 𝑔 𝑠 · 𝜎 𝐵 𝑗 ( 𝑥 , . . . , 𝑥 𝑛 ) = 𝜎 𝐴 ( 𝑔 𝑠 ( 𝑥 ) , . . . , 𝑔 𝑠 𝑛 ( 𝑥 𝑛 )) . Indeed, choose 𝑦 𝑘 ∈ 𝐵 𝑠 𝑘 with 𝑏 𝑠 𝑘 𝑖 ( 𝑦 𝑘 ) = ℎ 𝑠 𝑘 ( 𝑥 𝑘 ) , 𝑘 = , . . . , 𝑛 , using (2.2). Then 𝑔 𝑠 · 𝜎 𝐵 𝑗 ( 𝑥 , . . . , 𝑥 𝑛 ) = 𝑔 𝑠 · ℎ 𝑠 · 𝜎 𝐵 𝑗 ( 𝑥 , . . . , 𝑥 𝑛 ) 𝑔 = 𝑔 · ℎ = 𝑔 𝑠 · 𝜎 𝐵 𝑖 ( ℎ 𝑠 ( 𝑥 ) , . . . , ℎ 𝑠 𝑛 ( 𝑥 𝑛 )) ℎ a Σ -homomorphism = 𝑔 𝑠 · 𝜎 𝐵 𝑖 ( 𝑏 𝑠 𝑖 ( 𝑦 ) , . . . , 𝑏 𝑠 𝑛 𝑖 ( 𝑦 𝑛 )) 𝑏 𝑠 𝑘 𝑖 ( 𝑦 𝑘 ) = ℎ 𝑠 𝑘 ( 𝑥 𝑘 ) = 𝑔 𝑠 · 𝑏 𝑠𝑖 · 𝜎 𝐵 ( 𝑦 , . . . , 𝑦 𝑛 ) 𝑏 𝑖 a Σ -homomorphism = 𝜎 𝐴 ( 𝑔 𝑠 · 𝑏 𝑠 𝑖 ( 𝑦 ) , . . . , 𝑔 𝑠 𝑛 · 𝑏 𝑠 𝑛 𝑖 ( 𝑦 𝑛 )) 𝑔 · 𝑏 𝑖 = 𝑓 a Σ -homomorphism = 𝜎 𝐴 ( 𝑔 𝑠 · ℎ 𝑠 ( 𝑦 ) , . . . , 𝑔 𝑠 𝑛 · ℎ 𝑠 𝑛 ( 𝑦 𝑛 )) 𝑏 𝑠 𝑘 𝑖 ( 𝑦 𝑘 ) = ℎ 𝑠 𝑘 ( 𝑥 𝑘 ) = 𝜎 𝐴 ( 𝑔 𝑠 ( 𝑥 ) , . . . , 𝑔 𝑠 𝑛 ( 𝑥 𝑛 )) 𝑔 = 𝑔 · ℎ. (3b) For every relation symbol 𝑟 : 𝑠 , . . . , 𝑠 𝑛 in Σ , we have that ( 𝑥 , . . . , 𝑥 𝑛 ) ∈ 𝑟 𝐵 𝑗 implies ( 𝑔 𝑠 ( 𝑥 ) , . . . , 𝑔 𝑠 𝑛 ( 𝑥 𝑛 )) ∈ 𝑟 𝐴 . Indeed, using (2.3), we can choose ( 𝑦 , . . . , 𝑦 𝑛 ) ∈ 𝑟 𝐵 with ( 𝑏 𝑠 𝑖 ( 𝑦 ) , . . . , 𝑏 𝑠 𝑛 𝑖 ( 𝑦 𝑛 )) = ( ℎ 𝑠 ( 𝑥 ) , . . . , ℎ 𝑠 𝑛 ( 𝑥 𝑛 )) . Then the 𝑛 -tuple ( 𝑔 𝑠 ( 𝑥 ) , . . . , 𝑔 𝑠 𝑛 ( 𝑥 𝑛 )) = ( 𝑔 𝑠 · 𝑏 𝑠 𝑖 ( 𝑦 ) , . . . , 𝑔 𝑠 𝑛 · 𝑏 𝑠 𝑛 𝑖 ( 𝑦 𝑛 )) , Vol. 1, No. 1, Article . Publication date: January 2021. J. Adámek, L.-T. Chen, S. Milius, and H. Urbat lies in 𝑟 𝐴 because 𝑔 · 𝑏 𝑖 = 𝑓 is a Σ -homomorphism. (cid:3) Notation 2.11.
Let D be a full subcategory of Σ - Str . We denote by
Stone D the full subcategory of Stone ( Σ - Str ) on all Stone topological Σ -structures whose Σ -structure liesin D . Moreover, let D f denote the full subcategory of D on all finite objects, i.e. 𝐷 ∈ D f if each 𝐷 𝑠 is finite. Corollary 2.12.
Let D be a full subcategory of Σ - Str closed under cofiltered limits. Then
Pro D f is the full subcategory of Stone D given by all profinite D -structures, i.e. cofiltered limits of finite Σ -structures in D . The proof is completely analogous to that of Proposition 2.10: the only fact we used in that proofwas the description of cofiltered limits in Σ - Str . Example 2.13.
For D = Pos , we get an alternative description of the category
Priest of Ex-ample 2.3(2). For the signature Σ with a single binary relation, Pos is a full subcategory of Σ - Str .The category
Stone ( Σ - Str ) is that of graphs on Stone spaces. By Corollary 2.12, Pro ( Pos f ) is thecategory of all profinite posets, i.e. Stone graphs that are cofiltered limits of finite posets. Note thatevery such limit 𝐵 = ( 𝑉 , 𝐸 ) is a poset: given 𝑥 ∈ 𝑉 we have ( 𝑥, 𝑥 ) ∈ 𝐸 because every object ofthe given cofiltered diagram has its relation reflexive. Analogously, 𝐸 is transitive and (since limitcones are collectively monic) antisymmetric.Moreover, 𝐵 is a Priestley space: given 𝑥, 𝑦 ∈ 𝑉 with 𝑥 (cid:2) 𝑦 , then there exists a member 𝑏 𝑖 : 𝐵 → 𝐵 𝑖 of the limit cone with 𝑏 𝑖 ( 𝑥 ) (cid:2) 𝑏 𝑖 ( 𝑦 ) . Since 𝐵 𝑖 is finite, and thus carries the discrete topology,the upper set 𝑏 − 𝑖 (↑ 𝑥 ) is clopen, and it contains 𝑥 but not 𝑦 . Conversely, every Priestley space is aprofinite poset, as shown by Speed [22]. Example 2.14.
Johnstone [10, Thm. VI.2.9] proves that for a number of “everyday” varieties ofalgebras D , we simply have Pro D f = Stone D . This holds for semigroups, monoids, groups, vector spaces, semilattices, distributive lattices, etc.In contrast, for some important varieties
Pro D f is a proper subcategory of Stone D , e.g. for thevariety of lattices or the variety of Σ -algebras where Σ consists of a single unary operation. Remark 2.15. (1) The category Σ - Str has a factorization system (E , M) where E consists of allsurjective Σ -homomorphisms (more precisely, every sort is a surjective function) and M consistsof all injective Σ -homomorphisms reflecting all relations. That is, a Σ -homomorphism 𝑓 : 𝑋 → 𝑌 lies in M iff for every sort 𝑠 the function 𝑓 𝑠 : 𝑋 𝑠 → 𝑌 𝑠 is injective, and for every relation symbol 𝑟 : 𝑠 , . . . , 𝑠 𝑛 in Σ and every 𝑛 -tuple ( 𝑥 , . . . , 𝑥 𝑛 ) ∈ 𝑋 𝑠 × . . . × 𝑋 𝑠 𝑛 one has ( 𝑥 , . . . , 𝑥 𝑛 ) ∈ 𝑟 𝑋 iff ( 𝑓 𝑠 ( 𝑥 ) , . . . , 𝑓 𝑠 𝑛 ( 𝑥 𝑛 )) ∈ 𝑟 𝑌 . The (E , M) -factorization of a Σ -homomorphism 𝑔 : 𝑋 → 𝑍 is constructed as follows. Define a Σ -structure 𝑌 by 𝑌 𝑠 = 𝑔 𝑠 [ 𝑋 𝑠 ] for all sorts 𝑠 ∈ S , let the operations of 𝑌 be the domain-codomainrestriction of those of 𝑍 , and for every relation symbol 𝑟 : 𝑠 , . . . , 𝑠 𝑛 define 𝑟 𝑌 to be the restric-tion of 𝑟 𝑍 to 𝑌 , i.e. 𝑟 𝑌 = 𝑟 𝑍 ∩ 𝑌 𝑠 × . . . × 𝑌 𝑠 𝑛 . Then the codomain restriction of 𝑔 is a surjective Σ -homomorphism 𝑒 : 𝑋 ։ 𝑌 , and the embedding 𝑚 : 𝑌 𝑍 is a injective Σ -homomorphismreflecting all relations.(2) Similarly, the category Stone ( Σ - Str ) has the factorization system (E , M) where E consists ofall surjective morphisms and M of all relation-reflecting monomorphisms. Indeed, if 𝑓 : 𝑋 → 𝑍 is a continuous Σ -homomorphism, and if its factorization in Σ - Str is given by a Σ -structure 𝑌 and , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 9 Σ -homomorphisms 𝑒 : 𝑋 ։ 𝑌 (surjective) and 𝑚 : 𝑌 𝑍 (injective and relation-reflecting), thenthe Stone topology on 𝑌 inherited from 𝑍 yields, due to 𝑌 = 𝑒 [ 𝑋 ] being closed in 𝑍 , the desiredfactorization in Stone ( Σ - Str ) . Remark 2.16.
Recall that the arrow category A → of a category A has as objects all morphisms 𝑓 : 𝑋 → 𝑌 in A . A morphism from 𝑓 : 𝑋 → 𝑌 to 𝑔 : 𝑈 → 𝑉 in A → is given by a pair of morph-isms 𝑚 : 𝑋 → 𝑈 and 𝑛 : 𝑌 → 𝑉 in A with 𝑛 · 𝑓 = 𝑔 · 𝑚 . Identities and composition are definedcomponentwise. If A has limits of some type, then also A → has these limits, and the two projec-tion functors from A → to A mapping an arrow to its domain or codomain, respectively, preservethem. Lemma 2.17. (1)
For every cofiltered diagram 𝐷 in Set f with epic connecting maps, the limit coneof 𝐷 in Stone is formed by epimorphisms. (2)
For every cofiltered diagram 𝐷 in Stone → whose objects are epimorphisms in Stone , also lim 𝐷 isepic. Proof.
These properties follow easily from standard results about cofiltered limits in the cat-egory of compact Hausdorff spaces, see e.g. Ribes and Zalesskii [20, Sec. 1]. Here, we give an al-ternative proof using Stone duality, i.e. we verify that the category BA of boolean algebras satisfiesthe statements dual to (1) and (2).The dual of (1) states that a filtered diagram of finite boolean algebras with monic connectingmaps has a colimit in BA whose colimit maps are monic. This follows from the fact that filteredcolimits in BA are created by the forgetful functor to Set , and that filtered colimits of monics in
Set clearly have the desired property.Similarly, the dual of (2) states that a filtered colimit of monomorphisms in BA → is a mono-morphism, which follows from the corresponding property in Set → . (cid:3) In universal algebra, a pseudovariety of Σ -algebras is defined to be a class of finite algebras closedunder finite products, subalgebras, and quotient algebras. In the present section, we introduce anabstract concept of pseudovariety in a given category D with a specified full subcategory D f . Theobjects of D f are called “finite”, but this is just terminology. Our approach follows the footsteps ofBanaschewski and Herrlich [5] who introduced varieties of objects in a category D , and provedthat they are precisely the full subcategories of D presentable by an abstract notion of equation(see Definition 3.2). Here, we establish a similar result for pseudovarieties: they are precisely thefull subcategories of D f that can be presented by pseudoequations (Proposition 3.8), which areshown to be equivalent to profinite equations in many examples (Theorem 3.23). Assumption 3.1.
For the rest of our paper, we fix a complete category D with a proper factor-ization system (E , M) , that is, all morphisms in E are epic and all morphisms in M are monic. Quotients and subobjects in D are represented by morphisms in E and M , respectively, and de-noted by ։ and . Moreover, we fix a small full subcategory D f whose objects are called the finite objects of D , and denote by E f and M f the morphisms of D f in E or M , respectively. Weassume that(1) the category D f is closed under finite limits and subobjects, and(2) every object of D f is a quotient of some projective object of D .Here, recall that an object 𝑋 is called projective (more precisely, E -projective ) if for every quotient 𝑒 : 𝑃 ։ 𝑃 ′ and every morphism 𝑓 : 𝑋 → 𝑃 ′ there exists a morphism 𝑔 : 𝑋 → 𝑃 with 𝑒 · 𝑔 = 𝑓 . , Vol. 1, No. 1, Article . Publication date: January 2021. Definition 3.2 (Banaschewski and Herrlich [5]) . (1) A variety is a full subcategory of D closedunder products, subobjects, and quotients.(2) An equation is a quotient 𝑒 : 𝑋 ։ 𝐸 of a projective object 𝑋 . An object 𝐴 is said to satisfy theequation 𝑒 provided that 𝐴 is injective w.r.t. 𝑒 , that is, if for every morphism 𝑔 : 𝑋 → 𝐴 there existsa morphism ℎ : 𝐸 → 𝐴 making the triangle below commute: 𝑋 𝑒 / / / / ∀ 𝑔 (cid:25) (cid:25) ✷✷✷✷✷✷ 𝐸 ∃ ℎ (cid:6) (cid:6) ✌ ✌ ✌ 𝐴 We note that Banaschewski and Herrlich worked with the factorization system of regular epi-morphisms and monomorphisms. However, all their results and proofs apply to general properfactorization systems, as already pointed out in their paper [5].
Example 3.3.
Let Σ be a one-sorted signature of operation symbols. If D = Σ - Alg is the categoryof Σ -algebras with its usual factorization system ( E = surjective homomorphisms and M = inject-ive homomorphisms), then the above definition of a variety gives the usual concept in universalalgebra: a class of Σ -algebras closed under product algebras, subalgebras, and homomorphic im-ages. Moreover, equations in the above categorical sense are expressively equivalent to equations 𝑡 = 𝑡 ′ between Σ -terms in the usual sense:(1) Given a term equation 𝑡 = 𝑡 ′ , where 𝑡, 𝑡 ′ ∈ 𝑇 Σ 𝑋 are taken from the free algebra of all Σ -terms inthe set 𝑋 of variables, let ∼ denote the least congruence on 𝑇 Σ 𝑋 with 𝑡 ∼ 𝑡 ′ . The correspondingquotient morphism 𝑒 : 𝑇 Σ 𝑋 ։ 𝑇 Σ 𝑋 /∼ is a categorical equation satisfied by precisely those Σ -algebras that satisfy 𝑡 = 𝑡 ′ in the usual sense.(2) Conversely, given a projective Σ -algebra 𝑋 and a surjective homomorphism 𝑒 : 𝑋 ։ 𝐸 , then forany set 𝑋 of generators of 𝑋 we have a split epimorphism 𝑞 : 𝑇 Σ 𝑋 ։ 𝑋 using the projectivity of 𝑋 . Consider the set of term equations 𝑡 = 𝑡 ′ where ( 𝑡, 𝑡 ′ ) ranges over the kernel of 𝑒 · 𝑞 : 𝑇 Σ 𝑋 ։ 𝐸 .Then a Σ -algebra 𝐴 satisfies all these equations iff it satisfies 𝑒 in the categorical sense.Recall that the category D is E -co-well-powered if for every object 𝑋 of D the quotients withdomain 𝑋 form a small set. Theorem 3.4 (Banaschewski and Herrlich [5]) . Let D be a category with a proper factorizationsystem (E , M) . Suppose that D is complete, E -co-well-powered, and has enough projectives, i.e. everyobject is a quotient of a projective one. Then, a full subcategory of D is a variety iff it can be presentedby a class of equations. That is, it consists of precisely those objects satifying each of these equations. Note that the category of Σ -algebras satisfies all conditions of the theorem. Thus, in view ofExample 3.3, Banaschewski and Herrlich’s result subsumes Birkhoff’s HSP theorem [6]. In thefollowing, we are going to move from varieties in D to pseudovarieties in D f . Definition 3.5. A pseudovariety is a full subcategory of D f closed under finite products, subob-jects, and quotients. Remark 3.6.
Quotients of an object 𝑋 are ordered by factorization: given E -quotients 𝑒 , 𝑒 , weput 𝑒 ≤ 𝑒 if 𝑒 factorizes through 𝑒 𝑋 𝑒 (cid:5) (cid:5) (cid:5) (cid:5) ☛☛☛☛☛☛ 𝑒 (cid:25) (cid:25) (cid:25) (cid:25) ✸✸✸✸✸✸ 𝐸 𝐸 o o ❴ ❴ ❴ ❴ , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 11 Every pair of quotients 𝑒 𝑖 : 𝑋 ։ 𝐸 𝑖 has a least upper bound, or join , 𝑒 ∨ 𝑒 obtained by (E , M) -factorizing the mediating morphism h 𝑒 , 𝑒 i : 𝑋 → 𝐸 × 𝐸 as follows: 𝑋 𝑒 𝑖 (cid:15) (cid:15) (cid:15) (cid:15) 𝑒 ∨ 𝑒 / / / / h 𝑒 ,𝑒 i ❑❑❑❑❑ % % ❑❑❑❑ 𝐹 (cid:15) (cid:15) (cid:15) (cid:15) 𝐸 𝑖 𝐸 × 𝐸 . 𝜋 𝑖 o o (3.1)A nonempty collection of quotients closed under joins is called a semilattice of quotients . Definition 3.7. A pseudoequation is a semilattice 𝜌 𝑋 of quotients of a projective object 𝑋 (of“variables”). A finite object 𝐴 of D satisfies 𝜌 𝑋 if 𝐴 is cone-injective w.r.t. 𝜌 𝑋 , that is, for everymorphism ℎ : 𝑋 → 𝐴 , there exists a member 𝑒 : 𝑋 ։ 𝐸 of 𝜌 𝑋 through which ℎ factorizes: 𝑋 ∃ 𝑒 (cid:6) (cid:6) (cid:6) (cid:6) ☞☞☞☞☞☞ ∀ ℎ (cid:25) (cid:25) ✷✷✷✷✷✷ 𝐸 ∃ / / ❴❴❴❴ 𝐴 Proposition 3.8.
A collection of finite objects of D forms a pseudovariety iff it can be presented bypseudoequations, i.e. it consists of precisely those finite objects that satisfy each of the given pseudo-equations. Proof. (1) We first prove the if direction. Since the intersection of a family of pseudovarietiesis a pseudovariety, it suffices to prove that for every pseudoequation 𝜌 𝑋 over a projective object 𝑋 , the class V of all finite objects satisfying 𝜌 𝑋 forms a pseudovariety, i.e. is closed under finiteproducts, subobjects, and quotients.(1a) Finite products.
Let
𝐴, 𝐵 ∈ V . Since 𝐴 and 𝐵 satisfy 𝜌 𝑋 , for every morphism h ℎ, 𝑘 i : 𝑋 → 𝐴 × 𝐵 there exists 𝑒 : 𝑋 ։ 𝐸 in 𝜌 𝑋 such that both ℎ : 𝑋 → 𝐴 and 𝑘 : 𝑋 → 𝐵 factorize through 𝑒 – thisfollows from the closedness of pseudoequations under binary joins. Given ℎ = 𝑒 · ℎ ′ and 𝑘 = 𝑒 · 𝑘 ′ ,then h ℎ ′ , 𝑘 ′ i : 𝑋 → 𝐸 𝑖 is the desired factorization: h ℎ, 𝑘 i = 𝑒 · h ℎ ′ , 𝑘 ′ i . Thus 𝐴 × 𝐵 ∈ V . Since the terminal object 1 clearly satisfies every pseudoequation, we also have1 ∈ V .(1b) Subobjects.
Let 𝑚 : 𝐴 𝐵 be a morphism in M f with 𝐵 ∈ V . Then for every morphism ℎ : 𝑋 → 𝐴 we know that 𝑚 · ℎ factorizes as 𝑒 · 𝑘 for some 𝑒 : 𝑋 ։ 𝐸 in 𝜌 𝑋 and some 𝑘 : 𝐸 → 𝐵 . Thediagonal fill-in property then shows that ℎ factorizes through 𝑒 : 𝑋 ℎ (cid:15) (cid:15) 𝑒 / / / / 𝐸 𝑘 (cid:15) (cid:15) (cid:127) (cid:127) ⑧ ⑧ ⑧ ⑧ 𝐴 / / 𝑚 / / 𝐵 Thus, 𝐴 ∈ V .(1c) Quotients.
Let 𝑞 : 𝐵 ։ 𝐴 be a morphism in E f with 𝐵 ∈ V . Every morphism ℎ : 𝑋 → 𝐴 factorizes, since 𝑋 is projective, as ℎ = 𝑞 · 𝑘 for some 𝑘 : 𝑋 → 𝐵 Since 𝑘 factorizes through some 𝑒 ∈ 𝜌 𝑋 , so does ℎ . Thus, 𝐴 ∈ V . , Vol. 1, No. 1, Article . Publication date: January 2021. (2) For the “only if” direction, suppose that V is a pseudovariety. For every projective object 𝑋 we form the pseudoequation 𝜌 𝑋 consisting of all quotients 𝑒 : 𝑋 ։ 𝐸 with 𝐸 ∈ V . This is indeed asemilattice: given 𝑒, 𝑓 ∈ 𝜌 𝑋 we have 𝑒 ∨ 𝑓 ∈ 𝜌 𝑋 by (3.1), using that V is closed under finite productsand subobjects. We claim that V is presented by the collection of all the above pseudoequations 𝜌 𝑋 .(2a) Every object 𝐴 ∈ V satisfies all 𝜌 𝑋 . Indeed, given a morphism ℎ : 𝑋 → 𝐴 , factorize it as 𝑒 : 𝑋 ։ 𝐸 in E followed by 𝑚 : 𝐸 𝐴 in M . Then 𝐸 ∈ V because V is closed under subobjects,so 𝑒 is a member of 𝜌 𝑋 . Therefore ℎ = 𝑚 · 𝑒 is the desired factorization of ℎ , proving that 𝐴 satisfies 𝜌 𝑋 .(2b) Every finite object 𝐴 satisfying all the pseudoequations 𝜌 𝑋 lies in V . Indeed, by Assump-tion 3.1 there exists a quotient 𝑞 : 𝑋 ։ 𝐴 for some projective object 𝑋 . Since 𝐴 satisfies 𝜌 𝑋 , thereexists a factorization 𝑞 = ℎ · 𝑒 for some 𝑒 : 𝑋 ։ 𝐸 in 𝜌 𝑋 and some ℎ : 𝐸 → 𝐴 . We know that 𝐸 ∈ V ,and from 𝑞 ∈ E we deduce ℎ ∈ E . Thus 𝐴 , being a quotient of an object of V , lies in V . (cid:3) Remark 3.9. (1) Proposition 3.8 would remain valid if we defined pseudoequations as semilat-tices of finite quotients of a projective object. This follows immediately from the above proof.(2) Let us assume that a collection
Var of projective objects of D is given such that every finiteobject is a quotient of an object of Var (cf. Assumption 3.1(2)). Then we could define pseudoequa-tions as semilattices of quotients of members of
Var with finite codomains. Again, from the aboveproof we see that Proposition 3.8 would remain true.We would like to reduce pseudoequations to equations in the sense of Banaschewski and Herr-lich. For that we need to move from the category D to the pro-completion of D f . Notation 3.10.
Since D has (cofiltered) limits, the embedding D f D extends to an essentiallyunique cofinitary functor 𝑉 : Pro D f → D . Example 3.11. If D is a full subcategory of Σ - Str closed under cofiltered limits, we have seen that
Pro D f can be described as a full subcategory of Stone D by Corollary 2.12. The above functor 𝑉 : Pro D f → D is the functor forgetting the topology. Indeed, the corresponding forgetful functor from Stone ( Σ - Str ) to Σ - Str is cofinitary, hence, so is 𝑉 . Remark 3.12.
Recall, e.g. from Mac Lane [14], that the right Kan extension of a functor 𝐹 : A → C along 𝐾 : A → B is a functor 𝑅 = Ran 𝐾 𝐹 : B → C with a universal natural transformation 𝜀 : 𝑅𝐾 → 𝐹 , that is, for every functor 𝐺 : B → C and every natural transformation 𝛾 : 𝐺𝐾 → 𝐹 there exists a unique natural transformation 𝛾 † : 𝐺 → 𝑅 with 𝛾 = 𝜀 · 𝛾 † 𝐾 . If A is small and C iscomplete, then the right Kan extension exists [14, Theorem X.3.1, X.4.1], and the object 𝑅𝐵 ( 𝐵 ∈ B )can be constructed as the limit 𝑅𝐵 = lim ( 𝐵 / 𝐾 𝑄 𝐵 −−→ A 𝐹 −→ C ) , where 𝐵 / 𝐾 denotes the slice category of all morphisms 𝑓 : 𝐵 → 𝐾𝐴 ( 𝐴 ∈ A ) and 𝑄 𝐵 is theprojection functor 𝑓 ↦→ 𝐴 . Equivalently, 𝑅𝐵 is given by the end 𝑅𝐵 = ∫ 𝐴 ∈ A B ( 𝐵, 𝐾𝐴 ) ⋔ 𝐹𝐴, with 𝑆 ⋔ 𝐶 denoting 𝑆 -fold power of 𝐶 ∈ C . , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 13 Lemma 3.13.
The functor 𝑉 has a left adjoint c (−) = Ran 𝐽 𝐸 : D → Pro D f given by the right Kan extension of the embedding 𝐸 : D f Pro D f along the embedding 𝐽 : D f D and making the following triangle commute up to isomorphism: D f / / 𝐽 / / " " 𝐸 " " ❋❋❋❋❋❋❋❋ D d (−) | | ②②②②②②②②② Pro D f Proof.
Recall that, up to equivalence,
Pro D f is the full subcategory of [ D f , Set ] op on cofilteredlimits of representables with 𝐸𝐷 = D f ( 𝐷, −) for every 𝐷 ∈ D f (see Remark A.6), and the functor 𝑉 is given by 𝑉 = Ran 𝐸 𝐽 : Pro D f → D . Consider the following chain of isomorphisms natural in 𝐷 ∈ D and 𝐻 ∈ Pro D f : D ( 𝐷, 𝑉 𝐻 ) (cid:27) D ( 𝐷, ( Ran 𝐸 𝐽 ) 𝐻 ) (cid:27) D (cid:18) 𝐷, ∫ 𝑋 Pro D f ( 𝐻, 𝐸𝑋 ) ⋔ 𝐽 𝑋 (cid:19) by the end formula for
Ran , = D (cid:18) 𝐷, ∫ 𝑋 [ D f , Set ] (
𝐸𝑋, 𝐻 ) ⋔ 𝐽 𝑋 (cid:19)
Pro D f full subcategory of [ D f , Set ] op , (cid:27) D (cid:18) 𝐷, ∫ 𝑋 𝐻𝑋 ⋔ 𝐽 𝑋 (cid:19) by the Yoneda lemma, (cid:27) ∫ 𝑋 D ( 𝐷, 𝐻𝑋 ⋔ 𝐽 𝑋 ) D ( 𝐷, −) preserves ends, (cid:27) ∫ 𝑋 Set ( 𝐻𝑋, D ( 𝐷, 𝐽 𝑋 )) by the universal property of power, (cid:27) [ D f , Set ] ( 𝐻, D ( 𝐷, 𝐽 −)) the set of nat. trafo. as an end, = Pro D f ( D ( 𝐷, 𝐽 −) , 𝐻 ) Pro D f full subcategory of [ D f , Set ] op .Hence, the functor c (−) : 𝐷 ↦→ D ( 𝐷, 𝐽 −) is a left adjoint to 𝑉 . Moreover, c (−) extends 𝐸 : for each 𝐷 ∈ D f , we have b 𝐷 = D ( 𝐽 𝐷, 𝐽 −) = D f ( 𝐷, −) = 𝐸𝐷, and similarly on morphisms, since 𝐽 is a full inclusion. It remains to verify that the functor c (−) coincides with Ran 𝐽 𝐸 . This follows from the fact that every presheaf is a canonical colimit ofrepresentables expressed as a coend in [ D f , Set ] : D ( 𝐷, 𝐽 −) (cid:27) ∫ 𝑋 D ( 𝐷, 𝐽 𝑋 ) •
𝐸𝑋, with • denoting copowers. This corresponds to an end in [ D f , Set ] op : ∫ 𝑋 D ( 𝐷, 𝐽 𝑋 ) ⋔ 𝐸𝑋 = ( Ran 𝐽 𝐸 ) 𝐷. Thus c (−) = Ran 𝐽 𝐸 , as claimed. (cid:3) , Vol. 1, No. 1, Article . Publication date: January 2021. Construction 3.14.
By expressing the right Kan extension c (−) = Ran 𝐽 𝐸 : D → Pro D f as a limit, the action 𝐷 → b 𝐷 on objects, 𝑓 ↦→ b 𝑓 on morphisms, the unit, and the counit of theadjunction c (−) ⊣ 𝑉 are given as follows.(1) For every object 𝐷 of D , the object b 𝐷 ∈ Pro D f is a limit of the diagram 𝑃 𝐷 : 𝐷 / D f → Pro D f , 𝑃 𝐷 ( 𝐷 𝑎 −→ 𝐴 ) = 𝐴. We use the following notation for the limit cone of 𝑃 𝐷 : 𝐷 𝑎 −→ 𝐴 b 𝐷 b 𝑎 −→ 𝐴 where ( 𝐴, 𝑎 ) ranges over 𝐷 / D f . For finite 𝐷 we choose the trivial limit: b 𝐷 = 𝐷 and b 𝑎 = 𝑎 .(2) Given 𝑓 : 𝐷 → 𝐷 ′ in D , the morphisms d 𝑎 · 𝑓 with 𝑎 ranging over 𝐷 ′ / D f form a cone over 𝑃 𝐷 ′ .Define b 𝑓 : b 𝐷 → c 𝐷 ′ to be the unique morphism such that the following triangles commute for all 𝑎 : 𝐷 ′ → 𝐴 with 𝐴 ∈ D f : b 𝐷 b 𝑓 / / d 𝑎 · 𝑓 (cid:24) (cid:24) ✶✶✶✶✶✶ c 𝐷 ′ b 𝑎 (cid:5) (cid:5) ☛☛☛☛☛☛☛ 𝐴 Note that overloading the notation c (−) causes no problem because if c 𝐷 ′ = 𝐷 ′ ∈ D f then b 𝑓 is aprojection of the limit cone for 𝑃 𝐷 (see item (1)), since for 𝑎 = id 𝐷 ′ we have b 𝑎 = id 𝐷 ′ .(3) The unit 𝜂 at 𝐷 ∈ D is given by the unique morphism 𝜂 𝐷 : 𝐷 → 𝑉 b 𝐷 in D such that the following triangles commute for all ℎ : 𝐷 → 𝐴 with 𝐴 ∈ D f : 𝐷 𝜂 𝐷 / / ℎ ❆❆❆❆❆❆❆❆ 𝑉 b 𝐷 𝑉 b ℎ (cid:15) (cid:15) 𝐴 Here one uses that 𝑉 is cofinitary, and thus the morphisms 𝑉 b ℎ form a limit cone in D .(4) The counit 𝜀 at 𝐷 ∈ Pro D f is the unique morphism 𝜀 𝐷 : c 𝑉 𝐷 → 𝐷 in b D such that the following triangles commute, where 𝑎 : 𝐷 → 𝐴 ranges over the slice category 𝐷 / D f : c 𝑉 𝐷 𝜀 𝐷 / / c 𝑉 𝑎 (cid:26) (cid:26) ✹✹✹✹✹✹✹ 𝐷 𝑎 (cid:6) (cid:6) ✌✌✌✌✌✌ 𝐴 , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 15 Notation 3.15.
Recall that E f and M f are the morphisms of D f in E and M , respectively. Wedenote by b E and c M the collection of all morphisms of Pro D f that are cofiltered limits of members of E f or M f in thearrow category ( Pro D f ) → , respectively. Remark 3.16. (1) For every finite E -quotient 𝑒 : 𝑋 ։ 𝐸 in D , the corresponding limit projection b 𝑒 : b 𝑋 ։ 𝐸 lies in b E . Indeed, since 𝐸 is finitely copresentable in Pro D f , the morphism b 𝑒 factorizesthrough b ℎ for some ℎ in 𝑋 / D f , which can be assumed to be a quotient in E . Otherwise, take the (E , M) -factorization ℎ = 𝑚 · 𝑞 of ℎ and replace ℎ by 𝑞 .Thus, we obtain an initial subdiagram 𝑃 ′ 𝑋 : I → Pro D f of 𝑃 𝑋 : 𝑋 / D f → Pro D f by restricting 𝑃 𝑋 to the full subcategory of finite quotients ℎ : 𝑋 ։ 𝐴 in 𝑋 / D f through which 𝑒 factorizes,i.e. where 𝑒 = 𝑒 ℎ · ℎ for some 𝑒 ℎ : 𝐴 → 𝐸 . Note that 𝑒 ℎ ∈ E because 𝑒, ℎ ∈ E . The quotients 𝑒 ℎ ( ℎ ∈ I ) form a cofiltered diagram in ( Pro D f ) → with limit cone ( b ℎ, id 𝐸 ) : b 𝑋 b ℎ (cid:15) (cid:15) b 𝑒 / / 𝐸 id 𝑒 (cid:15) (cid:15) 𝐴 𝑒 ℎ / / / / 𝐸 Thus, b 𝑒 ∈ b E .(2) For every cofiltered diagram 𝐵 : 𝐼 → D f with connecting morphisms in E , the limit projectionsin Pro D f lie in b E . Indeed, let 𝑏 𝑖 : 𝑋 → 𝐵 𝑖 ( 𝑖 ∈ 𝐼 ) denote the limit cone. Given 𝑗 ∈ 𝐼 , we are to show 𝑏 𝑗 ∈ b E . Form the diagram in D → f whose objects are all connecting morphisms of 𝐵 with codomain 𝐵 𝑗 and whose morphisms from ℎ : 𝐵 𝑖 ։ 𝐵 𝑗 to ℎ ′ : 𝐵 𝑖 ′ ։ 𝐵 𝑗 are all connecting maps 𝑘 : 𝐵 𝑖 ։ 𝐵 𝑖 ′ of 𝐵 . This is a cofiltered diagram in Pro D f with limit 𝑏 𝑗 and the following limit cone: 𝑋 𝑏 𝑖 (cid:15) (cid:15) 𝑏 𝑗 / / 𝐵 𝑗 id (cid:15) (cid:15) 𝐵 𝑖 ℎ / / / / 𝐵 𝑗 Since each ℎ lies in E f , this proves 𝑏 𝑗 ∈ b E .(3) For every cone 𝑝 𝑖 : 𝑃 → 𝐵 𝑖 of the diagram 𝐵 in (2) with 𝑝 𝑖 ∈ b E for all 𝑖 ∈ 𝐼 , the uniquefactorization 𝑝 : 𝑃 → 𝑋 through the limit of 𝐵 lies in b E . Indeed, 𝑝 is the limit of 𝑝 𝑖 , 𝑖 ∈ 𝐼 , with thefollowing limit cone: 𝑃 id (cid:15) (cid:15) 𝑝 / / 𝑋 𝑏 𝑖 (cid:15) (cid:15) 𝑃 𝑝 𝑖 / / 𝐵 𝑖 Proposition 3.17.
The pair ( b E , c M) is a proper factorization system of Pro D f . Proof. (1) All morphisms of b E are epic. This follows from the dual of [3, Prop. 1.62]; however,we give a direct proof. Given 𝑒 : 𝑋 → 𝑌 in b E , we have a limit cone of a cofiltered diagram 𝐷 in , Vol. 1, No. 1, Article . Publication date: January 2021. ( Pro D f ) → as follows: 𝑋 𝑒 / / 𝑎 𝑖 (cid:15) (cid:15) 𝑌 𝑏 𝑖 (cid:15) (cid:15) 𝐴 𝑖 𝑒 𝑖 / / / / 𝐵 𝑖 ( 𝑖 ∈ 𝐼 ) where 𝑒 𝑖 ∈ E f for each 𝑖 ∈ 𝐼 . Let 𝑝, 𝑞 : 𝑌 → 𝑍 be two morphisms with 𝑝 · 𝑒 = 𝑞 · 𝑒 ; we need toshow 𝑝 = 𝑞 . Without loss of generality we can assume that the object 𝑍 is finite because D f islimit-dense in Pro D f . Since ( 𝑏 𝑖 ) is a cofiltered limit cone in Pro D f , there exists 𝑖 ∈ 𝐼 such that 𝑝 and 𝑞 factorize through 𝑏 𝑖 , i.e. there exists morphisms 𝑝 ′ , 𝑞 ′ with 𝑝 ′ · 𝑏 𝑖 = 𝑝 and 𝑞 ′ · 𝑏 𝑖 = 𝑞 . The limitprojection 𝑎 𝑖 of the cofiltered limit 𝑋 = lim 𝐴 𝑖 in Pro D f merges 𝑝 ′ · 𝑒 𝑖 and 𝑞 ′ · 𝑒 𝑖 (since 𝑒 merges 𝑝 and 𝑞 ). Since 𝑍 is finite, there exists a connecting morphism ( 𝑎 𝑗𝑖 , 𝑏 𝑗𝑖 ) of 𝐷 such that 𝑝 ′ · 𝑒 𝑖 and 𝑞 ′ · 𝑒 𝑖 are merged by 𝑎 𝑗𝑖 . 𝑋 𝑎 𝑗 (cid:18) (cid:18) 𝑒 / / 𝑎 𝑖 (cid:15) (cid:15) 𝑌 𝑏 𝑖 (cid:15) (cid:15) 𝑝 / / 𝑞 / / 𝑍𝐴 𝑖 𝑒 𝑖 / / / / 𝑒 𝑖 / / / / 𝐵 𝑖 𝑝 ′ L L 𝑞 ′ L L 𝐴 𝑗 𝑎 𝑗𝑖 B B ✆✆✆✆✆✆✆✆✆ 𝑒 𝑗 / / / / 𝐵 𝑗𝑏 𝑗𝑖 \ \ ✾✾✾✾✾✾✾✾✾ Therefore 𝑝 ′ · 𝑏 𝑗𝑖 · 𝑒 𝑗 = 𝑝 ′ · 𝑒 𝑖 · 𝑎 𝑗𝑖 = 𝑞 ′ · 𝑒 𝑖 · 𝑎 𝑗𝑖 = 𝑞 ′ · 𝑏 𝑗𝑖 · 𝑒 𝑗 . Since 𝑒 𝑗 is an epimorphism in D f , this implies 𝑝 ′ · 𝑏 𝑗𝑖 = 𝑞 ′ · 𝑏 𝑗𝑖 . Thus 𝑝 = 𝑝 ′ · 𝑏 𝑖 = 𝑝 ′ · 𝑏 𝑗𝑖 · 𝑏 𝑗 = 𝑞 ′ · 𝑏 𝑗𝑖 · 𝑏 𝑗 = 𝑞 ′ · 𝑏 𝑖 = 𝑞. (2) All morphisms of c M are monic. Indeed, given 𝑚 : 𝑋 → 𝑌 in c M , we have a limit cone of acofiltered diagram 𝐷 in ( Pro D f ) → as follows: 𝑋 𝑚 / / 𝑎 𝑖 (cid:15) (cid:15) 𝑌 𝑏 𝑖 (cid:15) (cid:15) 𝐴 𝑖 / / 𝑚 𝑖 / / 𝐵 𝑖 ( 𝑖 ∈ 𝐼 ) where 𝑚 𝑖 ∈ M f for each 𝑖 ∈ 𝐼 . Suppose that 𝑓 , 𝑔 : 𝑍 → 𝑋 with 𝑚 · 𝑓 = 𝑚 · 𝑔 are given. Express 𝑍 as a cofiltered limit 𝑧 𝑗 : 𝑍 ։ 𝑍 𝑗 ( 𝑗 ∈ 𝐽 ) of finite objects with epimorphic limit projections 𝑧 𝑗 . Foreach 𝑖 ∈ 𝐼 , since 𝐴 𝑖 is finitely copresentable, we obtain a factorization of 𝑎 𝑖 · 𝑓 and 𝑎 𝑖 · 𝑔 throughsome 𝑧 𝑗 𝑖 , say 𝑓 𝑖 · 𝑧 𝑗 𝑖 = 𝑓 and 𝑔 𝑖 · 𝑧 𝑗 𝑖 = 𝑔 . 𝑍 𝑓 / / 𝑔 / / 𝑧 𝑗𝑖 (cid:15) (cid:15) (cid:15) (cid:15) 𝑋 𝑚 / / 𝑎 𝑖 (cid:15) (cid:15) 𝑌 𝑏 𝑖 (cid:15) (cid:15) 𝑍 𝑗 𝑖 𝑓 𝑖 / / 𝑔 𝑖 / / 𝐴 𝑖 / / 𝑚 𝑖 / / 𝐵 𝑖 , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 17 From 𝑚 · 𝑓 = 𝑚 · 𝑔 it follows that 𝑚 𝑖 · 𝑓 𝑖 · 𝑧 𝑗 𝑖 = 𝑚 𝑖 · 𝑔 𝑖 · 𝑧 𝑗 𝑖 for each 𝑖 . This implies 𝑚 𝑖 · 𝑓 𝑖 = 𝑚 𝑖 · 𝑔 𝑖 because 𝑧 𝑗 𝑖 is epic, and thus 𝑓 𝑖 = 𝑔 𝑖 because 𝑚 𝑖 is monic in D f . Therefore, 𝑎 𝑖 · 𝑓 = 𝑎 𝑖 · 𝑔 for each 𝑖 ,thus 𝑓 = 𝑔 because the limit projections 𝑎 𝑖 are collectively monic.(3) Every morphism 𝑔 : 𝑋 → 𝑌 of Pro D f has an ( b E , c M) -factorization. Indeed, ( Pro D f ) → is thepro-completion of D → f ; see [3, Cor. 1.54] for the dual statement. Thus, there exists a cofiltereddiagram 𝑅 : 𝐼 → D → f with limit 𝑔 . Let the following morphisms 𝑋 𝑎 𝑖 (cid:15) (cid:15) 𝑔 / / 𝑌 𝑏 𝑖 (cid:15) (cid:15) 𝐴 𝑖 𝑔 𝑖 / / 𝐵 𝑖 ( 𝑖 ∈ 𝐼 ) form the limit cone. Factorize 𝑔 𝑖 into an E -morphism 𝑒 𝑖 : 𝐴 𝑖 ։ 𝐶 𝑖 and followed by an M -morphism 𝑚 𝑖 : 𝐶 𝑖 𝐵 𝑖 . Since D f is closed under subobjects, we have 𝑒 𝑖 ∈ E f and 𝑚 𝑖 ∈ M f . Diagonal fill-inyields a diagram 𝑅 : 𝐼 → D f with objects 𝐶 𝑖 , 𝑖 ∈ 𝐼 , and connecting morphisms derived from thoseof 𝑅 . Let 𝑍 ∈ Pro D f be a limit of 𝑅 with the limit cone 𝑐 𝑖 : 𝑍 → 𝐶 𝑖 ( 𝑖 ∈ 𝐼 ) . Then there are unique morphisms 𝑒 = lim 𝑒 𝑖 ∈ b E , and 𝑚 = lim 𝑚 𝑖 ∈ c M such that the followingdiagrams commute for all 𝑖 ∈ 𝐼 : 𝑋 𝑎 𝑖 (cid:15) (cid:15) 𝑔 / / 𝑒 $ $ ■■■■■■■ 𝑌 𝑏 𝑖 (cid:15) (cid:15) 𝑍 𝑚 : : ✉✉✉✉✉✉✉ 𝑐 𝑖 (cid:15) (cid:15) 𝐴 𝑖 𝑒 𝑖 / / / / 𝐶 𝑖 / / 𝑚 𝑖 / / 𝐵 𝑖 (4) We verify the diagonal fill-in property. Let a commutative square 𝑋 𝑒 / / 𝑢 (cid:15) (cid:15) 𝑌 𝑣 (cid:15) (cid:15) 𝑃 𝑚 / / 𝑄 with 𝑒 ∈ b E and 𝑚 ∈ c M be given.(4a) Assume first that 𝑚 ∈ M f . Express 𝑒 as a cofiltered limit of objects 𝑒 𝑖 ∈ E f with the followinglimit cone: 𝑋 𝑒 / / 𝑎 𝑖 (cid:15) (cid:15) 𝑌 𝑏 𝑖 (cid:15) (cid:15) 𝐴 𝑖 𝑒 𝑖 / / 𝐵 𝑖 Since 𝑃 is finite and 𝑋 = lim 𝐴 𝑖 is a cofiltered limit, 𝑢 factorizes through some 𝑎 𝑖 . Analogously for 𝑣 and some 𝑏 𝑖 ; the index 𝑖 can be chosen to be the same since the diagram is cofiltered. Thus wehave morphisms 𝑢 ′ , 𝑣 ′ such that in the following diagram the left-hand triangle and the right-hand , Vol. 1, No. 1, Article . Publication date: January 2021. one commute: 𝑋 𝑒 / / GF@A 𝑢 / / 𝑎 𝑖 (cid:15) (cid:15) 𝑌 EDBC 𝑣 / / 𝑏 𝑖 (cid:15) (cid:15) 𝐴 𝑖𝑢 ′ (cid:15) (cid:15) 𝑒 𝑖 / / / / 𝐵 𝑖𝑣 ′ (cid:15) (cid:15) 𝑃 𝑚 / / 𝑄 Without loss of generality, we can assume that the lower part also commutes. Indeed, 𝑄 is finiteand the limit map 𝑎 𝑖 merges the lower part: ( 𝑚 · 𝑢 ′ ) · 𝑎 𝑖 = 𝑚 · 𝑢 = 𝑣 · 𝑒 = 𝑣 ′ · 𝑏 𝑖 · 𝑒 = ( 𝑣 ′ · 𝑒 𝑖 ) · 𝑎 𝑖 . Since our diagram is cofiltered, some connecting morphism from 𝐴 𝑗 to 𝐴 𝑖 also merges the lowerpart. Hence, by choosing 𝑗 instead of 𝑖 we could get the lower part commutative.Since 𝑒 𝑖 ∈ E and 𝑚 ∈ M , we use diagonal fill-in to get a morphism 𝑑 : 𝐵 𝑖 → 𝐵 with 𝑑 · 𝑒 𝑖 = 𝑢 ′ and 𝑚 · 𝑑 = 𝑣 ′ . Then 𝑑 · 𝑏 𝑖 : 𝑌 → 𝑃 is the desired diagonal in the original square.(4b) Now suppose that 𝑚 ∈ c M is arbitrary, i.e. a cofiltered limit a diagram 𝐷 whose objects aremorphisms 𝑚 𝑡 of M f with a limit cone as follows: 𝑃 𝑚 / / 𝑝 𝑡 (cid:15) (cid:15) 𝑄 𝑞 𝑡 (cid:15) (cid:15) 𝑃 𝑡 / / 𝑚 𝑡 / / 𝑄 𝑡 ( 𝑡 ∈ 𝑇 ) For each 𝑡 we have, due to item (4a) above, a diagonal fill-in 𝑋 𝑒 / / 𝑢 (cid:15) (cid:15) 𝑌 𝑣 (cid:15) (cid:15) 𝑑 𝑡 ✍✍✍✍✍✍ (cid:7) (cid:7) ✍✍✍✍✍✍ 𝑃 𝑝 𝑡 (cid:15) (cid:15) 𝑄 𝑞 𝑡 (cid:15) (cid:15) 𝑃 𝑡 / / 𝑚 𝑡 / / 𝑄 𝑡 Given a connecting morphism ( 𝑝, 𝑞 ) : 𝑚 𝑡 → 𝑚 𝑠 ( 𝑡, 𝑠 ∈ 𝑇 ) of the diagram 𝐷 , the following triangle 𝑌 𝑑 𝑡 (cid:3) (cid:3) ✞✞✞✞✞✞✞ 𝑑 𝑠 (cid:27) (cid:27) ✼✼✼✼✼✼✼ 𝑃 𝑡 𝑝 / / 𝑃 𝑠 commutes, that is, all 𝑑 𝑡 form a cone of the diagram 𝐷 · 𝐷 , where 𝐷 : D → f → D f is the domainfunctor, with limit 𝑝 𝑡 : 𝑃 → 𝑃 𝑡 ( 𝑡 ∈ 𝑇 ). Indeed, 𝑒 is epic by item (1), and from the fact that 𝑝 𝑠 = 𝑝 · 𝑝 𝑡 we obtain ( 𝑝 · 𝑑 𝑡 ) · 𝑒 = 𝑝 · 𝑝 𝑡 · 𝑢 = 𝑝 𝑠 · 𝑢 = 𝑑 𝑠 · 𝑒. Thus, there exists a unique 𝑑 : 𝑌 → 𝑃 with 𝑑 𝑡 = 𝑝 𝑡 · 𝑑 for all 𝑡 ∈ 𝑇 . This is the desired diagonal: 𝑢 = 𝑑 · 𝑒 follows from ( 𝑝 𝑡 ) 𝑡 ∈ 𝑇 being collectively monic, since 𝑝 𝑡 · 𝑢 = 𝑑 𝑡 · 𝑒 = 𝑝 𝑡 · 𝑑 · 𝑒. (cid:3) , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 19 Proposition 3.18.
Let D be a full subcategory of Σ - Str closed under products and subobjects. Thenin
Pro D f ⊆ Stone D we have b E = surjective morphisms, and c M = relation-reflecting injective morphisms , cf. Remark 2.15(2). Proof. (1) Let 𝑒 : 𝑋 → 𝑌 be a surjective morphism of Pro D f . We shall prove that 𝑒 ∈ b E byexpressing it as a cofiltered limit of a diagram of quotients in D → f . In Stone D we have the fac-torization system (E , M ) where E = surjective homomorphisms, and M = injective relation-reflecting homomorphisms. This follows from Remark 2.15 and the fact that D , being closed undersubobjects in Σ - Str , inherits the factorization system Σ - Str .Since
Pro D f is the closure of D f under cofiltered limits in Stone ( D ) by Corollary 2.12, also ( Pro D f ) → = Pro ( D → f ) is the closure of D → f under cofiltered limits in ( Stone D ) → . Thus for 𝑒 there exists a cofiltered diagram 𝐷 in D → f of morphisms ℎ 𝑖 : 𝐴 𝑖 → 𝐵 𝑖 ( 𝑖 ∈ 𝐼 ) of D f with a limitcone in ( Stone D ) → as follows: 𝑋 𝑒 / / 𝑎 𝑖 (cid:15) (cid:15) 𝑌 𝑏 𝑖 (cid:15) (cid:15) 𝐴 𝑖 ℎ 𝑖 / / 𝐵 𝑖 Using the factorization system (E , M ) we factorize 𝑎 𝑖 = 𝑚 𝑖 · 𝑎 𝑖 and 𝑏 𝑖 = 𝑛 𝑖 · 𝑏 𝑖 for 𝑖 ∈ 𝐼, and use the diagonal fill-in to define morphisms ℎ 𝑖 as follows: 𝑋 𝑒 / / / / 𝑎 𝑖 (cid:15) (cid:15) (cid:15) (cid:15) 𝑌 𝑏 𝑖 (cid:15) (cid:15) (cid:15) (cid:15) 𝐴 𝑖 ℎ 𝑖 / / ❴❴❴ (cid:15) (cid:15) 𝑚 𝑖 (cid:15) (cid:15) 𝐵 𝑖 (cid:15) (cid:15) 𝑛 𝑖 (cid:15) (cid:15) 𝐴 𝑖 ℎ 𝑖 / / 𝐵 𝑖 We obtain a diagram 𝐷 with objects ℎ 𝑖 : 𝐴 𝑖 → 𝐵 𝑖 ( 𝑖 ∈ 𝐼 ) in ( Stone D ) → . Connecting morphismsare derived from those of 𝐷 : given ( 𝑝, 𝑞 ) : ℎ 𝑖 → ℎ 𝑗 in 𝐷𝐴 𝑖 ℎ 𝑖 / / 𝑝 (cid:15) (cid:15) 𝐵 𝑖𝑞 (cid:15) (cid:15) 𝐴 𝑗 ℎ 𝑗 / / 𝐵 𝑗 , Vol. 1, No. 1, Article . Publication date: January 2021. the diagonal fill-in property yields morphisms 𝑝 and 𝑞 as follows: 𝑋 𝑎 𝑗 (cid:15) (cid:15) 𝑎 𝑖 / / / / 𝐴 𝑖𝑚 𝑖 (cid:15) (cid:15) 𝑝 (cid:4) (cid:4) ✠ ✠ ✠ ✠ ✠ ✠ 𝐴 𝑖𝑝 (cid:15) (cid:15) 𝐴 𝑗 / / 𝑚 𝑗 / / 𝐴 𝑗 𝑌 𝑎 𝑗 (cid:15) (cid:15) 𝑏 𝑖 / / / / 𝐵 𝑖𝑛 𝑖 (cid:15) (cid:15) 𝑞 (cid:4) (cid:4) ✡ ✡ ✡ ✡ ✡ ✡ 𝐵 𝑖𝑞 (cid:15) (cid:15) 𝐵 𝑗 / / 𝑛 𝑗 / / 𝐵 𝑗 It is easy to see that ( 𝑝, 𝑞 ) is a morphism from ℎ 𝑖 to ℎ 𝑗 in ( Stone D ) → . This yields a cofiltereddiagram 𝐷 . Since ℎ 𝑖 · 𝑎 𝑖 = 𝑏 𝑖 · 𝑒 is surjective, it follows that ℎ 𝑖 is also surjective. We claim that themorphisms ( 𝑎 𝑖 , 𝑏 𝑖 ) : 𝑒 → ℎ 𝑖 ( 𝑖 ∈ 𝐼 ) form a limit cone of 𝐷 . To see this, note first that since the morphisms ( 𝑎 𝑖 , 𝑏 𝑖 ) : 𝑒 → ℎ 𝑖 , 𝑖 ∈ 𝐼 , forma cone of 𝐷 and all 𝑚 𝑖 and 𝑛 𝑖 are monic, the morphisms ( 𝑎 𝑖 , 𝑏 𝑖 ) , 𝑖 ∈ 𝐼 , form a cone of 𝐷 . Now letanother cone be given with domain 𝑟 : 𝑈 → 𝑉 as follows: 𝑈 𝑟 / / 𝑢 𝑖 (cid:15) (cid:15) 𝑉 𝑣 𝑖 (cid:15) (cid:15) 𝐴 𝑖 (cid:15) (cid:15) 𝑚 𝑖 (cid:15) (cid:15) ℎ 𝑖 / / 𝐵 𝑖 (cid:15) (cid:15) 𝑛 𝑖 (cid:15) (cid:15) 𝐴 𝑖 ℎ 𝑖 / / 𝐵 𝑖 Then we get a cone of 𝐷 for all 𝑖 ∈ 𝐼 by the morphisms ( 𝑚 𝑖 𝑢 𝑖 , 𝑛 𝑖 𝑣 𝑖 ) : 𝑟 → ℎ 𝑖 . The unique factoriza-tion ( 𝑢, 𝑣 ) through the limit cone of 𝐷 : 𝑈 𝑟 / / 𝑢 (cid:15) (cid:15) ?>89 𝑚 𝑖 𝑢 𝑖 ✥✥✥✥✥✥✥ 𝑉 𝑣 (cid:15) (cid:15) EDBC 𝑛 𝑖 𝑣 𝑖 o o 𝑋 𝑒 / / 𝑎 𝑖 (cid:15) (cid:15) 𝑌 𝑏 𝑖 (cid:15) (cid:15) 𝐴 𝑖 ℎ 𝑖 / / 𝐵 𝑖 is a factorization of ( 𝑢 𝑖 , 𝑣 𝑖 ) through the cone ( 𝑎 𝑖 , 𝑏 𝑖 ) . Indeed, in the following diagram 𝑈 𝑟 / / 𝑢 (cid:15) (cid:15) ?>89 𝑢 𝑖 ✥✥✥✥✥✥✥✥ 𝑉 𝑣 (cid:15) (cid:15) 𝑋 𝑒 / / 𝑎 𝑖 (cid:15) (cid:15) 𝑌 𝑏 𝑖 (cid:15) (cid:15) 𝐴 𝑖𝑚 𝑖 (cid:15) (cid:15) ℎ 𝑖 / / 𝐵 𝑖𝑛 𝑖 (cid:15) (cid:15) BC o o ED 𝑣 𝑖 𝐴 𝑖 ℎ 𝑖 / / 𝐵 𝑖 the desired equality 𝑣 𝑖 = 𝑏 𝑖 𝑣 follows since 𝑛 𝑖 is monic; analogously for 𝑢 𝑖 = 𝑎 𝑖 𝑢 . The uniquenessof the factorization ( 𝑢, 𝑣 ) also follows from the last diagram: if the upper left-hand and right-handparts commute, then ( 𝑢, 𝑣 ) is a factorization of the cone ( 𝑚 𝑖 𝑢 𝑖 , 𝑛 𝑖 𝑣 𝑖 ) through the limit cone of 𝐷 .Thus, it is unique. , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 21 (2) Conversely, every cofiltered limit of quotients in D → f is surjective in Pro D f . Indeed, cofilteredlimits in Pro D f are formed in Stone ( Σ - Str ) by Corollary 2.12, and the forgetful functor into Stone thus preserves them. Hence the same is true about the forgetful functor from ( Pro D f ) → to Stone → .Thus, the claim follows from Lemma 2.17.(3) We show that every morphism of Pro D f which is monic and reflects relations is an element of c M .(3a) We first prove a property of filtered colimits in BA → . Let 𝐷 be a filtered diagram with objects ℎ 𝑖 : 𝐴 𝑖 → 𝐵 𝑖 ( 𝑖 ∈ 𝐼 ) in BA → . Let ℎ 𝑖 = 𝑚 𝑖 · 𝑒 𝑖 be the factorization of ℎ 𝑖 into an epimorphism 𝑒 𝑖 : 𝐴 𝑖 ։ 𝐵 𝑖 followed by a monomorphism 𝑚 𝑖 : 𝐵 𝑖 𝐵 𝑖 in BA . Using diagonal fill-in we get afiltered diagram 𝐷 with objects 𝑒 𝑖 ( 𝑖 ∈ 𝐼 ) and with connecting morphisms ( 𝑢, 𝑣 ) : 𝑒 𝑖 → 𝑒 𝑗 derivedfrom the connecting morphisms ( 𝑢, 𝑣 ) : ℎ 𝑖 → ℎ 𝑗 of 𝐷 using diagonal fill-in: 𝐴 𝑖 ℎ 𝑖 / / 𝑢 (cid:15) (cid:15) 𝑒 𝑖 (cid:31) (cid:31) (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ 𝐵 𝑖𝑣 (cid:15) (cid:15) 𝐵 𝑖𝑣 (cid:15) (cid:15) ? ? 𝑚 𝑖 ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ 𝐵 𝑗 (cid:31) (cid:31) 𝑚 𝑗 (cid:31) (cid:31) ❃❃❃❃❃❃❃❃ 𝐴 𝑗 𝑒 𝑗 ? ? ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ℎ 𝑗 / / 𝐵 𝑗 Our claim is that if the colimit ℎ = colim ℎ 𝑖 in BA → is an epimorphism of BA , then one has ℎ = colim 𝑒 𝑖 . To see this, suppose that a colimit cocone of 𝐷 is given as follows: 𝐴 𝑖 𝑒 𝑖 (cid:31) (cid:31) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ ℎ 𝑖 / / 𝑎 𝑖 (cid:15) (cid:15) 𝐵 𝑖𝑏 𝑖 (cid:15) (cid:15) 𝐵 𝑖 ? ? 𝑚 𝑖 ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) 𝐴 ℎ / / 𝐵 Then we prove that 𝐷 has the colimit cocone ( 𝑎 𝑖 , 𝑏 𝑖 · 𝑚 𝑖 ) , 𝑖 ∈ 𝐼 . Indeed, since 𝐴 = colim 𝐴 𝑖 withcolimit cocone ( 𝑎 𝑖 ) , all we need to verify is that 𝐵 = colim 𝐵 𝑖 with cocone ( 𝑏 𝑖 · 𝑚 𝑖 ) . This cocone iscollectively epic because every element 𝑥 of 𝐵 has the form 𝑥 = ℎ ( 𝑦 ) for some 𝑦 ∈ 𝐴 , using that ℎ is epic by hypothesis, and that the cocone ( 𝑎 𝑖 ) is collectively epic. The diagram 𝐷 is filtered, thus,to prove that 𝐵 = colim 𝐵 𝑖 , we only need to verify that whenever a pair 𝑥 , 𝑥 ∈ 𝐵 𝑖 (for some 𝑖 ∈ 𝐼 )is merged by 𝑏 𝑖 · 𝑚 𝑖 , there exists a connecting morphism 𝑣 : 𝐵 𝑖 → 𝐵 𝑗 merging 𝑥 , 𝑥 . Since 𝑚 𝑖 ismonic and 𝐵 = colim 𝐵 𝑖 , some connecting morphism 𝑣 : 𝐵 𝑖 → 𝐵 𝑗 merges 𝑚 𝑖 ( 𝑥 ) and 𝑚 𝑖 ( 𝑥 ) . Then 𝑚 𝑗 · 𝑣 ( 𝑥 ) = 𝑣 · 𝑚 𝑖 ( 𝑥 ) = 𝑣 · 𝑚 𝑖 ( 𝑥 ) = 𝑚 𝑗 · 𝑣 ( 𝑥 ) , whence 𝑣 ( 𝑥 ) = 𝑣 ( 𝑥 ) because 𝑚 𝑗 is monic. , Vol. 1, No. 1, Article . Publication date: January 2021. (3b) Denote by 𝑊 : Stone ( Σ - Str ) →
Set S the forgetful functor mapping a Stone-topological Σ -structure to its underlying sorted set. Moreover, letting Σ rel ⊆ Σ denote the set of all relationsymbols in Σ , we have the forgetful functors 𝑊 𝑟 : Stone ( Σ - Str ) →
Set ( 𝑟 ∈ Σ rel ) assigning to every object 𝐴 the corresponding subset 𝑟 𝐴 ⊆ 𝐴 𝑠 × · · · × 𝐴 𝑠 𝑛 . From the description oflimits in Σ - Str in Remark 2.6, it follows that the functors 𝑊 and 𝑊 𝑟 ( 𝑟 ∈ Σ rel ) collectively preserveand reflect limits. That is, given a diagram 𝐷 in Stone ( Σ - Str ) , a cocone of 𝐷 is a limit cone if andonly if its image under 𝑊 is a limit cone of 𝑊 · 𝐷 and its image under 𝑊 𝑟 is a limit cone of 𝑊 𝑟 · 𝐷 for all 𝑟 ∈ Σ rel .(3c) We are ready to prove that if ℎ : 𝐴 → 𝐵 in Pro D f is a relation-reflecting monomorphism,then ℎ ∈ c M . We have a cofiltered diagram 𝐷 in D → f with objects ℎ 𝑖 : 𝐴 𝑖 → 𝐵 𝑖 and a limit cone ( 𝑎 𝑖 , 𝑏 𝑖 ) : ℎ 𝑖 → ℎ ( 𝑖 ∈ 𝐼 ). Let ℎ 𝑖 = 𝑚 𝑖 · 𝑒 𝑖 be the image factorization in Σ - Str . 𝐴 ℎ / / 𝑎 𝑖 (cid:15) (cid:15) 𝐵 𝑏 𝑖 (cid:15) (cid:15) 𝐴 𝑖 𝑒 𝑖 / / / / @A BC ℎ 𝑖 (cid:15) (cid:15) 𝐴 𝑖 / / 𝑚 𝑖 / / 𝐵 𝑖 It is our goal to prove that ℎ = lim 𝑖 ∈ 𝐼 𝑚 𝑖 . More precisely: we have 𝑚 𝑖 in D → f and diagonal fill-inyields a cofiltered diagram 𝐷 of these objects in D → f . We will prove that ( 𝑒 𝑖 · 𝑎 𝑖 , 𝑏 𝑖 ) : ℎ → 𝑚 𝑖 ( 𝑖 ∈ 𝐼 )is a limit cone. By part (3b) above it suffices to show that the images of that cone under 𝑊 → and 𝑊 → 𝑟 ( 𝑟 ∈ Σ rel ) are limit cones.For 𝑊 → just dualize (3a): from the fact that 𝑊 ℎ = lim 𝑊 ℎ 𝑖 we derive 𝑊 ℎ = lim 𝑊 𝑚 𝑖 . Given 𝑟 : 𝑠 , . . . , 𝑠 𝑛 in Σ rel , we know that 𝑟 𝐴 consists of the 𝑛 -tuples ( 𝑥 , . . . , 𝑥 𝑛 ) with ( 𝑎 𝑖 ( 𝑥 ) , . . . , 𝑎 𝑖 ( 𝑥 𝑛 )) ∈ 𝑟 𝐴 𝑖 for every 𝑖 ∈ 𝐼 (see Remark 2.6). In particular, for ( 𝑥 , . . . , 𝑥 𝑛 ) ∈ 𝑟 𝐴 we have ( 𝑒 𝑖 · 𝑎 𝑖 ( 𝑥 ) , . . . , 𝑒 𝑖 · 𝑎 𝑖 ( 𝑥 𝑛 )) ∈ 𝑟 𝐴 𝑖 . Conversely, given ( 𝑥 , . . . , 𝑥 𝑛 ) with the latter property, then ( 𝑚 𝑖 · 𝑒 𝑖 · 𝑎 𝑖 ( 𝑥 ) , . . . , 𝑚 𝑖 · 𝑒 𝑖 · 𝑎 𝑖 ( 𝑥 𝑛 )) ∈ 𝑟 𝐵 𝑖 , i.e. ( 𝑏 𝑖 · ℎ ( 𝑥 ) , . . . , 𝑏 𝑖 · ℎ ( 𝑥 𝑛 )) ∈ 𝑟 𝐵 𝑖 for all 𝑖 ∈ 𝐼 . Since 𝐵 = lim 𝐵 𝑖 , this implies ( ℎ ( 𝑥 ) , . . . , ℎ ( 𝑥 𝑛 )) ∈ 𝑟 𝐵 , whence ( 𝑥 , . . . , 𝑥 𝑛 ) ∈ 𝑟 𝐴 because ℎ is relation-reflecting.(4) It remains to prove that every morphism 𝑚 ∈ c M is a relation-reflecting monomorphism. Let acofiltered limit cone be given as follows: 𝐴 𝑚 / / 𝑎 𝑖 (cid:15) (cid:15) 𝐵 𝑏 𝑖 (cid:15) (cid:15) 𝐴 𝑖 / / 𝑚 𝑖 / / 𝐵 𝑖 ( 𝑖 ∈ 𝐼 ) where each 𝑚 𝑖 lies in M f , i.e. is a relation-reflecting monomorphism in D f . Then 𝑚 is monic: given 𝑥 ≠ 𝑦 in 𝐴 , there exists 𝑖 ∈ 𝐼 with 𝑎 𝑖 ( 𝑥 ) ≠ 𝑎 𝑖 ( 𝑦 ) because the limit projections 𝑎 𝑖 are collectivelymonic. Since 𝑚 𝑖 is monic, this implies 𝑏 𝑖 · 𝑚 ( 𝑥 ) ≠ 𝑏 𝑖 · 𝑚 ( 𝑦 ) , whence 𝑚 ( 𝑥 ) ≠ 𝑚 ( 𝑦 ) .Moreover, for every relation symbol 𝑟 : 𝑠 , . . . , 𝑠 𝑛 in Σ and ( 𝑥 , . . . , 𝑥 𝑛 ) ∈ 𝐴 𝑠 × · · · × 𝐴 𝑠 𝑛 , we havethat ( 𝑥 , . . . , 𝑥 𝑛 ) ∈ 𝑟 𝐴 iff ( 𝑚 ( 𝑥 ) , . . . , 𝑚 ( 𝑥 𝑛 )) ∈ 𝑟 𝐵 . Indeed, the only if direction follows from the fact that the maps 𝑚 𝑖 · 𝑎 𝑖 preserve relations and themaps 𝑏 𝑖 collectively reflect them. For the if direction, suppose that ( 𝑚 ( 𝑥 ) , . . . 𝑚 ( 𝑥 𝑛 )) ∈ 𝑟 𝐵 . Since , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 23 for every 𝑖 ∈ 𝐼 the morphism 𝑏 𝑖 preserves relations and 𝑚 𝑖 reflects them, we get ( 𝑎 𝑖 ( 𝑥 ) , . . . , 𝑎 𝑖 ( 𝑥 𝑛 )) ∈ 𝑟 𝐴 𝑖 for every 𝑖 . Since the maps 𝑎 𝑖 collectively reflect relations, this implies ( 𝑥 , . . . , 𝑥 𝑛 ) ∈ 𝑟 𝐴 . (cid:3) Definition 3.19.
The factorization system (E , M) of D is called profinite if E is closed in D → under cofiltered limits of finite quotients; that is, for every cofiltered diagram 𝐷 in D → whoseobjects are elements of E f , the limit of 𝐷 in D → lies in E . Example 3.20.
For every full subcategory D ⊆ Σ - Str closed under limits and subobjects, the fac-torization system of surjective morphisms and relation-reflecting injective morphisms is profinite.This follows from Lemma 2.17 and the fact that limits in D are formed at the level of underlyingsets (see Remark 2.6). Proposition 3.21.
If the factorization system (E , M) of D is profinite, the following holds: (1) The forgetful functor 𝑉 : Pro D f → D is faithful and satisfies 𝑉 ( b E) ⊆ E . (2) For every E -projective object 𝑋 ∈ D , the object b 𝑋 ∈ b D is b E -projective. (3) Every object of D f is an b E -quotient of some b E -projective object in Pro D f . Proof. (1) 𝑉 ( b E) ⊆ E is clear: given 𝑒 ∈ b E expressed as a cofiltered limit of finite quotients 𝑒 𝑖 , 𝑖 ∈ 𝐼 , in ( Pro D f ) → , then since 𝑉 is cofinitary, we see that 𝑉 𝑒 is a cofiltered limit of
𝑉 𝑒 𝑖 = 𝑒 𝑖 in D → ,thus 𝑉 𝑒 ∈ E by the definition of a profinite factorization system.To prove that 𝑉 is faithful, recall that a right adjoint is faithful if and only if each component ofits counit is epic. Thus, it suffices to prove that 𝜀 𝐷 ∈ b E (and use that by Proposition 3.17 every b E -morphism is epic). The triangles defining 𝜀 𝐷 in Construction 3.14(4) can be restricted to those with 𝑎 ∈ b E . Indeed, in the slice category 𝐷 / D f all objects 𝑎 : 𝐷 → 𝐴 in b E form an initial subcategory.Now given such a triangle with 𝑎 ∈ b E we know that 𝑉 𝑎 ∈ E . Thus all those objects 𝐴 form acofiltered diagram with connecting morphisms in E . Moreover, c 𝑉 𝑎 ∈ b E by Remark 3.16(1). Thisimplies 𝜀 𝐷 ∈ b E by Remark 3.16(3) .(2) Let 𝑋 be an E -projective object. To show that b 𝑋 is b E -projective, suppose that a quotient 𝑒 : 𝐴 ։ 𝐵 in b E and a morphism 𝑓 : b 𝑋 → 𝐵 are given. Since c (−) is left adjoint to 𝑉 and 𝑉 ( b E) ⊆ E , themorphism 𝑓 has an adjoint transpose 𝑓 ∗ : 𝑋 → 𝑉 𝐵 that factorizes through
𝑉 𝐸 via 𝑔 ∗ for some 𝑔 : b 𝑋 → 𝐴 . Then 𝑒 · 𝑔 = 𝑓 , which proves that b 𝑋 is projective. 𝑋 𝑔 ∗ ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤ 𝑓 ∗ ❇❇❇❇❇❇❇❇ 𝑉 𝐴
𝑉 𝑒 / / / / 𝑉 𝐵 iff b 𝑋 𝑔 (cid:0) (cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) 𝑓 (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ 𝐴 𝑒 / / / / 𝐵 (3) Given 𝐴 ∈ D f , by Assumption 3.1 there exists an E -projective object 𝑋 ∈ D and a quotient 𝑒 : 𝑋 ։ 𝐴 . The limit projection b 𝑒 : b 𝑋 ։ 𝐴 lies in b E by Remark 3.16(1), and item (2) above showsthat b 𝑋 is b E -projective. (cid:3) We are ready to prove the following general form of the Reiterman Theorem: given the factor-ization system ( b E , c M) on the pro-completion of D f , we have the concept of an equation in Pro D f .We call it a profinite equation for D , and prove that pseudovarieties in D are precisely the classesin D f that can be presented by profinite equations. Definition 3.22. A profinite equation is an equation in Pro D f , i.e. a morphism 𝑒 : 𝑋 ։ 𝐸 in b E whose domain 𝑋 is b E -projective. It is satisfied by a finite object 𝐷 provided that 𝐷 is injective w.r.t. 𝑒 . , Vol. 1, No. 1, Article . Publication date: January 2021. Theorem 3.23 (Generalized Reiterman Theorem) . Given a profinite factorization system on D , aclass of finite objects is a pseudovariety iff it can be presented by profinite equations. Proof.
Every class
V ⊆ D f presented by profinite equations is a pseudovariety: this is provedprecisely as (1) in Proposition 3.8.Conversely, every pseudovariety can be presented by profinite equations. Indeed, following thesame proposition, it suffices to construct, for every pseudoequation 𝑒 𝑖 : 𝑋 ։ 𝐸 𝑖 ( 𝑖 ∈ 𝐼 ), a profiniteequation satisfied by the same finite objects.For every 𝑖 ∈ 𝐼 , we have the corresponding limit projection b 𝑒 𝑖 : b 𝑋 ։ 𝐸 𝑖 with 𝑒 𝑖 = 𝑉 b 𝑒 𝑖 · 𝜂 𝑋 . Let 𝑅 be the diagram in D f of objects 𝐸 𝑖 . The connecting morphism 𝑘 : 𝐸 𝑖 → 𝐸 𝑗 are given by thefactorization 𝑋 𝑒 𝑗 (cid:27) (cid:27) (cid:27) (cid:27) ✼✼✼✼✼✼✼ 𝑒 𝑖 (cid:3) (cid:3) (cid:3) (cid:3) ✟✟✟✟✟✟✟ 𝐸 𝑖 𝑘 / / / / 𝐸 𝑗 iff 𝑒 𝑗 ≤ 𝑒 𝑖 . Since the pseudoequation is closed under finite joins, 𝑅 is cofiltered. Form the limit of 𝑅 in Pro D f with the limit cone 𝑝 𝑖 : 𝐸 ։ 𝐸 𝑖 ( 𝑖 ∈ 𝐼 ) . The morphisms b 𝑒 𝑖 above form a cone of 𝑅 : given 𝑒 𝑗 = 𝑘 · 𝑒 𝑖 , then 𝑉 b 𝑒 𝑗 · 𝜂 𝑋 = 𝑉 ( š 𝑘 · 𝑒 𝑖 ) · 𝜂 𝑋 = 𝑘 · 𝑉 b 𝑒 𝑖 · 𝜂 𝑋 implies b 𝑒 𝑗 = 𝑘 · b 𝑒 𝑖 by the universal property of 𝜂 𝑋 . Thus we have a unique morphism 𝑒 : b 𝑋 ։ 𝐸 making the following triangles commutative: b 𝑋 𝑒 / / / / b 𝑒 𝑖 (cid:31) (cid:31) (cid:31) (cid:31) ❃❃❃❃❃❃❃❃ 𝐸 𝑝 𝑖 (cid:15) (cid:15) (cid:15) (cid:15) 𝐸 𝑖 ( 𝑖 ∈ 𝐼 ) The connecting morphisms of 𝑅 lie in E (since 𝑘 · 𝑒 𝑖 ∈ E implies 𝑘 ∈ E ). Thus each b 𝑒 𝑖 lies in b E since 𝑒 𝑖 ∈ E , see Remark 3.16(2). Therefore, 𝑒 ∈ b E by Remark 3.16(3). Since b 𝑋 is b E -projective byProposition 3.21, we have thus obtained a profinite equation 𝑒 : b 𝑋 ։ 𝐸 .We are going to prove that a finite object 𝐴 satisfies the pseudoequation ( 𝑒 𝑖 ) 𝑖 ∈ 𝐼 iff it satisfies theprofinite equation 𝑒 .(1) Let 𝐴 satisfy the pseudoequation ( 𝑒 𝑖 ) . For every morphism 𝑓 : b 𝑋 → 𝐴 we present a factoriza-tion through 𝑒 . The morphism 𝑉 𝑓 · 𝜂 𝑋 factorizes through some 𝑒 𝑗 , 𝑗 ∈ 𝐼 : 𝑋 𝜂 𝑋 / / 𝑒 𝑗 (cid:15) (cid:15) (cid:15) (cid:15) 𝑉 b 𝑋 𝑉 𝑓 (cid:15) (cid:15) 𝐸 𝑗 𝑔 / / 𝐴 Since 𝑒 𝑗 = 𝑉 b 𝑒 𝑗 · 𝜂 𝑋 , we get 𝑉 ( 𝑔 · b 𝑒 𝑗 ) · 𝜂 𝑋 = 𝑉 𝑓 · 𝜂 𝑋 . By the universal property of 𝜂 𝑋 this implies 𝑔 · b 𝑒 𝑗 = 𝑓 . , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 25 The desired factorization is 𝑔 · 𝑝 𝑗 : b 𝑋 𝑓 (cid:15) (cid:15) 𝑒 / / / / b 𝑒 𝑗 ❃❃ (cid:31) (cid:31) (cid:31) (cid:31) ❃❃ 𝐸 𝑝 𝑗 (cid:15) (cid:15) (cid:15) (cid:15) 𝐴 𝐸 𝑗𝑔 o o (2) Let 𝐴 satisfy the profinite equation 𝑒 . For every morphism ℎ : 𝑋 → 𝐴 we find a factorizationthrough some 𝑒 𝑗 . The morphism b ℎ : b 𝑋 → 𝐴 factorizes through 𝑒 : b ℎ = 𝑢 · 𝑒 with 𝑢 : 𝐸 → 𝐴. The codomain of 𝑢 is finite, thus, 𝑢 factorizes through one of the limit projection of 𝐸 , i.e. 𝑢 = 𝑣 · 𝑝 𝑗 with 𝑗 ∈ 𝐼 and 𝑣 : 𝐸 𝑗 → 𝐴. This gives the following commutative diagram: b 𝑋 b ℎ (cid:15) (cid:15) 𝑒 / / 𝐸 𝑝 𝑗 (cid:15) (cid:15) 𝑢 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) 𝐴 𝐸 𝑗𝑣 o o (3.2)That 𝑣 is the desired factorization of ℎ is now shown using the following diagram: 𝑋 @A ℎ / / 𝜂 𝑋 / / 𝑉 b 𝑋 𝑉 b ℎ ( ( PPPPPPPPPPPPPPPP
𝑉 𝑒 / / 𝑉 𝐸
𝑉 𝑝 𝑗 / / 𝐸 𝑗𝑣 (cid:15) (cid:15) ED (cid:15) (cid:15) (cid:15) (cid:15) GF 𝑒 𝑗 𝐴 Indeed, the upper part commutes since since 𝑒 𝑗 = 𝑉 b 𝑒 𝑗 · 𝜂 𝑋 = 𝑉 𝑝 𝑗 · 𝑉 𝑒 · 𝜂 𝑋 , the lower left-hand part commutes since ℎ = 𝑉 b ℎ · 𝜂 𝑋 , and for the remaining lower right-hand partapply 𝑉 to (3.2) and use that 𝑉 𝑣 = 𝑣 since 𝑣 lies in D f . (cid:3) In the present section we establish the main result of our paper: a generalization of Reiterman’stheorem from algebras over a signature to algebras for a given monad T in a category D (The-orem 4.20). To this end, we introduce and investigate the profinite monad b T associated to themonad T . It provides an abstract perspective on the formation of spaces of profinite words orprofinite terms and serves as key technical tool for our categorical approach to profinite algebras. Assumption 4.1.
Throughout this section, D is a category satisfying Assumption 3.1, and T = ( 𝑇 , 𝜇, 𝜂 ) is a monad on D preserving quotients, i.e. 𝑇 (E) ⊆ E .We denote by D T the category of T -algebras and T -homomorphisms, and by D T f the full subcat-egory of all finite algebras , i.e. T -algebras whose underlying object lies in D f . Remark 4.2.
The category D T satisfies Assumption 3.1. More precisely: , Vol. 1, No. 1, Article . Publication date: January 2021. (1) Since T preserves quotients, the factorization system of D lifts to D T : every homomorphismin D T factorizes as a homomorphism in E followed by one in M . When speaking about quotientalgebras and subalgebras of T -algebras, we refer to this lifted factorization system (E T , M T ) .(2) Since D is complete, so is D T with limits created by the forgetful functor into D .(3) The category D T f is closed under finite products and subalgebras, since D f is closed under finiteproducts and subobjects.(4) For every E -projective object 𝑋 , the free algebra ( 𝑇 𝑋, 𝜇 𝑋 ) is E T -projective. Indeed, given T -homomorphisms 𝑒 : ( 𝐴, 𝛼 ) ։ ( 𝐵, 𝛽 ) and ℎ : ( 𝑇 𝑋, 𝜇 𝑋 ) → ( 𝐵, 𝛽 ) with 𝑒 ∈ E , then ℎ · 𝜂 𝑋 : 𝑋 → 𝐵 factorizes through 𝑒 in D , i.e. ℎ · 𝜂 𝑋 = 𝑒 · 𝑘 for some 𝑘 . Then the T -homomorphism 𝑘 : ( 𝑇 𝑋, 𝜇 𝑋 ) →( 𝐴, 𝛼 ) extending 𝑘 fulfils 𝑒 · 𝑘 · 𝜂 𝑋 = ℎ · 𝜂 𝑋 , hence, 𝑒 · 𝑘 = ℎ by the universal property of 𝜂 𝑋 . 𝑋 𝜂 𝑋 / / 𝑘 (cid:15) (cid:15) ✤✤✤ ( 𝑇 𝑋, 𝜇 𝑋 ) 𝑘 y y t t t t t ℎ (cid:15) (cid:15) ( 𝐴, 𝛼 ) 𝑒 / / / / ( 𝐵, 𝛽 ) It follows that every finite algebra is a quotient of an E T -projective T -algebra. Notation 4.3.
The forgetful functor of D T f into Pro D f is denoted by 𝐾 : D T f → Pro D f For example, if D = Σ - Str , then 𝐾 assigns to every finite T -algebra its underlying Σ -structure,equipped with the discrete topology. Remark 4.4.
For any functor 𝐾 : A → C , the right Kan extension 𝑅 = Ran 𝐾 𝐾 : C → C can be naturally equipped with the structure of a monad. Its unit and multiplication are given by b 𝜂 = ( id 𝐾 ) † : Id → 𝑅 and b 𝜇 = ( 𝜀 · 𝑅𝜀 ) † : 𝑅𝑅 → 𝑅, where 𝜀 : 𝑅𝐾 → 𝐾 denotes the universal natural transformation and (−) † is defined as in Re-mark 3.12. The monad ( 𝑅, b 𝜂, b 𝜇 ) is called the codensity monad of 𝐾 , see e.g. Linton [13]. Definition 4.5.
The profinite monad b T = ( b 𝑇 , b 𝜇, b 𝜂 ) of the monad T is the codensity monad of the forgetful functor 𝐾 : D T f → Pro D f . Construction 4.6.
Since
Pro D f is complete and D T f is small, the limit formula for right Kanextensions (see Remark 3.12) yields the following concrete description of the profinite monad:(1) To define the action of b 𝑇 on an object 𝑋 , form the coslice category 𝑋 / 𝐾 of all morphisms 𝑎 : 𝑋 → 𝐾 ( 𝐴, 𝛼 ) with ( 𝐴, 𝛼 ) ∈ D T f . The projection functor 𝑄 𝑋 : 𝑋 / 𝐾 → Pro D f , mapping 𝑎 to 𝐴 ,has a limit b 𝑇 𝑋 = lim 𝑄 𝑋 . The limit cone is denoted as follows: 𝑋 𝑎 −→ 𝐾 ( 𝐴, 𝛼 ) b 𝑇 𝑋 𝛼 + 𝑎 −−→ 𝐴 For every finite T -algebra ( 𝐴, 𝛼 ) , we write 𝛼 + : b 𝑇 𝐴 → 𝐴 , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 27 instead of 𝛼 + id 𝐴 .(2) The action of b 𝑇 on morphisms 𝑓 : 𝑌 → 𝑋 is given by the following commutative triangles b 𝑇 𝑌 b 𝑇 𝑓 / / 𝛼 + 𝑎𝑓 (cid:27) (cid:27) ✽✽✽✽✽✽✽ b 𝑇 𝑋 𝛼 + 𝑎 (cid:2) (cid:2) ✝✝✝✝✝✝✝ 𝐴 for all 𝑎 : 𝑋 → 𝐾 ( 𝐴, 𝛼 ) .(3) The unit b 𝜂 : Id → b 𝑇 is given by the following commutative triangles 𝑋 b 𝜂 𝑋 / / 𝑎 (cid:26) (cid:26) ✻✻✻✻✻✻✻ b 𝑇 𝑋 𝛼 + 𝑎 (cid:2) (cid:2) ✝✝✝✝✝✝✝ 𝐴 for all 𝑎 : 𝑋 → 𝐾 ( 𝐴, 𝛼 ) .and the multiplication by the following commutative squares b 𝑇 b 𝑇 𝑋 b 𝜇 𝑋 / / b 𝑇𝛼 + 𝑎 (cid:15) (cid:15) b 𝑇 𝑋 𝛼 + 𝑎 (cid:15) (cid:15) b 𝑇 𝐴 𝛼 + / / 𝐴 for all 𝑎 : 𝑋 → 𝐾 ( 𝐴, 𝛼 ) . Remark 4.7.
A concept related to the profinite monad was studied by Bojańczyk [7] who associ-ates to every monad T on Set a monad T on Set (rather than on
Pro
Set f = Stone as in our setting).Specifically, T is the monad induced by the composite right adjoint Stone b T → Stone 𝑉 −→ Set . Itsconstruction also appears in the work of Kennison and Gildenhuys [11] who investigated codensitymonads for
Set -valued functors and their connection with profinite algebras.
Remark 4.8. (1) Every finite T -algebra ( 𝐴, 𝛼 ) yields a finite b T -algebra ( 𝐴, 𝛼 + ) . Indeed, the unitlaw and the associative law for 𝛼 + follow from Construction 4.6(3) with 𝑋 = 𝐴 and 𝑎 = id 𝐴 .(2) The monad b 𝑇 is cofinitary. To see this, let 𝑥 𝑖 : 𝑋 → 𝑋 𝑖 ( 𝑖 ∈ 𝐼 ) be a cofiltered limit cone in Pro D f .For each object of 𝑄 𝑋 given by an algebra ( 𝐴, 𝛼 ) and morphism 𝑎 : 𝑋 → 𝐴 , due to 𝐴 ∈ D f thereexists 𝑖 ∈ 𝐼 and a morphism 𝑏 : 𝑋 𝑖 → 𝐴 with 𝑎 = 𝑏 · 𝑥 𝑖 . From the definition of b 𝑇 on morphisms weget 𝛼 + 𝑎 = ( b 𝑇 𝑋 𝑖 b 𝑇𝑥 𝑖 −−−−→ b 𝑇 𝐴 𝛼 + 𝑏 −−−→ 𝐵 ) . To prove that b 𝑇 𝑥 𝑖 : b 𝑇 𝑋 → b 𝑇 𝑋 𝑖 ( 𝑖 ∈ 𝐼 ) forms a limit cone, suppose that any cone 𝑐 𝑖 : 𝐶 → b 𝑇 𝑋 𝑖 ( 𝑖 ∈ 𝐼 ) is given. It is easy to verify that then the cone of 𝑄 𝑋 (see Construction 4.6(1)) assigningto the above 𝑎 the morphism 𝛼 + 𝑏 · 𝑐 𝑖 is well-defined, i.e. independent of the choice of 𝑖 and 𝑏 andcompatible with 𝑄 𝑋 . The unique morphism 𝑐 : 𝐶 → b 𝑇 𝑋 factorizing that cone fulfils 𝑐 𝑖 = b 𝑇 𝑥 𝑖 · 𝑐 because this equation holds when postcomposed with the members of the limit cone of 𝑄 𝑋 𝑖 . Thisproves the claim.(3) The free b T -algebra ( b 𝑇 𝑋, b 𝜇 𝑋 ) on an object 𝑋 of Pro D f is a cofiltered limit of finite b T -algebras. Infact, for the squares in Construction 4.6(3) defining b 𝜇 𝑋 we have the limit cone ( 𝛼 + 𝑎 ) in Pro D f , andsince all 𝛼 + 𝑎 are homomorphisms of b T -algebras and the forgetful functor from ( Pro D f ) b T to Pro D f reflects limits, it follows that ( b 𝑇 𝑋, b 𝜇 𝑋 ) is a limit of the algebras ( 𝐴, 𝛼 + ) . , Vol. 1, No. 1, Article . Publication date: January 2021. (4) For “free” objects of Pro D f , i.e. those of the form ˆ 𝑋 for 𝑋 ∈ D (cf. Lemma 3.13), the definition of b 𝑇 b 𝑋 can be stated in a more convenient form: b 𝑇 b 𝑋 is the cofiltered limit of all finite quotient algebrasof the free T -algebra ( 𝑇 𝑋, 𝜇 𝑋 ) . More precisely, let ( 𝑇 𝑋, 𝜇 𝑋 ) և D T f denote the full subcategory of theslice category ( 𝑇 𝑋, 𝜇 𝑋 )/ D T f on all finite quotient algebras of ( 𝑇 𝑋, 𝜇 𝑋 ) , and consider the diagram 𝐷 𝑋 : ( 𝑇 𝑋, 𝜇 𝑋 ) և D T f → Pro D f that maps 𝑒 : ( 𝑇 𝑋, 𝜇 𝑋 ) ։ ( 𝐴, 𝛼 ) to 𝐴 . Then we have the following Lemma 4.9.
For every object 𝑋 of D , one has b 𝑇 b 𝑋 = lim 𝐷 𝑋 . Proof.
The diagram 𝐷 𝑋 is the composite ( 𝑇 𝑋, 𝜇 𝑋 ) և D T f ( 𝑇 𝑋, 𝜇 𝑋 )/ D T f (cid:27) b 𝑋 / 𝐾 𝑄 ˆ 𝑋 −−→ Pro D f , where the isomorphism ( 𝑇 𝑋, 𝜇 𝑋 )/ D T f (cid:27) b 𝑋 / 𝐾 maps 𝑒 : ( 𝑇 𝑋, 𝜇 𝑋 ) → ( 𝐴, 𝛼 ) to š 𝑒 · 𝜂 𝑋 : b 𝑋 → 𝐴 . Sinceevery T -homomorphism has an (E T , M T ) -factorization, ( 𝑇 𝑋, 𝜇 𝑋 ) և D T f is an initial subcategory of ( 𝑇 𝑋, 𝜇 𝑋 )/ D T f . Thus, b 𝑇 𝑋 = lim 𝑄 ˆ 𝑋 = lim 𝐷 . (cid:3) Notation 4.10.
The above proof gives, for every object 𝑋 ∈ D , the limit cone 𝛼 + š 𝑒 · 𝜂 𝑋 : b 𝑇 b 𝑋 ։ 𝐴 with 𝑒 : ( 𝑇 𝑋, 𝜇 𝑋 ) ։ ( 𝐴, 𝛼 ) ranging over ( 𝑇 𝑋, 𝜇 𝑋 ) և D T f . In the following, we abuse notation andsimply write 𝛼 + 𝑒 for 𝛼 + š 𝑒 · 𝜂 𝑋 . Example 4.11.
Given the monad
𝑇 𝑋 = 𝑋 ∗ of monoids on D = Set , the profinite monad is themonad of monoids in
Stone b 𝑇 𝑋 = free monoid in Stone on the space
𝑋 .
For a finite set 𝑋 , the elements of b 𝑇 𝑋 are called the profinite words over 𝑋 . A profinite word is acompatible choice of a congruence class of 𝑋 ∗ /∼ for every congruence ∼ of finite rank. Compat-ibility means that given another congruence ≈ containing ∼ , the class chosen for ≈ contains theabove class as a subset. Lemma 4.12.
The monad b 𝑇 preserves quotients, i.e. b 𝑇 ( b E) ⊆ b E . Proof.
Suppose that 𝑒 : 𝑋 → 𝑌 is a morphism im b E . This means that it can be expressed as acofiltered limit in b D → of morphisms 𝑒 𝑖 ∈ E f ( 𝑖 ∈ 𝐼 ): 𝑋 𝑒 / / / / 𝑝 𝑖 (cid:15) (cid:15) 𝑌 𝑞 𝑖 (cid:15) (cid:15) 𝑋 𝑖 𝑒 𝑖 / / / / 𝑌 𝑖 Since b 𝑇 is cofinitary by Remark 4.8(2), it follows that b 𝑇 𝑒 is the limit of b 𝑇 𝑒 𝑖 = 𝑇 𝑒 𝑖 ( 𝑖 ∈ 𝐼 ) in b D → .Since 𝑇 preserves E , we have 𝑇 𝑒 𝑖 ∈ E for all 𝑖 ∈ 𝐼 , which proves that b 𝑇 𝑒 ∈ b E . (cid:3) It follows that the factorization system ( b E , c M) of Pro D f lifts to the category ( Pro D f ) b T . Moreover,this category with the choice ( Pro D f ) b T f = all b T -algebras ( 𝐴, 𝛼 ) with 𝐴 ∈ D f satisfies all the requirements of Assumption 3.1; this is analogous to the corresponding observa-tions for D T in Remark 4.2. Note that we are ultimately interested in finite T -algebras, not finite b T -algebras. However, there is no clash: we shall prove in Proposition 4.16 that they coincide. , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 29 Notation 4.13.
Recall from Construction 4.6 the definition of b 𝑇 𝑋 as a cofiltered limit 𝛼 + 𝑎 : b 𝑇 𝑋 → 𝐴 of 𝑄 𝑋 : 𝑋 / 𝐾 → Pro D f . Since the functor 𝑉 : Pro D f → D (see Notation 3.10) preserves that limit,and since all morphisms 𝑇𝑉 𝑋
𝑇𝑉 𝑎 −−−→
𝑇 𝐴 𝛼 −→ 𝐴 form a cone of 𝑉 · 𝑄 𝑋 , there is a unique morphism 𝜑 𝑋 such the squares below commute for forevery finite T -algebra ( 𝐴, 𝛼 ) : 𝑇𝑉 𝑋 𝜑 𝑋 / / ❴❴❴❴ 𝑇𝑉𝑎 (cid:15) (cid:15) 𝑉 b 𝑇 𝑋 𝑉𝛼 + 𝑎 (cid:15) (cid:15) 𝑇 𝐴 𝛼 / / 𝐴 (4.1) Example 4.14.
For the monoid monad
𝑇 𝑋 = 𝑋 ∗ on Set , the map 𝜑 𝑋 : ( 𝑉 𝑋 ) ∗ → 𝑉 b 𝑇 𝑋 is the embedding of finite words into profinite words. More precisely, by representing elementsof b 𝑇 𝑋 as compatible choices of congruences classes (see Example 4.11), 𝜑 𝑋 maps 𝑤 ∈ 𝑋 ∗ to thecompatible family of all congruence classes [ 𝑤 ] ∼ of 𝑤 , where ∼ ranges over all congruences on 𝑋 ∗ of finite rank.We now prove that the morphisms 𝜑 𝑋 are the components of a monad morphism from T to b T in the sense of Street [23]. Lemma 4.15.
The morphisms 𝜑 𝑋 form a natural transformation 𝜑 : 𝑇𝑉 → 𝑉 b 𝑇 such that the following diagrams commute: 𝑉 𝜂𝑉 (cid:5) (cid:5) ✡✡✡✡✡✡✡ 𝑉 b 𝜂 (cid:25) (cid:25) ✹✹✹✹✹✹ 𝑇𝑉 𝜑 / / 𝑉 b 𝑇 𝑇𝑇𝑉 𝑇𝜑 / / 𝜇𝑉 (cid:15) (cid:15) 𝑇𝑉 b 𝑇 𝜑 b 𝑇 / / 𝑉 b 𝑇 b 𝑇 𝑉 b 𝜇 (cid:15) (cid:15) 𝑇𝑉 𝜑 / / 𝑉 b 𝑇 Proof. (1) We first prove that 𝜑 is natural. Given a morphism 𝑓 : 𝑋 → 𝑌 in Pro D f , consider anarbitrary object 𝑎 : 𝑌 → 𝐾 ( 𝐴, 𝛼 ) of 𝑄 𝑌 (see Construction 4.6(1)) and recall that by the definitionof b 𝑇 on the morphism 𝑓 we have 𝛼 + 𝑎 · b 𝑇 𝑓 = 𝛼 + 𝑎 · 𝑓 . Consider the following diagram:
𝑇𝑉 𝑋 𝜑 𝑋 / / 𝑇𝑉 𝑓 (cid:15) (cid:15) 𝑇𝑉 ( 𝑎 · 𝑓 ) " " ❊❊❊❊❊❊❊❊ 𝑉 b 𝑇 𝑋 𝑉𝛼 + 𝑎 · 𝑓 } } ④④④④④④④④④ 𝑉 b 𝑇 𝑓 (cid:15) (cid:15)
𝑇 𝐴 𝛼 / / 𝐴𝑇𝑉 𝑌
𝑇𝑉𝑎 < < ②②②②②②②② 𝜑 𝑌 / / 𝑉 b 𝑇 𝑌
𝑉 𝛼 + 𝑎 a a ❈❈❈❈❈❈❈ Since all inner parts commute by definition, and the morphisms
𝑉 𝛼 + 𝑎 form a collectively moniccone using that 𝑉 is cofinitary, we see that the outside commutes, i.e. 𝜑 is natural. , Vol. 1, No. 1, Article . Publication date: January 2021. (2) To prove 𝑉 b 𝜂 𝑋 = 𝜑 𝑋 · 𝜂 𝑉𝑋 , use the collectively monic cone 𝑉 𝛼 + 𝑎 : 𝑉 b 𝑇 𝑋 → 𝑉 𝐴 , where 𝑎 : 𝑋 → 𝐾 ( 𝐴, 𝛼 ) ranges over 𝑄 𝑋 . Using the triangle in Construction 4.6(3), we see that the following dia-gram 𝑉 𝑋 𝜂 𝑉𝑋 / / 𝑉 𝑎 (cid:15) (cid:15)
𝑇𝑉 𝑋
𝑇𝑉𝑎 (cid:15) (cid:15) 𝜑 𝑋 / / 𝑉 b 𝑇 𝑋 𝑉𝛼 + 𝑎 (cid:15) (cid:15) ED (cid:15) (cid:15) GF 𝑉 b 𝜂 𝑋 𝐴 𝜂 𝐴 / / 𝑇 𝐴 𝛼 / / 𝐴 has the desired upper part commutative, since it commutes when post-composed by every 𝑉 𝛼 + 𝑎 ,which follows from the fact that the two lower squares and the outside clearly commute.(3) To prove 𝑉 b 𝜇 𝑋 · 𝜑 b 𝑇𝑋 · 𝑇 𝜑 𝑋 = 𝜑 𝑋 · 𝜇 𝑉 𝑋 , we again use the collectively monic cone
𝑉 𝛼 + 𝑎 . The squarein Construction 4.6(3) makes it clear that in the following diagram 𝑇𝑇𝑉 𝑋
𝑇𝑇𝑉 𝑎 & & ▼▼▼▼▼▼▼▼▼▼ 𝑇𝜑 𝑋 (cid:15) (cid:15) 𝜇 𝑉𝑋 / / 𝑇𝑉 𝑋
𝑇𝑉𝑎 y y ssssssssss 𝜑 𝑋 (cid:15) (cid:15) 𝑇𝑇 𝐴 𝜇 𝐴 / / 𝑇𝛼 (cid:15) (cid:15) 𝑇 𝐴 𝛼 (cid:15) (cid:15) 𝑇𝑉 b 𝑇 𝑋
𝑇𝑉 𝛼 + 𝑎 / / 𝜑 b 𝑇𝑋 (cid:15) (cid:15) 𝑇 𝐴 𝛼 / / 𝜑 𝐴 (cid:15) (cid:15) 𝐴𝑉 b 𝑇 𝐴 𝑉𝛼 + / / 𝐴𝑉 b 𝑇 b 𝑇 𝑋 𝑉 b 𝑇𝛼 + 𝑎 ssssssssss 𝑉 b 𝜇 𝑋 / / 𝑉 b 𝑇 𝑋
𝑉 𝛼 + 𝑎 d d ■■■■■■■■■■ the outside commutes, since it does when post-composed by all 𝑉 𝛼 + 𝑎 . (cid:3) Proposition 4.16.
The categories of finite T -algebras and finite b T -algebras are isomorphic: the func-tor taking ( 𝐴, 𝛼 ) to ( 𝐴, 𝛼 + ) and being the identity map on morphisms is an isomorphism. Proof. (1) We first prove that, given finite T -algebras ( 𝐴, 𝛼 ) and ( 𝐵, 𝛽 ) , a morphism ℎ : 𝐴 → 𝐵 is a homomorphism for T iff ℎ : ( 𝐴, 𝛼 + ) → ( 𝐵, 𝛽 + ) is a homomorphism for b T . If the latter holds,then the naturality of 𝜑 yields a commutative diagram as follows 𝑇 𝐴 𝑇ℎ (cid:15) (cid:15) 𝜑 𝐴 / / 𝑉 b 𝑇 𝐴 𝑉 b 𝑇ℎ (cid:15) (cid:15) 𝑉𝛼 + / / 𝑉 𝐴 𝑉ℎ (cid:15) (cid:15) 𝐴 ℎ (cid:15) (cid:15) 𝑇 𝐵 𝜑 𝐵 / / 𝑉 b 𝑇 𝐵
𝑉 𝛽 + / / 𝑉 𝐵 𝐵
Thus ℎ is a homomorphism for T , since the horizontal morphisms are 𝛼 and 𝛽 , respectively. , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 31 Conversely, if ℎ is a homomorphism for T , then the diagram 𝑄 𝐴 of Construction 4.6(1) has thefollowing connecting morphism 𝐴 id 𝐴 (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦ ℎ ❅❅❅❅❅❅❅❅ 𝐾 ( 𝐴, 𝛼 ) 𝐾ℎ / / 𝐾 ( 𝐵, 𝛽 ) . This implies ℎ · 𝛼 + = 𝛽 + ℎ . The definition of b 𝑇ℎ yields 𝛽 + · b 𝑇ℎ = 𝛽 + ℎ (see Construction 4.6(1) again).Thus, ℎ is a homomorphism for b T : b 𝑇 𝐴 𝛽 + ℎ ❄❄❄ (cid:31) (cid:31) ❄❄ 𝛼 + / / b 𝑇ℎ (cid:15) (cid:15) 𝐴 ℎ (cid:15) (cid:15) b 𝑇 𝐵 𝛽 + / / 𝐵 Note that the only if part implies that the object assignment ( 𝐴, 𝛼 ) ↦→ (
𝐴, 𝛼 + ) is indeed functorial.(2) For every finite b T -algebra ( 𝐴, 𝛿 ) we prove that the composite 𝛼 = 𝑇 𝐴 𝜑 𝐴 −−→ 𝑉 b 𝑇 𝐴
𝑉 𝛿 −−→
𝑉 𝐴 = 𝐴 (4.2)defines a T -algebra with 𝛼 + = 𝛿 .The unit law follows from that of 𝛿 , 𝛿 · b 𝜂 𝐴 = id and from 𝜑 𝐴 · 𝜂 𝐴 = 𝑉 b 𝜂 𝐴 (see Lemma 4.15): 𝐴 𝜂 𝐴 (cid:15) (cid:15) 𝑉 b 𝜂 𝐴 " " ❉❉❉❉❉❉❉❉ ED id 𝐴 (cid:15) (cid:15) 𝑇 𝐴 𝜑 𝐴 / / 𝑉 b 𝑇 𝐴 𝑉𝛿 / / 𝐴 BC O O @A 𝛼 The associative law follows from that of 𝛿 , 𝛿 · b 𝜇 𝐴 = 𝛿 · b 𝑇 𝛿 and from 𝜑 𝐴 · 𝜇 𝐴 = 𝑉 b 𝜇 𝐴 · 𝜑 b 𝑇𝐴 · 𝑇 𝜑 𝐴 (seeLemma 4.15): 𝑇𝑇 𝐴 𝑇𝜑 𝐴 (cid:15) (cid:15) GF@A 𝑇𝛼 / / 𝜇 𝐴 / / 𝑇 𝐴 𝜑 𝐴 (cid:15) (cid:15) EDBC 𝛼 o o 𝑇𝑉 b 𝑇 𝐴
𝑇𝑉 𝛿 (cid:15) (cid:15) 𝜑 b 𝑇 𝐴 / / 𝑉 b 𝑇 b 𝑇 𝐴 𝑉 b 𝜇 𝐴 / / 𝑉 b 𝑇𝛿 (cid:15) (cid:15) 𝑉 b 𝑇 𝐴
𝑉 𝛿 (cid:15) (cid:15)
𝑇 𝐴 𝜑 𝐴 / / 𝑉 b 𝑇 𝐴
𝑉 𝛿 / / 𝐴 BC O O @A 𝛼 To prove that 𝛼 + = 𝛿, recall from Lemma 4.9 and Notation 4.10 that b 𝑇 𝐴 is a cofiltered limit of all finite quotients 𝑏 : ( 𝑇 𝐴, 𝜇 𝐴 ) ։ ( 𝐵, 𝛽 ) in D T with the limit cone 𝛽 + 𝑏 : b 𝑇 𝐴 ։ 𝐵 . Since 𝐴 is finite, both 𝛼 + and 𝛿 factorize through , Vol. 1, No. 1, Article . Publication date: January 2021. one of the limit projections 𝛽 + 𝑏 , i.e. we have commutative triangles as follows: 𝐴 b 𝑇 𝐴 𝛽 + 𝑏 (cid:15) (cid:15) 𝛿 / / 𝛼 + o o 𝐴𝐵 𝛿 A A ✂✂✂✂✂✂✂✂✂ 𝛼 ] ] ❁❁❁❁❁❁❁❁❁ (4.3)Recall from Notation 4.10 that 𝛽 + 𝑏 denotes 𝛽 + š 𝑏 · 𝜂 𝐴 , and by Lemma 3.13 we have š 𝑏 · 𝜂 𝐴 = 𝑏 · 𝜂 𝐴 : 𝐴 → 𝐵 since this morphism lies in D f . Combining this with the definition (4.1) of 𝜑 𝐴 we have a commut-ative square 𝑇𝑉 𝐴 𝜑 𝐴 / / 𝑇𝑉 ( 𝑏 · 𝜂 𝐴 ) (cid:15) (cid:15) 𝑉 b 𝑇 𝐴 𝛽 + 𝑏 (cid:15) (cid:15) 𝑇 𝐵 𝛽 / / 𝐵 (4.4)Now we compute 𝛿 · 𝛽 · 𝑇 ( 𝑏 · 𝜂 𝐴 ) = 𝛿 · 𝛽 · 𝑇𝑉 ( 𝑏 · 𝜂 𝐴 ) since 𝑏 · 𝜂 𝐴 lies in D f = 𝛿 · 𝑉 𝛽 + 𝑏 · 𝜑 𝐴 by (4.4) = 𝑉 𝛿 · 𝑉 𝛽 + 𝑏 · 𝜑 𝐴 since 𝛿 lies in D f = 𝑉 𝛿 · 𝜑 𝐴 by (4.3).Analogously, we obtain 𝑉 𝛼 + · 𝜑 𝐴 = 𝛼 · 𝛽 · 𝑇 ( 𝑏 · 𝜂 𝐴 ) . (4.5)From the definition (4.1) of 𝜑 𝐴 , we also get 𝑉 𝛼 + · 𝜑 𝐴 = 𝑉 𝛼 + 𝑖𝑑 · 𝜑 𝐴 = 𝛼 · 𝑇𝑉 id 𝐴 = 𝛼 = 𝑉 𝛿 · 𝜑 𝐴 , (4.6)where we use (4.2) in the last step. Therefore, we can compute 𝛿 · 𝑏 = 𝛿 · 𝑏 · 𝜇 𝐴 · 𝑇 𝜂 𝐴 since 𝜇 𝐴 · 𝑇 𝜂 𝐴 = id = 𝛿 · 𝛽 · 𝑇𝑏 · 𝑇 𝜂 𝐴 since 𝑏 is a T -homomorphism = 𝑉 𝛿 · 𝜑 𝐴 shown previously = 𝑉 𝛼 · 𝜑 𝐴 by (4.6) = 𝛼 · 𝛽 · 𝑇𝑏 · 𝑇 𝜂 𝐴 by (4.5) = 𝛼 · 𝑏 · 𝜇 𝐴 · 𝑇 𝜂 𝐴 since 𝑏 is a T -homomorphism = 𝛼 · 𝑏. since 𝜇 𝐴 · 𝑇 𝜂 𝐴 = id .Since 𝑏 is epic, this implies 𝛼 = 𝛿 , whence 𝛼 + = 𝛿 .(3) Uniqueness of 𝛼 . Let ( 𝐴, 𝛼 ) be a finite T -algebra with 𝛼 + = 𝛿 . By the definition of 𝜑 𝐴 thisimplies 𝛼 = 𝑉 𝛼 + · 𝜑 𝐴 = 𝑉 𝛿 · 𝜑 𝐴 , so 𝛼 is unique. (cid:3) From now on, we identify finite algebras for T and for b T . Proposition 4.17.
The pro-completion of the category D T f of finite T -algebras is the full subcategoryof the category of b T -algebras given by all cofiltered limits of finite T -algebras. , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 33 Proof.
Let L denote the full subcategory of ( Pro D f ) b T given by all cofiltered limits of finite T -algebras. To show that L forms the pro-completion of D T f , we verify the three conditions ofCorollary A.5. By definition L satisfies condition (2), and condition (1) follows from the fact thatsince Pro D f has cofiltered limits, so does ( Pro D f ) b T . Thus, it only remains to prove condition (3):every algebra ( 𝐴, 𝛼 + ) with ( 𝐴, 𝛼 ) ∈ D T f is finitely copresentable in L . Let 𝑏 𝑖 : ( 𝐵, 𝛽 ) → ( 𝐵 𝑖 , 𝛽 𝑖 ) , ( 𝑖 ∈ 𝐼 ) , be a limit cone of a cofiltered diagram 𝐷 in L . Our task is to prove for every morphism 𝑓 : ( 𝐵, 𝛽 ) → (
𝐴, 𝛼 + ) that(a) a factorization through a limit projection exists, i.e. 𝑓 = 𝑓 ′ · 𝑏 𝑖 for some 𝑖 ∈ 𝐼 and 𝑓 ′ : ( 𝐵 𝑖 , 𝛽 𝑖 ) →( 𝐴, 𝛼 + ) , and(b) given another factorization 𝑓 = 𝑓 ′′ · 𝑏 𝑖 in L , then 𝑓 ′ and 𝑓 ′′ are merged by a connectingmorphism 𝑏 𝑗𝑖 : ( 𝐵 𝑗 , 𝛽 𝑗 ) → ( 𝐵 𝑖 , 𝛽 𝑖 ) of 𝐷 (for some 𝑗 ∈ 𝐼 ).Ad (a), since 𝑏 𝑖 : 𝐵 → 𝐵 𝑖 is a limit of a cofiltered diagram in Pro D f and 𝐴 is as an object of D f finitely copresentable in Pro D f , we have 𝑖 ∈ 𝐼 and a factorization 𝑓 = 𝑓 ′ · 𝑏 𝑖 , for some 𝑓 ′ : 𝐵 𝑖 → 𝐴 in Pro D f . If 𝑓 ′ is a T -homomorphism, i.e. if the following diagram b 𝑇 𝐵 𝛽 / / b 𝑇𝑏 𝑖 (cid:15) (cid:15) GF@A b 𝑇 𝑓 o o 𝐵 𝑏 𝑖 (cid:15) (cid:15) EDBC 𝑓 / / b 𝑇 𝐵 𝑖 𝛽 𝑖 / / b 𝑇 𝑓 ′ (cid:15) (cid:15) 𝐵 𝑖𝑓 ′ (cid:15) (cid:15) b 𝑇 𝐴 𝛼 + / / 𝐴 (4.7)commutes, we are done. In general, we have to change the choice of 𝑖 : from Construction 4.6(2)recall that b 𝑇 is cofinitary, thus ( b 𝑇𝑏 𝑖 ) 𝑖 ∈ 𝐼 is a limit cone. The parallel pair 𝑓 ′ · 𝛽 𝑖 , 𝛼 + · b 𝑇 𝑓 ′ : b 𝑇 𝐵 𝑖 → 𝐴 has a finitely copresentable codomain (in Pro D f ) and is merged by b 𝑇𝑏 𝑖 . Indeed, the outside of theabove diagram (4.7) commutes since 𝑓 = 𝑓 ′ · 𝑏 𝑖 is a homomorphism. Consequently, that parallelpair is also merged by b 𝑇𝑏 𝑗𝑖 for some connecting morphism 𝑏 𝑗𝑖 : ( 𝐵 𝑗 , 𝛽 𝑗 ) → ( 𝐵 𝑖 , 𝛽 𝑖 ) of the diagram 𝐷 : ( 𝛼 + · b 𝑇 𝑓 ′ ) · b 𝑇 𝑏 𝑗𝑖 = ( 𝑓 ′ · 𝛽 𝑖 ) · b 𝑇𝑏 𝑗𝑖 . From 𝑏 𝑖 = 𝑏 𝑗𝑖 · 𝑏 𝑗 we get another factorization of 𝑓 : 𝑓 = ( 𝑓 ′ · 𝑏 𝑗𝑖 ) · 𝑏 𝑗 , Vol. 1, No. 1, Article . Publication date: January 2021. and this tells us that the factorization morphism 𝑓 = 𝑓 ′ · 𝑏 𝑗𝑖 is a homomorphism as desired: b 𝑇 𝐵 𝛽 / / b 𝑇𝑏 𝑗 (cid:15) (cid:15) 𝐵 𝑏 𝑗 (cid:15) (cid:15) b 𝑇 𝐵 𝑗 𝛽 𝑗 / / GF@A b 𝑇 𝑓 / / b 𝑇𝑏 𝑗𝑖 (cid:15) (cid:15) 𝐵 𝑗𝑏 𝑗𝑖 (cid:15) (cid:15) EDBC 𝑓 o o b 𝑇 𝐵 𝑖 𝛽 𝑖 / / b 𝑇 𝑓 ′ (cid:15) (cid:15) 𝐵 𝑖𝑓 ′ (cid:15) (cid:15) b 𝑇 𝐴 𝛼 + / / 𝐴 Ad (b), suppose that 𝑓 ′ , 𝑓 ′′ : ( 𝐵 𝑖 , 𝛽 𝑖 ) → ( 𝐴, 𝛼 ) are homomorphisms satisfying 𝑓 = 𝑓 ′ · 𝑏 𝑖 = 𝑓 ′′ · 𝑏 𝑖 .Since 𝐵 = lim 𝐵 𝑖 is a cofiltered limit in Pro D f and the limit projection 𝑏 𝑖 merges 𝑓 ′ , 𝑓 ′′ : 𝐵 𝑖 → 𝐴 , itfollows that some connecting morphism 𝑏 𝑗𝑖 : ( 𝐵 𝑗 , 𝛽 𝑗 ) → ( 𝐵 𝑖 , 𝛽 𝑖 ) also merges 𝑓 ′ , 𝑓 ′′ , as desired. (cid:3) Remark 4.18. If (E , M) is a profinite factorization system on D , then (E T , M T ) is a profinitefactorization system on D T . Indeed, since E is closed in D → under cofiltered limits of finite quo-tients, and since the forgetful functor from ( D T ) → to D → creates limits, it follows that E T is alsoclosed under cofiltered limits of finite quotients. Definition 4.19. A b T -equation is an equation in the category of b T -algebras, i.e. a b T -homomorphism 𝑒 in b E b T with b E b T -projective domain. A finite T -algebra satisfies 𝑒 if it is injective with respect to 𝑒 in ( Pro D f ) b T . Theorem 4.20 (Generalized Reiterman Theorem for Monads) . Let D be a category with a profinitefactorization system (E , M) , and suppose that T is a monad preserving quotients. Then a class offinite T -algebras is a pseudovariety in D T f iff it can be presented by b T -equations. Remark 4.21.
We will see in the proof that the b T -equations presenting a given pseudovariety canbe chosen to be of the form 𝑒 : ( b 𝑇 b 𝑋, b 𝜇 b 𝑋 ) ։ ( 𝐴, 𝛼 ) where 𝑒 ∈ b E , the object 𝑋 is E -projective in D ,and 𝐴 is finite. Moreover, we can assume 𝑋 ∈ Var for any class
Var of objects as in Remark 3.9.
Proof of Theorem 4.20.
Every class of finite T -algebras presented by b T -equations is a pseudo-variety – this is analogous to Proposition 3.8.Conversely, let V be a pseudovariety in D T f . For every finite T -algebra ( 𝐴, 𝛼 ) we have an E -projective object 𝑋 in D and a quotient 𝑒 : 𝑋 ։ 𝐴 (see Assumption 3.1). Since b 𝑒 ∈ b E by Re-mark 3.16(1), we have b 𝑇 b 𝑒 ∈ b E by Lemma 4.12. Therefore the homomorphism 𝑒 : ( b 𝑇 b 𝑋, b 𝜇 𝑋 ) →( 𝐴, 𝛼 + ) extending b 𝑒 lies in b E : we have 𝑒 = 𝛼 + · b 𝑇 b 𝑒 , and 𝛼 + is a split epimorphism by the unit law 𝛼 + · b 𝜂 𝐴 = id 𝐴 . Since ( b E , c M) is a proper factorization system and Pro D f has finite coproducts,every split epimorphism lies in b E [2, Thm. 14.11], whence 𝛼 + ∈ b E . Thus, we see that every finite T -algebra is a quotient, in the category of b T -algebras, of ( b 𝑇 b 𝑋, b 𝜇 b 𝑋 ) for an E -projective object 𝑋 of D . Each such quotient lies in Pro D T f . Indeed, the codomain, being a finite T -algebra, does. To seethat the domain also does, combine Remark 4.8(3) and Proposition 4.16.In Remark 3.9 we can thus denote by Var the collection of all free algebras ( b 𝑇 b 𝑋, b 𝜇 b 𝑋 ) where 𝑋 ranges over E -projective objects of D . Then Theorem 3.23 and Remark 3.9 yield our claim that , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 35 every pseudovariety in D T f can be presented by b T -equations which are finite quotients of freealgebras ( b 𝑇 b 𝑋, b 𝜇 b 𝑋 ) where 𝑋 is E -projective in D . (cid:3) In our presentation so far, we have worked with an abstract categorical notion of equations givenby quotients of projective objects. In Reiterman’s original paper [19] on pseudovarieties of Σ -algebras, a different concept is used: equations between implicit operations , or equivalently, equa-tions between profinite terms . This raises a natural question: which categories D allow the simpli-fication of equations in the sense of Definition 4.19 to equations between profinite terms? It turnsout to be sufficient that D is cocomplete and has a finite dense set S of objects that are projectivew.r.t. strong epimorphisms. Recall that density of S means that every object 𝐷 of D is a canonicalcolimit of all morphisms from objects of S to 𝐷 . More precisely, if we view S as a full subcategoryof D , then 𝐷 is the colimit of the diagram S/ 𝐷 → D given by (cid:18) 𝑠 𝑓 −→ 𝐷 (cid:19) ↦→ 𝑠 with colimit cocone given by the morphisms 𝑓 . Assumption 5.1.
Throughout this section D is a cocomplete category with a finite dense set S of objects projective w.r.t. strong epimorphisms. It follows (see Proposition 5.4 below) that D has ( StrongEpi , Mono ) -factorizations, and we work with this factorization system. We denote by D f the collection of all objects 𝐷 such that D ( 𝑠, 𝐷 ) is finite for every object 𝑠 ∈ S . (5.1)We will show in Proposition 5.4 below that every category D satisfying the above assumptionscan be presented as a category of algebras over an S -sorted signature. Throughout this section,let Σ be an S -sorted algebraic signature, i.e. a signature without relation symbols. We denote by Alg Σ the category of Σ -algebras and homomorphisms. Example 5.2. (1) The category
Set S satisfies Assumption 5.1. A finite dense set in Set S is givenby the objects 𝑠 ( 𝑠 ∈ S) where 𝑠 is the S -sorted set that is empty in all sorts except 𝑠 , and has a single element ∗ in sort 𝑠 .Indeed, let 𝐴 and 𝐵 be S -sorted sets and let a cocone of the canonical diagram for 𝐴 be given: 𝑠 𝑓 −→ 𝐴 𝑠 𝑓 ∗ −−→ 𝐵 By this we mean that we have morphisms 𝑓 ∗ : 𝑠 → 𝐵 for every 𝑓 : 𝑠 → 𝐴 (and observe that thecocone condition is void in this case because there are no connecting morphisms → 𝑡 for 𝑠 ≠ 𝑡 ).Then we are to prove that there exists a unique S -sorted function ℎ : 𝐴 → 𝐵 with 𝑓 ∗ = ℎ · 𝑓 forall 𝑓 . Uniqueness is clear: given 𝑥 ∈ 𝐴 of sort 𝑠 , let 𝑓 𝑥 : 𝑠 → 𝐴 be the map with 𝑓 𝑥 (∗) = 𝑥 . Then ℎ · 𝑓 𝑥 = 𝑓 ∗ 𝑥 implies ℎ ( 𝑥 ) = 𝑓 ∗ 𝑥 (∗) . Conversely, if ℎ is defined by the above equation, then for every 𝑠 ∈ S and 𝑓 : 𝑠 → 𝐴 we have 𝑓 ∗ = ℎ · 𝑓 because 𝑓 = 𝑓 𝑥 for 𝑥 = 𝑓 (∗) .More generally, every set of objects K 𝑠 ( 𝑠 ∈ S ), where K 𝑠 is nonempty in sort 𝑠 and empty in allother sorts, is dense in Set S . , Vol. 1, No. 1, Article . Publication date: January 2021. (2) The category Alg Σ satisfies Assumption 5.1. Recall that strong epimorphisms are preciselythe homomorphisms with surjective components, and monomorphisms are the homomorphismswith injective components. It follows easily that for the free-algebra functor 𝐹 Σ : Set S → Alg Σ all algebras 𝐹 Σ 𝑋 are projective w.r.t. strong epimorphisms. We present a finite dense set of freealgebras.Assume first that Σ is a unary signature, i.e. all operation symbols in Σ are of the form 𝜎 : 𝑠 → 𝑡 .Then the free algebras 𝐹 Σ 𝑠 ( 𝑠 ∈ S) form a dense set in Alg Σ . Indeed, let 𝑈 Σ : Alg Σ → Set S denote the forgetful functor and 𝜂 : Id → 𝑈 Σ 𝐹 Σ the unit of the adjunction 𝐹 Σ ⊣ 𝑈 Σ . Given Σ -algebras 𝐴 and 𝐵 and a cocone of the canonicaldiagram as follows: 𝐹 Σ 𝑠 𝑓 −→ 𝐴𝐹 Σ 𝑠 𝑓 ∗ −−→ 𝐵 We are to prove that there exists a unique homomorphism ℎ : 𝐴 → 𝐵 with 𝑓 ∗ = ℎ · 𝑓 for every 𝑓 .We obtain a corresponding cocone in Set S as follows: 𝑠 𝜂 −→ 𝑈 Σ 𝐹 Σ 𝑠 𝑈 Σ 𝑓 −−−→ 𝑈 Σ 𝐴 𝑠 𝜂 −→ 𝑈 Σ 𝐹 Σ 𝑠 𝑈 Σ 𝑓 ∗ −−−−→ 𝑈 Σ 𝐵 Due to (1) there exists a unique function 𝑘 : 𝑈 Σ 𝐴 → 𝑈 Σ 𝐵 with 𝑈 Σ 𝑓 ∗ · 𝜂 = ( 𝑘 · 𝑈 Σ 𝑓 ) · 𝜂 for all 𝑓 . (5.2)Here and in the following we drop the subscripts indicating components of 𝜂 . It remains to provethat 𝑘 is a homomorphism from 𝐴 to 𝐵 ; then the universal property of 𝜂 implies 𝑓 ∗ = 𝑘 · 𝑓 . Thus,given 𝜎 : 𝑠 → 𝑡 in Σ and 𝑎 ∈ 𝐴 𝑠 we need to prove 𝑘 ( 𝜎 𝐴 ( 𝑎 )) = 𝜎 𝐵 ( 𝑘 ( 𝑎 )) . Consider the uniquehomomorphisms 𝑓 : 𝐹 Σ 𝑡 → 𝐴, 𝑓 (∗) = 𝜎 𝐴 ( 𝑎 ) ,𝑔 : 𝐹 Σ 𝑠 → 𝐴, 𝑔 (∗) = 𝑎,𝑗 : 𝐹 Σ 𝑡 → 𝐹 Σ 𝑠 , 𝑗 (∗) = 𝜎 (∗) . Then 𝑓 = 𝑔 · 𝑗 and thus 𝑓 ∗ = 𝑔 ∗ · 𝑗 because the morphisms (−) ∗ form a cocone of the canonicaldiagram of 𝐴 . It follows that 𝑘 ( 𝜎 𝐴 ( 𝑎 )) = 𝑘 ( 𝑓 (∗)) = 𝑓 ∗ (∗) = 𝑔 ∗ ( 𝑗 (∗)) = 𝑔 ∗ ( 𝜎 (∗)) = 𝜎 𝐵 ( 𝑔 ∗ (∗)) = 𝜎 𝐵 ( 𝑘 ( 𝑔 (∗))) = 𝜎 𝐵 ( 𝑘 ( 𝑎 )) , where the last but one equation holds by (5.2). Thus, 𝑘 is a homomorphism as desired.For a general signature Σ , let 𝑘 ∈ N ∪ { 𝜔 } be an upper bound of the arities of operation symbolsin Σ and let for every set 𝑇 ⊆ S the following S -sorted set 𝑋 𝑇 be given: 𝑋 𝑇 is empty for every sortoutside of 𝑇 , and for sorts 𝑠 ∈ 𝑇 the elements are ( 𝑋 𝑇 ) 𝑠 = { 𝑖 | 𝑖 < 𝑘 } . Then the set 𝐹 Σ 𝑋 𝑇 ( 𝑇 ⊆ S) is dense in Alg Σ . The proof is analogous to the unary case.(3) The category of graphs, i.e. sets with a binary relation, and graph homomorphisms satisfies As-sumption 5.1. Strong epimorphisms are precisely the surjective homomorphisms which are alsosurjective on all edges. Thus the two graphs shown below are clearly projective w.r.t. strong epi-morphisms. Moreover, they form a dense set: every graph is a canonical colimit of all of its verticesand all of its edges. , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 37 (4) Every variety, and even every quasivariety of Σ -algebras (presented by implications) satisfiesAssumption 5.1. This will follow from Proposition 5.4 below. Definition 5.3.
A full subcategory D of Alg Σ is said to be closed under ( StrongEpi , Mono ) -factorizations if for every morphism 𝑓 : 𝐴 → 𝐵 of D with factorization 𝑓 = 𝐴 𝑒 / / / / 𝐶 / / 𝑚 / / 𝐵 ,the object 𝐶 lies in D . Proposition 5.4.
For every category D the following two statements are equivalent: (1) D is cocomplete and has a finite dense set of objects which are projective w.r.t. strong epimorphisms. (2) There exists a signature Σ such that D is equivalent to a full reflective subcategory of Alg Σ closedunder ( StrongEpi , Mono ) -factorizations.Moreover, Σ can always be chosen to be a unary signature. Proof. (2) ⇒ (1) Suppose that D ⊆ Alg Σ is a full reflective subcategory closed under ( StrongEpi , Mono ) -factorizations. Denote by (−) @ : Alg Σ → D the reflector (i.e. the left adjoint to the inclusionfunctor D ↩ → Alg Σ ) and by 𝜂 𝑋 : 𝑋 → 𝑋 @ the universal maps. From Example 5.2 we know Alg Σ has a finite dense set of projective objects 𝐴 𝑖 , 𝑖 ∈ 𝐼 . We prove that the objects 𝐴 @ 𝑖 , 𝑖 ∈ 𝐼 , form adense set in D .To verify the density, let A be the full subcategory of Alg Σ on { 𝐴 𝑖 } 𝑖 ∈ 𝐼 . For every algebra 𝐷 ∈ D the canonical diagram A / 𝐷 → Alg Σ assigning 𝐴 𝑖 to each 𝑓 : 𝐴 𝑖 → 𝐷 has the canonical colimit 𝐷 . Since the left adjoint (−) @ preserves that colimit, we have that 𝐷 = 𝐷 @ is a canonical colimitof all 𝑓 @ : 𝐴 @ 𝑖 → 𝐷 for 𝑓 ranging over A / 𝐷 , as required. (Indeed, observe that every morphism 𝑓 : 𝐴 @ 𝑖 → 𝐷 in D has the form 𝑓 = 𝑓 @ because the subcategory D is full and contains the domainand codomain of 𝑓 .)Next, we observe that every strong epimorphism 𝑒 of D is strongly epic in Alg Σ . Indeed, takethe ( StrongEpi , Mono ) -factorization 𝑒 = 𝑚 · 𝑒 ′ of 𝑒 in Alg Σ . Since D is closed under factorizations,we have that 𝑒 ′ , 𝑚 ∈ D . Moreover, the morphism 𝑚 is monic in D because it is monic in Alg Σ .Since 𝑒 is a strong (and thus extremal) epimorphism in D , it follows that 𝑚 is an isomorphism.Thus 𝑒 (cid:27) 𝑒 ′ is a strong epimorphism in Alg Σ .Since each 𝐴 𝑖 is projective w.r.t. strong epimorphisms in Alg Σ , it thus follows that 𝐴 @ 𝑖 is project-ive w.r.t. strong epimorphisms 𝑒 : 𝐵 ։ 𝐶 in D . Indeed, given a morphism ℎ : 𝐴 @ 𝑖 → 𝐶 , compose itwith the universal arrow 𝜂 : 𝐴 𝑖 → 𝐴 @ 𝑖 . Thus, ℎ · 𝜂 factorizes in Alg Σ through 𝑒 : 𝐴 𝑖 𝜂 / / 𝑘 (cid:15) (cid:15) ✤✤✤ 𝐴 @ 𝑖ℎ (cid:15) (cid:15) 𝑘 ⑥⑥ ~ ~ ⑥⑥⑥ 𝐵 𝑒 / / / / 𝐶 The unique morphism 𝑘 : 𝐴 @ 𝑖 → 𝐵 of D with 𝑘 = 𝑘 · 𝜂 then fulfils the desired equality ℎ = 𝑒 · 𝑘 since ℎ · 𝜂 = 𝑒 · 𝑘 · 𝜂 .(1) ⇒ (2) Let S be a finite dense set of objects projective w.r.t. strong epimorphisms, and consider S as a full subcategory of D . Define an S -sorted signature of unary symbols Σ = Mor (S op ) \ { id 𝑠 | 𝑠 ∈ S } . Every morphism 𝜎 : 𝑠 → 𝑡 of S op has arity as indicated: the corresponding unary operation hasinputs of sort 𝑠 and yields values of sort 𝑡 . Define a functor 𝐸 : D → Alg Σ , Vol. 1, No. 1, Article . Publication date: January 2021. by assigning to every object 𝐷 the S -sorted set with sorts ( 𝐸𝐷 ) 𝑠 = D ( 𝑠, 𝐷 ) for 𝑠 ∈ S endowed with the operations 𝜎 𝐸𝐷 : D ( 𝑠, 𝐷 ) → D ( 𝑠 ′ , 𝐷 ) given by precomposing with 𝜎 : 𝑠 ′ → 𝑠 in S ⊆ D . To every morphism 𝑓 : 𝐷 → 𝐷 of D assignthe Σ -homomorphism 𝐸 𝑓 with sorts ( 𝐸 𝑓 ) 𝑠 : D ( 𝑠, 𝐷 ) → D ( 𝑠, 𝐷 ) given by postcomposing with 𝑓 . To say that S is a dense set is equivalent to saying that 𝐸 is fulland faithful [3, Prop. 1.26]. Moreover, since D is cocomplete, 𝐸 is a right adjoint [3, Prop. 1.27].Thus, D is equivalent to a full reflective subcategory of Alg Σ .Next we show that D has the factorization system ( StrongEpi , Mono ) . Indeed, being reflective in Alg Σ , it is a complete category. Moreover, D is well-powered because the right adjoint D ↩ → Alg Σ preserves monomorphisms and Alg Σ is well-powered. Consequently, the factorization system ex-ists [2, Cor. 14.21].To prove closure under factorizations, observe first that a morphism 𝑒 : 𝐷 → 𝐷 is stronglyepic in D iff 𝐸𝑒 is strongly epic in Alg Σ . Indeed, if 𝑒 is strongly epic, then 𝐸𝑒 has surjective sorts ( 𝐸𝑒 ) 𝑠 because 𝑠 is projective w.r.t. 𝑒 . Thus, 𝐸𝑒 is a strong epimorphism in Alg Σ . Conversely, if 𝐸𝑒 is strongly epic in Alg Σ , then for every commutative square 𝑔 · 𝑒 = 𝑚 · 𝑓 in D with 𝑚 monic, themorphism 𝐸𝑚 is monic in Alg Σ because 𝐸 is a right adjoint, and thus a diagonal exists.Now let 𝑓 : 𝐴 → 𝐵 be a morphism in D and let 𝑓 = 𝐴 𝑒 / / / / 𝐶 / / 𝑚 / / 𝐵 be its ( StrongEpi , Mono ) -factorization in D . Thus 𝐶 ∈ D and since by the above argument 𝐸𝑒 and 𝐸𝑚 are strong epimorph-isms and monomorphisms in Alg Σ , respectively, 𝐶 is the image of 𝑓 w.r.t. to the factorizationsystem of Alg Σ . (cid:3) Example 5.5. (1) If D = Set , we can take 𝑆 = { } where is a singleton set. The one-sortedsignature Σ in the above proof is empty, thus, Alg Σ = Set .(2) In the category
Gra of graphs we can take 𝑆 = { 𝐺 , 𝐺 } , see Example 5.2(3). Here Σ is a 2-sortedsignature with two operations 𝑠, 𝑡 : 𝐺 → 𝐺 . A graph 𝐺 = ( 𝑉 , 𝐸 ) is represented as an algebra 𝐴 with sorts 𝐴 𝐺 = 𝑉 and 𝐴 𝐺 = 𝐸 and 𝑠, 𝑡 given by the source and target of edges, respectively. Moreprecisely, Gra is equivalent to the full subcategory of all Σ -algebras ( 𝑉 , 𝐸 ) where for all 𝑒, 𝑒 ′ ∈ 𝐸 with 𝑠 ( 𝑒 ) = 𝑠 ( 𝑒 ′ ) and 𝑡 ( 𝑒 ) = 𝑡 ( 𝑒 ′ ) , one has 𝑒 = 𝑒 ′ . Assumption 5.6.
From now on we assume that(1) The category D is a full reflective subcategory of Σ -algebras closed under ( StrongEpi , Mono ) -factorizations; the reflecting of a Σ -algebra 𝐴 into D is denoted by 𝐴 @ .(2) The category D f consists of all Σ -algebras in D of finite cardinality in all sorts.In the case where the arities of operations in Σ are bounded, our present choice of D f corres-ponds well with the previous one in Assumption 5.1: choosing the set S as in Example 5.2(2), a Σ -algebra 𝐷 has finite cardinality iff the set of all morphisms from 𝑠 to 𝐷 (for 𝑠 ∈ 𝑆 ) is finite. Notation 5.7.
For the profinite monad b T of Definition 4.5 we denote by 𝑈 : ( Pro D f ) b T → Set S the forgetful functor that assigns to a b T -algebra ( 𝐴, 𝛼 ) the underlying S -sorted set of 𝐴 . , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 39 Recall from Corollary 2.12 that
Pro D f is a full subcategory of Stone ( Alg Σ ) , the category ofStone Σ -algebras and continuous homomorphisms, closed under limits. From Example 3.20 andProposition 3.18, we get the following Lemma 5.8.
The factorization system ( StrongEpi , Mono ) on D is profinite and yields the factoriz-ation system on Pro D f given by b E = continuous homomorphisms surjective in every sort, and c M = continuous homomorphisms injective in every sort. Notation 5.9.
Let 𝑋 be a finite S -sorted set of variables.(1) Denote by 𝐹 Σ 𝑋 the free Σ -algebra of terms . It is carried by the smallest S -sorted set containing 𝑋 and such thatfor every operation symbol 𝜎 : 𝑠 , . . . , 𝑠 𝑛 → 𝑠 and every 𝑛 -tuple of terms 𝑡 𝑖 of sorts 𝑠 𝑖 we have aterm 𝜎 ( 𝑡 , . . . , 𝑡 𝑛 ) of sort 𝑠. (2) For the reflection ( 𝐹 Σ 𝑋 ) @ , the free object of D on 𝑋 , we put 𝑋 ⊕ = œ ( 𝐹 Σ 𝑋 ) @ . This is a free object of
Pro D f on 𝑋 , see Lemma 3.13.(3) Let ( 𝐴, 𝛼 ) be a finite T -algebra. An interpretation of the given variables in ( 𝐴, 𝛼 ) is an 𝑆 -sortedfunction 𝑓 from 𝑋 to the underlying sorted set 𝑈 ( 𝐴, 𝛼 ) . We denote by 𝑓 @ : ( 𝐹 Σ 𝑋 ) @ → 𝐴 the corresponding morphism of D . It extends to a unique homomorphism of b T -algebras (since ( 𝐴, 𝛼 + ) is a b T -algebra by Proposition 4.16) that we denote by 𝑓 ⊕ : (cid:16)b 𝑇 𝑋 ⊕ , 𝜇 𝑋 ⊕ (cid:17) → ( 𝐴, 𝛼 + ) . Definition 5.10. A profinite term over a finite 𝑆 -sorted set 𝑋 (of variables) is an element of b 𝑇 𝑋 ⊕ . Example 5.11.
Let D = Set and
𝑇 𝑋 = 𝑋 ∗ be the monoid monad. For every finite set 𝑋 = 𝑋 @ wehave that b 𝑇 𝑋 ⊕ is the set of profinite words over 𝑋 (see Example 4.11). Definition 5.12.
Let 𝑡 , 𝑡 be profinite terms of the same sort in b 𝑇 𝑋 ⊕ . A finite T -algebra is said to satisfy the equation 𝑡 = 𝑡 provided that for every interpretation 𝑓 of 𝑋 we have 𝑓 ⊕ ( 𝑡 ) = 𝑓 ⊕ ( 𝑡 ) . Remark 5.13.
In order to distinguish equations being pairs of profinite terms according to Defin-ition 5.12 from equations being quotients according to Definition 4.19, we shall sometimes call thelatter equation morphisms . Theorem 5.14 (Generalized Reiterman Theorem for Monads on Σ -algebras) . Let D be a full re-flective subcategory of Alg Σ closed under ( StrongEpi , Mono ) -factorizations, and let T be a monadon D preserving strong epimorphisms. Then a collection of finite T -algebras is a pseudovariety iff itcan be presented by equations between profinite terms. Proof. (1) We first verify that all assumptions needed for applying Theorem 4.20 and Remark 4.21are satisfied. Put
Var ≔ { ( 𝐹 Σ 𝑋 ) @ | 𝑋 a finite S -sorted set } , the set of all free objects of D on finitely many generators. We know from Lemma 5.8 that thefactorization system ( StrongEpi , Mono ) is profinite. , Vol. 1, No. 1, Article . Publication date: January 2021. (1a) Every object ( 𝐹 Σ 𝑋 ) @ of Var is projective w.r.t. strong epimorphisms. Indeed, given a strongepimorphism 𝑒 : 𝐴 ։ 𝐵 in D , it is a strong epimorphism in Alg Σ , i.e. 𝑒 has a splitting 𝑖 : 𝐵 𝐴 in Set S with 𝑒 · 𝑖 = id . For every morphism 𝑓 : ( 𝐹 Σ 𝑋 ) @ → 𝐵 of D we are to prove that 𝑓 factorizesthrough 𝑒 . The S -sorted function 𝑋 → 𝐴 which is the domain-restriction of 𝑖 · 𝑓 : ( 𝐹 Σ 𝑋 ) @ → 𝐴 has a unique extension to a morphism 𝑔 : ( 𝐹 Σ 𝑋 ) @ → 𝐴 of D . It is easy to see that 𝑒 · 𝑖 = id implies 𝑒 · 𝑔 = 𝑓 , as required.(1b) Every object 𝐷 ∈ D f is a strong quotient 𝑒 : ( 𝐹 Σ 𝑋 ) @ ։ 𝐷 of some ( 𝐹 Σ 𝑋 ) @ in Var . Indeed, let 𝑋 be the underlying set of 𝐷 . Then the underlying function of id : 𝑋 → 𝐷 is a split epimorphismin Set S , hence, id @ : ( 𝐹 Σ 𝑋 ) @ ։ 𝐷 is a strong epimorphism.(2) By applying Theorem 4.20 and Remark 4.21, all we need to prove is that the presentation offinite T -algebras by equation morphisms 𝑒 : ( b 𝑇 𝑋 ⊕ , b 𝜇 𝑋 ⊕ ) ։ ( 𝐴, 𝛼 ) , 𝑋 finite and 𝑒 strongly epic , is equivalent to their presentation by equations between profinite terms.(2a) Let V be a collection in D T f presented by equations 𝑡 𝑖 = 𝑡 ′ 𝑖 in b 𝑇 𝑋 ⊕ 𝑖 , 𝑖 ∈ 𝐼 . Using Theorem 4.20,we just need proving that V is a pseudovariety:(i) Closure under finite products Î 𝑘 ∈ 𝐾 ( 𝐴 𝑘 , 𝛼 𝑘 ) : Let 𝑓 be an interpretation of 𝑋 𝑖 in the product.Then we have 𝑓 = h 𝑓 𝑘 i 𝑘 ∈ 𝐾 for interpretations 𝑓 𝑘 of 𝑋 𝑖 in ( 𝐴 𝑘 , 𝛼 𝑘 ) . By assumption 𝑓 ⊕ 𝑘 ( 𝑡 𝑖 ) = 𝑓 ⊕ 𝑘 ( 𝑡 ′ 𝑖 ) for every 𝑘 ∈ 𝐾 . Since the forgetful functor from b T -algebras to Set S preserves products, we have 𝑓 ⊕ = h 𝑓 ⊕ 𝑘 i 𝑘 ∈ 𝐾 , hence 𝑓 ⊕ ( 𝑡 𝑖 ) = 𝑓 ⊕ ( 𝑡 ′ 𝑖 ) .(ii) Closure under subobjects 𝑚 : ( 𝐴, 𝛼 ) ( 𝐵, 𝛽 ) : Let 𝑓 be an interpretation of 𝑋 𝑖 in ( 𝐴, 𝛼 ) . Then 𝑔 = ( 𝑈 𝑚 ) · 𝑓 is an interpretation in ( 𝐵, 𝛽 ) , thus 𝑔 ⊕ ( 𝑡 𝑖 ) = 𝑔 ⊕ ( 𝑡 ′ 𝑖 ) . Since 𝑚 is a homomorphism of b T -algebras, we have 𝑔 ⊕ = 𝑚 · 𝑓 ⊕ . Moreover, 𝑚 is monic in every sort, whence 𝑓 ⊕ ( 𝑡 𝑖 ) = 𝑓 ⊕ ( 𝑡 ′ 𝑖 ) .(iii) Closure under quotients 𝑒 : ( 𝐵, 𝛽 ) ։ ( 𝐴, 𝛼 ) : Let 𝑓 be an interpretation of 𝑋 𝑖 in 𝐴 . Since 𝑈 𝑒 isa split epimorphism in
Set S , we can choose 𝑚 : 𝑈 𝐴 → 𝑈 𝐵 with ( 𝑈 𝑒 ) · 𝑚 = id . Then 𝑔 = 𝑚 · 𝑓 isan interpretation of 𝑋 𝑖 in ( 𝐵, 𝛽 ) , thus, 𝑔 ⊕ ( 𝑡 𝑖 ) = 𝑔 ⊕ ( 𝑡 ′ 𝑖 ) . Since 𝑒 is a homomorphism of b T -algebras,we have 𝑒 · 𝑔 ⊕ = ( 𝑈 𝑒 · 𝑔 ) ⊕ = ( 𝑈 𝑒 · 𝑚 · 𝑓 ) ⊕ = 𝑓 ⊕ . Using this, we obtain 𝑓 ⊕ ( 𝑡 𝑖 ) = 𝑓 ⊕ ( 𝑡 ′ 𝑖 ) .(2b) For every equation morphism 𝑒 : ( b 𝑇 𝑋 ⊕ , b 𝜇 𝑋 ⊕ ) ։ ( 𝐴, 𝛼 ) we consider the set of all profinite equations 𝑡 = 𝑡 ′ where 𝑡, 𝑡 ′ ∈ b 𝑇 𝑋 ⊕ have the same sort andfulfil 𝑒 ( 𝑡 ) = 𝑒 ( 𝑡 ′ ) . We prove that given a finite algebra ( 𝐵, 𝛽 ) , it satisfies 𝑒 iff it satisfies all of thoseequations.(i) Let ( 𝐵, 𝛽 ) satisfy 𝑒 and let 𝑓 be an interpretation of 𝑋 in it. Then the homomorphism 𝑓 ⊕ factor-izes through 𝑒 : ( b 𝑇 𝑋 ⊕ , b 𝜇 𝑋 ⊕ ) 𝑓 ⊕ / / 𝑒 % % % % ❑❑❑❑❑❑❑❑❑❑ ( 𝐵, 𝛽 )( 𝐴, 𝛼 ) ℎ O O Thus, 𝑓 ⊕ ( 𝑡 ) = 𝑓 ⊕ ( 𝑡 ′ ) whenever 𝑒 ( 𝑡 ) = 𝑒 ( 𝑡 ′ ) , as required. , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 41 (ii) Let ( 𝐵, 𝛽 ) satisfy the given equations 𝑡 = 𝑡 ′ . We prove that every homomorphism ℎ : ( b 𝑇 𝑋 ⊕ , b 𝜇 𝑋 ⊕ ) →( 𝐵, 𝛽 ) factorizes through the given 𝑒 , which lies in ( Pro D f ) b T . We clearly have ℎ = 𝑓 ⊕ for the interpretation 𝑓 : 𝑋 → 𝑈 ( 𝐵, 𝛽 ) obtained by the domain-restriction of 𝑈 ℎ . Consequently,for all 𝑡, 𝑡 ′ ∈ b 𝑇 𝑋 ⊕ of the same sort, we know that 𝑒 ( 𝑡 ) = 𝑒 ( 𝑡 ′ ) implies ℎ ( 𝑡 ) = ℎ ( 𝑡 ′ ) . This tells us precisely that
𝑈 ℎ factorizes in
Set S through 𝑈 𝑒 : 𝑈 ( b 𝑇 𝑋 ⊕ , b 𝜇 𝑋 ⊕ ) 𝑈𝑒 ~ ~ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ 𝑈ℎ ❅❅❅❅❅❅❅❅ 𝑈 ( 𝐵, 𝛽 ) 𝑘 / / 𝑈 ( 𝐴, 𝛼 ) It remains to prove that 𝑘 is a homomorphism of b T -algebras. Firstly, 𝑘 preserves the operationsof Σ and is thus a morphism 𝑘 : 𝐵 → 𝐴 in D . This follows from 𝑈 𝑒 being epic in
Set S : given 𝜎 : 𝑠 , . . . , 𝑠 𝑛 → 𝑠 in Σ and elements 𝑥 𝑖 of sort 𝑠 𝑖 in 𝐵 , choose 𝑦 𝑖 of sort 𝑠 𝑖 in 𝑈 ( b 𝑇 𝑋 ⊕ , b 𝜇 𝑋 ⊕ ) with 𝑈 𝑒 ( 𝑦 𝑖 ) = 𝑥 𝑖 . Using that 𝑒 and ℎ are Σ -homomorphism we obtain the desired equation 𝑘 ( 𝜎 ( 𝑥 𝑖 )) = 𝑘 ( 𝜎 𝐵 ( 𝑈 𝑒 ( 𝑦 𝑖 )) = 𝑘 · 𝑈 𝑒 ( 𝜎 ( 𝑦 𝑖 )) = 𝑈 ℎ ( 𝜎 ( 𝑦 𝑖 )) . = 𝜎 𝐴 ( ℎ ( 𝑦 𝑖 )) = 𝜎 𝐴 ( 𝑘 ( 𝑥 𝑖 )) . Moreover, b 𝑇 𝑒 is epic by Lemma 5.8. In the following diagram b 𝑇 b 𝑇 𝑋 ⊕ b 𝜇 𝑋 ⊕ / / b 𝑇𝑒 (cid:15) (cid:15) GF@A b 𝑇ℎ / / b 𝑇 𝑋 ⊕ 𝑒 (cid:15) (cid:15) EDBC ℎ o o b 𝑇 𝐵 𝛽 / / b 𝑇𝑘 (cid:15) (cid:15) 𝐵 𝑘 (cid:15) (cid:15) b 𝑇 𝐴 𝛼 / / 𝐴 the outside and upper square commute because ℎ and 𝑒 are a homomorphism of b T -algebras, re-spectively, and the left hand and right hand parts commute because 𝑘 · 𝑒 = ℎ . Since b 𝑇 𝑒 is epic, itfollows that the lower square also commutes. (cid:3)
Remark 5.15.
We now show that profinite terms are just another view of the implicit opera-tions that Reiterman used in his paper [19]. We start with a one-sorted signature Σ (for notationalsimplicity) and then return to the general case. We denote by 𝑊 : D T f → Set the forgetful functor assigning to every finite algebra ( 𝐴, 𝛼 ) the underlying set 𝐴 . Definition 5.16. An 𝑛 -ary implicit operation is a natural transformation 𝜚 : 𝑊 𝑛 → 𝑊 for 𝑛 ∈ N .Thus if 𝑈 : D f → Set denotes the forgetful functor, then 𝜚 assigns to every finite T -algebra ( 𝐴, 𝛼 ) an 𝑛 -ary operation on 𝑈 𝐴 such that every homomorphism in D T f preserves that operation. , Vol. 1, No. 1, Article . Publication date: January 2021. For the case of finitary Σ -algebras, i.e. finitary monads T on Set , the above concept is due toReiterman [19, Sec. 2].
Example 5.17.
Let D = Set and
𝑇 𝑋 = 𝑋 ∗ be the monoid monad. Every element 𝑥 of a finitemonoid ( 𝐴, 𝛼 ) has a unique idempotent power 𝑥 𝑘 for some 𝑘 >
0, denoted by 𝑥 𝜔 . Since monoidmorphisms preserve idempotent powers, this yields a unary implicit operation 𝜚 with components 𝜚 ( 𝐴,𝑎 ) : 𝑥 ↦→ 𝑥 𝜔 . Notation 5.18.
Consider 𝑛 as the set { , . . . , 𝑛 − } . Every profinite term 𝑡 ∈ b 𝑇 𝑛 ⊕ defines an 𝑛 -aryimplicit operation 𝜚 𝑡 as follows: Given a finite T -algebra ( 𝐴, 𝛼 ) and an 𝑛 -tuple 𝑓 : 𝑛 → 𝑈 𝐴 , we getthe homomorphism 𝑓 ⊕ : ( b 𝑇 𝑛 ⊕ , b 𝜇 𝑛 ⊕ ) → ( 𝐴, 𝛼 ) , and 𝜚 𝑡 assigns to 𝑓 the value 𝜚 𝑡 ( 𝑓 ) = 𝑓 ⊕ ( 𝑡 ) . The naturality of 𝜚 𝑡 is easy to verify. Lemma 5.19.
Implicit 𝑛 -ary operations correspond bijectively to profinite terms in b 𝑇 𝑛 ⊕ via 𝑡 ↦→ 𝜚 𝑡 . Proof.
Recall from Corollary 2.12 that
Pro D f is a full subcategory of Stone ( Alg Σ ) closed underlimits. The forgetful functor of the latter preserves limits, hence, so does the forgetful functor 𝑈 : Pro D f → Set . Recall further from Construction 4.6 that b 𝑇 𝑛 ⊕ = lim 𝑄 𝑛 ⊕ where 𝑄 𝑛 ⊕ : 𝑛 ⊕ / 𝐾 → Pro D f is the diagram of all morphisms 𝑎 : 𝑛 ⊕ → 𝐾 ( 𝐴, 𝛼 ) = 𝐴 of Pro D f . Thus, profinite terms 𝑡 ∈ b 𝑇 𝑛 ⊕ are elements of the limit of 𝑈 · 𝑄 𝑛 ⊕ : 𝑛 ⊕ / 𝐾 → Set
By the well-known description of limits in
Set , to give 𝑡 means to give a compatible collection ofelements of 𝑈 𝐴 , i.e. for every 𝑛 ⊕ 𝑎 −→ 𝐾 ( 𝐴, 𝛼 ) one gives 𝑡 𝑎 ∈ 𝑈 𝐴 such that for every morphism of 𝑛 ⊕ / 𝐾 : 𝑛 ⊕ 𝑏 (cid:28) (cid:28) ✽✽✽✽✽✽✽ 𝑎 (cid:2) (cid:2) ✆✆✆✆✆✆✆ 𝐾 ( 𝐴, 𝛼 ) 𝐾ℎ / / 𝐾 ( 𝐵, 𝛽 ) we have 𝑈 ℎ ( 𝑡 𝑎 ) = 𝑡 𝑏 .Now observe that an object of 𝑛 ⊕ / 𝐾 is precisely a finite T -algebra ( 𝐴, 𝛼 ) together with an 𝑛 -tuple 𝑎 of elements of 𝑈 𝐴 . Thus, the given collection 𝑎 ↦→ 𝑡 𝑎 is precisely an 𝑛 -ary operation on 𝑈 𝐴 forevery finite algebra ( 𝐴, 𝛼 ) . Moreover, the compatibility means precisely that every homomorphism ℎ : ( 𝐴, 𝛼 ) → (
𝐵, 𝛽 ) of finite T -algebras preserves that operation. Thus, b 𝑇 𝑛 ⊕ consists of precisely the 𝑛 -ary implicit operations. Finally, it is easy to see that the resulting operation is 𝜚 𝑡 of Notation 5.18for every 𝑡 ∈ b 𝑇 𝑛 ⊕ . (cid:3) Remark 5.20. (1) For S -sorted signatures this is completely analogous. Let 𝑊 𝑠 : D T f → Set as-sign to every finite T -algebra ( 𝐴, 𝛼 ) the component of sort 𝑠 of the underlying S -sorted set 𝑈 𝐴 .An implicit operation of arity 𝜚 : 𝑠 , . . . , 𝑠 𝑛 → 𝑠 is a natural transformation 𝜚 : 𝑊 𝑠 × · · · × 𝑊 𝑠 𝑛 → 𝑊 𝑠 , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 43 Thus 𝜚 assigns to every finite T -algebra ( 𝐴, 𝛼 ) an operation 𝜚 ( 𝐴,𝑎 ) : 𝑈 𝐴 𝑠 × . . . 𝑈 𝐴 𝑠 𝑛 → 𝑈 𝐴 𝑠 that all homomorphisms in D T f preserve.(2) Recall that we identify every natural number 𝑛 with the set { , . . . , 𝑛 − } . For every arity 𝑠 , . . . , 𝑠 𝑛 → 𝑠 we choose a finite S -sorted set 𝑋 such that for every sort 𝑡 we have 𝑋 𝑡 = { 𝑖 ∈ { , . . . , 𝑛 } | 𝑡 = 𝑠 𝑖 } . Then for every finite T -algebra ( 𝐴, 𝛼 ) , to give an 𝑛 -tuple 𝑎 𝑖 ∈ 𝐴 𝑠 𝑖 is the same as to give S -sortedfunction 𝑓 : 𝑋 → 𝑈 𝐴 .(3) Notation 5.18 has the following generalization: given a profinite term 𝑡 ∈ b 𝑇 𝑋 ⊕ over 𝑋 of sort 𝑠 ,we define an implicit operation 𝜚 𝑡 : 𝑠 , . . . , 𝑠 𝑛 → 𝑠 by its components at all finite T -algebras ( 𝐴, 𝛼 ) as follows: 𝜚 𝑡 ( 𝑓 ) = 𝑓 ⊕ ( 𝑡 ) for all 𝑓 : 𝑋 → 𝑈 𝐴.
This yields a bijection between b 𝑇 𝑋 ⊕ and implicit operations of arity 𝑠 , . . . , 𝑠 𝑛 → 𝑠 for 𝑋 in (2).The proof is completely analogous to that of Lemma 5.19. Definition 5.21.
Let 𝜚 and 𝜚 ′ be implicit operations of the same arity. A finite algebra ( 𝐴, 𝛼 ) satisfies the equation 𝜚 = 𝜚 ′ if their components 𝜚 ( 𝐴,𝛼 ) and 𝜚 ′( 𝐴,𝛼 ) coincide.The above formula 𝜚 𝑡 ( 𝑓 ) = 𝑓 ⊕ ( 𝑡 ) shows that given profinite terms 𝑡, 𝑡 ′ ∈ b 𝑇 𝑋 ⊕ of the same sort,a finite algebra satisfies the profinite equation 𝑡 = 𝑡 ′ if and only if it satisfies the implicit equation 𝜚 𝑡 = 𝜚 𝑡 ′ . Consequently: Corollary 5.22.
Under the hypotheses of Theorem 5.14, a collection of finite T -algebras is a pseudo-variety iff it can be presented by equations between implicit operations. Whereas for varieties D of algebras the equation morphisms in the Reiterman Theorem 4.20 canbe substituted by equations 𝑡 = 𝑡 ′ between profinite terms, this does not hold for varieties D ofordered algebras (i.e. classes of ordered Σ -algebras specified by inequations 𝑡 ≤ 𝑡 ′ between terms).The problem is that Pos does not have a set dense of objects projective w.r.t. strong epimorphisms.Indeed, only discrete posets are projective w.r.t. the following regular epimorphism:We are going to show that for D = Pos (and more generally varieties D of ordered algebras) achange of the factorization system from ( StrongEpi , Mono ) to (surjective, order-reflecting) enablesus to apply the results of Section 4 to the proof that pseudovarieties of finite ordered T -algebras arepresentable by inequations between profinite terms. This generalizes results of Pin and Weil [17]who proved a version of Reiterman’s theorem (without monads) for ordered algebras, in fact, for , Vol. 1, No. 1, Article . Publication date: January 2021. general first-order structures. We begin with monads on Pos , and then show how this yields resultsfor monads on varieties D of ordered algebras. Notation 6.1.
Given an S -sorted signature Σ of operation symbols, let Σ ≤ denote the S -sortedfirst-order signature with operation symbols Σ and a binary relation symbol ≤ 𝑠 for every 𝑠 ∈ S .Moreover, let Alg Σ ≤ be the full subcategory of Σ ≤ - Str for which ≤ 𝑠 is interpreted as a partial order on the sort 𝑠 forevery 𝑠 ∈ S , and moreover every Σ -operation is monotone w.r.t. these orders. Thus, objects areordered Σ -algebras, morphisms are monotone Σ -homomorphisms. Recall from Remark 2.15 ourfactorization system with E = morphisms surjective in all sorts, and M = morphisms order-reflecting in all sorts.Thus a Σ -homomorphisms 𝑚 lies in M iff for all 𝑥, 𝑦 in the same sort of its domain we have 𝑥 ≤ 𝑦 iff 𝑚 ( 𝑥 ) ≤ 𝑚 ( 𝑦 ) . The notion of a subcategory D of Alg Σ ≤ being closed under factorizations isanalogous to Definition 5.3. Assumption 6.2.
Throughout this section, D denotes a full reflective subcategory of Alg Σ ≤ closed under factorizations. Moreover, D f is the full subcategory of D given by all algebras whichare finite in every sort.Thus, every variety of ordered algebras (presented by inequations 𝑡 ≤ 𝑡 ′ betweens terms) canserve as D , as well as every quasivariety (presented by implications between inequations). Remark 6.3. (1) Recall from Corollary 2.12 that
Pro D f is a full subcategory of Stone ( Alg Σ ≤ ) ,the category of ordered Stone Σ -algebras.(2) The factorization system on D inherited from Alg Σ ≤ is profinite, see Example 3.20. Moreover,the induced factorization system b E and c M of Pro D f is given by the surjective and order-reflectingmorphisms of Pro D f , respectively (see Proposition 3.18). Notation 6.4. (1) We again denote by (−) @ : Alg Σ ≤ → D the reflector.(2) For every finite S -sorted set 𝑋 we have the free algebra 𝐹 Σ 𝑋 (discretely ordered).(3) The free object of Pro D f on a sorted set 𝑋 is again denoted by 𝑋 ⊕ (in lieu of œ ( 𝐹 Σ 𝑋 ) @ ). Forevery finite T -algebra ( 𝐴, 𝛼 ) , given an interpretation 𝑓 of 𝑋 in ( 𝐴, 𝛼 ) , we obtain a homomorphism 𝑓 ⊕ : ( b 𝑇 𝑋 ⊕ , b 𝜇 𝑋 ⊕ ) → ( 𝐴, 𝛼 ) Definition 6.5.
By a profinite term on a finite S -sorted set 𝑋 of variables is meant an element of b 𝑇 𝑋 ⊕ .Given profinite terms 𝑡 , 𝑡 of the same sort 𝑠 , a finite T -algebra ( 𝐴, 𝛼 ) is said to satisfy theinequation 𝑡 ≤ 𝑡 provided that for every interpretation 𝑓 of 𝑋 we have 𝑓 ⊕ ( 𝑡 ) ≤ 𝑓 ⊕ ( 𝑡 ) . Theorem 6.6.
Let D be a full reflective subcategory of Alg Σ ≤ closed under factorizations, and let T be a monad on D preserving sortwise surjective morphisms. Then a collection of finite T -algebras is apseudovariety iff it can be presented by inequations between profinite terms. , Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 45 Proof.
In complete analogy to the proof of Theorem 5.14, we put
Var = { ( 𝐹 Σ 𝑋 ) @ | 𝑋 a finite S -sorted set } . and observe that Theorem 4.20 and Remark 4.21 can be applied.(1) If V is a collection of finite T -algebras presented by inequations 𝑡 𝑖 ≤ 𝑡 ′ 𝑖 , we need to verify that V is a pseudovariety. This is analogous to the proof of Theorem 5.14; in part (2) we use that 𝑚 reflects the relation symbols ≤ 𝑠 , hence from 𝑚 · 𝑓 ⊕ ( 𝑡 𝑖 ) ≤ 𝑠 𝑚 · 𝑓 ⊕ ( 𝑡 ′ 𝑖 ) we derive 𝑓 ⊕ ( 𝑡 𝑖 ) ≤ 𝑠 𝑓 ⊕ ( 𝑡 ′ 𝑖 ) .(2) Given an equation morphism 𝑒 : ( b 𝑇 𝑋 ⊕ , b 𝜇 𝑋 ⊕ ) ։ ( 𝐴, 𝛼 ) , consider all inequations 𝑡 ≤ 𝑠 𝑡 ′ where 𝑡 and 𝑡 ′ are profinite terms of sort 𝑠 with 𝑈 𝑒 ( 𝑡 ) ≤ 𝑈 𝑒 ( 𝑡 ′ ) in 𝐴 . We verify that a finite b T -algebra ( 𝐵, 𝛽 ) satisfies those inequations iff it satisfies 𝑒 . This is again completely analogous to the correspondingargument in the proof of Theorem 5.14; just at the end we need to verify, additionally, that 𝑥 ≤ 𝑠 𝑥 ′ in 𝐵 implies ℎ ( 𝑥 ) ≤ 𝑠 ℎ ( 𝑥 ′ ) in 𝐴. Denote by 𝑈 : ( Pro D f ) b T → Pos S the forgetful functor. Since 𝑈 𝑒 has surjective components, wehave terms 𝑡, 𝑡 ′ in b 𝑇 𝑋 ⊕ of sort 𝑠 with 𝑥 = 𝑈 𝑒 ( 𝑡 ) and 𝑥 ′ = 𝑈 𝑒 ( 𝑡 ′ ) , thus 𝑡 ≤ 𝑡 ′ is one of the aboveinequations. The algebra ( 𝐵, 𝛽 ) satisfies 𝑡 ≤ 𝑡 ′ and (like in Theorem 5.14) we get ℎ = 𝑓 ⊕ , hence 𝑈 ℎ ( 𝑡 ) ≤ 𝑈 ℎ ( 𝑡 ′ ) . From 𝑈 ℎ = 𝑘 · 𝑈 𝑒 , this yields 𝑘 ( 𝑥 ) ≤ 𝑠 𝑘 ( 𝑥 ′ ) . (cid:3) Remark 6.7.
In particular, if D is a variety of ordered one-sorted Σ -algebras and T a monadpreserving surjective morphisms, pseudovarieties of T -algebras can be described by inequationsbetween profinite terms. This generalizes the result of Pin and Weil [17]. REFERENCES [1] Jiří Adámek. 1977. Colimits of algebras revisited.
Bull. Aust. Math. Soc.
17, 3 (1977), 433–450.[2] Jiří Adámek, Horst Herrlich, and George E. Strecker. 2009.
Abstract and Concrete Categories: The Joy of Cats (2nd ed.).Dover Publications.[3] Jiří Adámek and Jiří Rosický. 1994.
Locally Presentable and Accessible Categories . Cambridge University Press. 332pages.[4] Jiří Adámek, Jiří Rosický, and Enrico Vitale. 2011.
Algebraic Theories . Cambridge University Press.[5] Bernhard Banaschewski and Horst Herrlich. 1976. Subcategories defined by implications.
Houston Journal Mathem-atics
Proceedings of the Cambridge Philosophical Society
10 (1935),433—-454.[7] Mikołaj Bojańczyk. 2015. Recognisable languages over monads. In
Proc. DLT , Igor Potapov (Ed.). LNCS, Vol. 9168.Springer, 1–13. Full version: http://arxiv.org/abs/1502.04898.[8] L.-T. Chen, J. Adámek, S. Milius, and H. Urbat. 2016. Profinite Monads, Profinite Equations and Reiterman’s Theorem.In
Proc. FoSSaCS’16 (Lecture Notes Comput. Sci.) , B. Jacobs and C. Löding (Eds.), Vol. 9634. Springer.[9] Samuel Eilenberg. 1976.
Automata, Languages, and Machines . Vol. 2. Academic Press, New York.[10] Peter T. Johnstone. 1982.
Stone spaces . Cambridge University Press. 398 pages.[11] John F. Kennison and Dion Gildenhuys. 1971. Equational completion, model induced triples and pro-objects.
J. PureAppl. Algebr.
1, 4 (1971), 317–346.[12] F. W. Lawvere. 1963.
Functorial Semantics of Algebraic Theories . Ph.D. Dissertation. Columbia University.[13] F. E. J. Linton. 1969. An outline of functorial semantics. In
Semin. Triples Categ. Homol. Theory , B. Eckmann (Ed.).LNM, Vol. 80. Springer Berlin Heidelberg, 7–52.[14] Saunders Mac Lane. 1998.
Categories for the working mathematician (2 ed.). Springer.[15] E. G. Manes. 1976.
Algebraic Theories . Graduate Texts in Mathematics, Vol. 26. Springer.[16] Stefan Milius and Henning Urbat. 2019. Equational Axiomatization of Algebras with Structure. In
Proc. Foundationsof Software Science and Computation Structures (FoSSaCS 2019) , Mikołaj Bojańczyk and Alex Simpson (Eds.). Springer,400–417.[17] Jean-Éric Pin and Pascal Weil. 1996. A Reiterman theorem for pseudovarieties of finite first-order structures.
AlgebraUniversalis
35 (1996), 577–595.[18] Hilary A. Priestley. 1972. Ordered topological spaces and the representation of distributive lattices.
Proc. London Math.Soc.
3, 3 (1972), 507. , Vol. 1, No. 1, Article . Publication date: January 2021. [19] Jan Reiterman. 1982. The Birkhoff theorem for finite algebras.
Algebra Universalis
14, 1 (1982), 1–10.[20] Luis Ribes and Pavel Zalesskii. 2010.
Profinite Groups . Springer Berlin Heidelberg.[21] Marcel Paul Schützenberger. 1965. On finite monoids having only trivial subgroups.
Inform. and Control
Bull. Austral. Math. Soc.
J. Pure Appl. Algebr.
2, 2 (1972), 149–168.[24] Henning Urbat, Jiří Adámek, Liang-Ting Chen, and Stefan Milius. 2017. Eilenberg Theorems for Free. In
Proc. 42ndInternational Symposium on Mathematical Foundations of Computer Science (MFCS 2017) (LIPIcs) , Kim G. Larsen, Hans L.Bodlaender, and Jean-François Raskin (Eds.), Vol. 83. Schloss Dagstuhl., Vol. 1, No. 1, Article . Publication date: January 2021. eiterman’s Theorem on Finite Algebras for a Monad 47
A IND- AND PRO-COMPLETIONS
The aim of this appendix is to characterize, for an arbitrary small category C , the free completion Pro C under cofiltered limits and its dual concept, the free completion Ind C under filtered colimits(see Notation 2.2). Let us first recall the construction of the latter: Remark A.1.
For any small category C , the ind-completion is given up to equivalence by the fullsubcategory L of the presheaf category [ C op , Set ] on filtered colimits of representables, and theYoneda embedding 𝐸 : C L , 𝐶 ↦→ C (− , 𝐶 ) . We usually leave the embedding 𝐸 implicit and view C as a full subcategory of L .Dually to Remark 2.1, an object 𝐴 of a category C is called finitely presentable if the functor A ( 𝐴, −) : C → Set is finitary, i.e. preserves filtered colimits.
Definition A.2.
Let 𝐿 be an object of a category L . Its canonical diagram w.r.t. a full subcategory C of L is the diagram 𝐷 𝐿 of all morphisms from objects of C to 𝐿 : 𝐷 𝐿 : C / 𝐿 → L , ( 𝐶 𝑐 −→ 𝐿 ) ↦→ 𝐿. Lemma A.3.
Let C be a full subcategory of L such that each object 𝐶 ∈ C is finitely presentable in L . An object 𝐿 of L is a colimit of some filtered diagram in C if and only if its canonical diagramis filtered and the canonical cocone ( 𝐶 𝑐 −→ 𝐿 ) 𝑐 ∈ C / 𝐿 is a colimit. Proof sketch.
The if part is trivial. Conversely, if 𝐿 is a colimit of some filtered diagram, thenwe can view it as a final subdiagram of its canonical diagram. Therefore, their colimits coincide. (cid:3) Theorem A.4.
Let C be a small category. A category L containing C as a full subcategory is anind-completion of C if and only if the following conditions hold: (1) L has filtered colimits, (2) every object of L is the colimit of a filtered diagram in C , and (3) every object of C is finitely presentable in L . Proof. (1) The only if part follows immediately from the construction of
Ind C in Remark A.1: (1)is obvious, (3) follows from the Yoneda Lemma, and (2) follows from Lemma A.3 and the fact that C is dense in [ C op , Set ] .(2) We now prove the if part. Suppose that (1)–(3) hold. Let 𝐹 : C → K be any functor to acategory K with filtered colimits.(2a) First, define the extension 𝐹 : L → K of 𝐹 as follows. For any object 𝐿 ∈ L expressed as thecanonical colimit ( 𝐶 𝑐 −→ 𝐿 ) 𝑐 ∈ C / 𝐿 , the colimit of 𝐹𝐷 𝐿 exists since the canonical diagram is filtered bycondition (2) and K has filtered colimits. Thus 𝐹 on objects can be given by a choice of a colimit: 𝐹 𝐿 ≔ colim (cid:18) C / 𝐿 𝐷 𝐿 −−→ C 𝐹 −→ K (cid:19) We choose the colimits such that
𝐹 𝐿 = 𝐿 if 𝐿 is in C . For any morphism 𝑓 : 𝐿 → 𝐿 ′ , each colimitinjection 𝜏 𝑐 : 𝐹𝐶 → 𝐹 𝐿 , for 𝐶 𝑐 −→ 𝐿 , associates with another colimit injection 𝜏 ′ 𝑓 · 𝑐 : 𝐹𝐶 → 𝐹 𝐿 ′ .Hence, there is a unique morphism 𝐹 𝑓 : 𝐹 𝐿 → 𝐹 𝐿 ′ such that 𝜏 ′ 𝑓 · 𝑐 = 𝐹 𝑓 · 𝜏 𝑐 . By the uniqueness ofmediating morphisms, 𝐹 preserves identities and composition. Therefore, 𝐹 extends 𝐹 . , Vol. 1, No. 1, Article . Publication date: January 2021. (2b) Second, we show that 𝐹 is finitary. Observe that 𝐹 is in fact a pointwise left Kan extensionof 𝐹 along the embedding 𝐸 : C L . By [14, Cor. X.5.4] we have, equivalently, that for every 𝐿 ∈ C and 𝐾 ∈ K the following map from K ( 𝐹 − , 𝐾 ) to the set of natural transformationsfrom L ( 𝐸 − , 𝐿 ) to K ( 𝐹 − , 𝐾 ) is a bijection: it assigns to a morphism 𝑓 : 𝐹 𝐿 → 𝐾 the naturaltransformation whose components are (cid:16) 𝐸𝐶 𝑐 −→ 𝐿 (cid:17) ↦→ (cid:18) 𝐹𝐶 = 𝐹 𝐸𝐶 𝐹𝑐 −−→ 𝐹 𝐿 𝑓 −→ 𝐾 (cid:19) . Hence, given any colimit cocone ( 𝐶 𝑖 → 𝐿 ) 𝑖 ∈I of a filtered diagram, we have the following chainof isomorphisms, natural in 𝐾 : K ( 𝐹 𝐿, 𝐾 ) (cid:27) [ C op , Set ] ( L ( 𝐸 − , 𝐿 ) , K ( 𝐹 − , 𝐾 )) see above (cid:27) [ C op , Set ] ( colim 𝑖 L ( 𝐸 − , 𝐶 𝑖 ) , K ( 𝐹 − , 𝐾 )) by condition (3) (cid:27) lim 𝑖 [ C op , Set ] ( L ( 𝐸 − , 𝐶 𝑖 ) , K ( 𝐹 − , 𝐾 )) (cid:27) lim 𝑖 K ( 𝐹𝐶 𝑖 , 𝐾 ) see above (cid:27) K ( colim 𝐹𝐶 𝑖 , 𝐾 ) Thus, by Yoneda Lemma, colim 𝐹𝐶 𝑖 = 𝐹 𝐿 , i.e. 𝐹 is finitary.(2c) The essential uniqueness of 𝐹 is clear, since this functor is given by a colimit construction. (cid:3) By dualizing Theorem A.4, we obtain an analogous characterization of pro-completions:
Corollary A.5.
Let C be a small category. The pro-completion of C is characterized, up to equival-ence of categories, as a category L containing C as a full subcategory such that (1) L has cofiltered limits, (2) every object of L is a cofiltered limit of a diagram in C , and (3) every object of C is finitely copresentable in L . Remark A.6.
Let C be a small category.(1) Pro C is unique up to equivalence.(2) Pro C can be constructed as the full subcategory of [ C , Set ] op given by all cofiltered limits ofrepresentable functors. The category C has a full embedding into Pro C via the Yoneda embed-ding 𝐸 : 𝐶 ↦→ C ( 𝐶, −) . This follows from the description of Ind-completions in Remark A.1 andthe fact that Pro C = ( Ind C op ) op . (3) If the category C is finitely complete, then Pro C can also be described as the dual of the cat-egory of all functors in [ C , Set ] preserving finite limits. Again, 𝐸 is given by the Yoneda embedding.This is dual to [3, Thm. 1.46]. Moreover, it follows that Pro C is complete and cocomplete.(4) Given a small category K with cofiltered limits, denote by [ Pro C , K ] cfin the full subcategoryof [ Pro C , K ] given by cofinitary functors. Then the pre-composition by 𝐸 defines an equivalenceof categories (−) · 𝐸 : [ Pro C , K ] cfin ≃ −−→ [ C , K ] , where the inverse is given by right Kan extension along 𝐸 ..