Applied Categorical Structures 4: 81-85, 1996. 81 (~ 1996 Kluwer Academic Publishers. Printed in the Netherlands.

Topological Spaces and Quasi-Varieties*

MICHAEL BARR and M. CRISTINA PEDICCHIO McGill University, Department of Mathematics, Burnside Hall, 805 Sherbrooke St. West, R Q. H3A 2K6, Montreal, Canada; and Dipartimento di Mathematica, Universit?t 34100 Trieste, Italy

(Received: 7 December 1994; accepted: 27 September 1995)

Abstract. We describe Top °p and ~qob °p as quasi-varieties by means of suitable "schizophrenic" objects.

Mathematics Subject Classifications (1991). 18A40, 18C15, 54B10.

Key words: category, topological space, variety, quasi-variety.

Introduction

The fact that the dual category of sober spaces is a quasi-variety, where quasi- varieties are intended in an infinitary sense, can be easily proved by using the duality between the category Sob of sober spaces and that one of spatial frames [7]. In such a duality theorem, the role played by the Sierpinski space 2 is essential, in fact the adjointness between spaces and frames is determined by representable functors, represented by 2, where 2 is thought of as a topological space or as a frame depending on the context. This double nature of 2 as a topological frame (schizophrenic object [11, 6]) allows to define a topological or algebraic enrichment on the induced hom sets.

Furthermore, the simple remark that 2 is an injective cogenerator [9] in Sob, will suffice to reconstruct the whole category of frames from 2 itself.

It appears quite surprising that arguments, similar to those developped for ~qob, still apply to arbitrary topological spaces, simply by replacing 2 by 3, where 3 is a three points space {0, 1,2} with {1,2} as the only non-trivial open subset.

The fact that 3 is an injective cogenerator in the category Top of topological spaces [1] will suffice to define a variety, containing Top ~ as a (regular epi)- reflective subcategory (= quasi-variety). Again, all the information are given by representable functors represented by 3 and 3 will have a double nature as a space and as an algebra, more precisely as a topological grid [4].

If we replace 3 by any other injective cogenerator of TOP, we get a new triple on Set; we show that the corresponding varieties of algebras are categorically equivalent.

* Research of the first author supported by grants from the NSERC of Canada and the FCAR du Qu6bec. Research of the second author supported by the Topology grant 40% and by the NATO grant CRG 941330.

82 MICHAEL BARR AND M. CRISTINA PEDICCHIO

We are grateful to H. Herrlich for interesting suggestions on the subject.

Duality Theorems

We say that a category E is a variety iff it is monadic over Set, i.e. it is equivalent to a category Se t T of T-algebras for a triple T over Set (see [8]); similarly, E is a quasi-variety or quasi-monadic over Se t iff, up to equivalences, it is a (regular epi)-reflective subcategory of a variety.

Algebraicity and quasi-algebraicity for a category can be easily characterized in terms of properties of the "free object on one generator" as in Theorems 2 and 3.

We will need the following definition, where P E E admits all copowers H j P , ij: P --+ Lid P denotes the canonical inclusion and £(P, X) is the horn set in ~.

DEFINITION 1. An object P E C is a regular generator (or regular separator) iff for any X E C the morphism ex: I]e(P,X) P ~ X , defined by ire X = f , for any f E E(P, X) , is a regular epi.

THEOREM 2 ([12]). C is a variety iff it satisfies the foUowing properties: (1) ~ is exact in the sense o f Barr [3]; (2) there exists a regular projective, P E C, with copowers; (3) P is a regular generator.

THEOREM 3 ([10]). E is a quasi-variety iff it satisfies the following properties: (1) £ has finite limits and coequalizers o f equivalence relations; (2) there exists a regular projective, P C E, with copowers; (3) P is a regular generator.

In particular, we are interested to the case where £ is a dual category, C = C°P; then conditions (2) and (3) dualize to (2 ~) there exists a regular injective I ~ ¢, with powers, and I is a regular

cogenerator. Now, let C be Top, the category of topological spaces and continuous maps;

our aim is to show that Theorem 3 applies to Top °p (see also [4]), and moreover that the proof reduces to a simple computation of contravariant adjoint functors determined by I considered as a "schizophrenic object".

Recall that a topological space is a regular cogenerator iff it contains an indiscrete subspace with two elements and a Sierpinski subspace, and that a topological space is a regular injective iff it is a retract of a power of 3, where 3 = {0, 1 ,2) with opens f~(3) = {~b, {1,2) , {0, 1,2}) (see [1]).

Choose I = 3, and define G: Top °p --+ Set by G ( X ) = Top(X , 3) for any X C Top, and F: Set -+ Top °p by F ( Y ) = F l y 3, for any Y C Set, where the product is clearly the topological one.

TOPOLOGICAL SPACES AND QUASI-VARIETIES 83

By construction F - [ G, with e x as counit, for any X E Top. Observe that F has an underlying representable functor given by F = S e t ( - , 3)

where 3 now denotes the three points set. In the following we will often use the same notation for an object and its corresponding underlying set; the hom set notation will denote in which category the object must be considered.

Now, define the triple T = GF on Set induced by the adjunction and consider the corresponding category Set T of T-algebras, with forgetful and free functors given by GT: Set T --+ Set and FT: Set --+ Set T respectively.

Let ~: Top °p --+ Set T be the canonical comparison functor defined by

~ ( X ) = (C(X) ,V(~x ) ) . If we apply @ to the topological space {,}, we get a T-algebra with underlying

set 3 and structural map h: Top(II3 3, 3) --+ 3 defined by h(u) = u(0, 1,2), for any continuous map u: 3 x 3 x 3 ~ 3. For any algebra A E Set T we can then define a functor ~: Set T --¢ Top °p by putting ~(A) equal to the set of T- homomorphisms Set T (A, 3), topological as a subspace of the topological product

1-Ia 3. The double nature of 3 as a space and as an algebra (more precisely as a

topological algebra) implies the following:

LEMMA 4. The functors • and e/are adjoint functors: ~-[e2. Proof. It suffices to show that

Top(X, ~(A)) ~- SetT(A, ~(X)) .

By applying the definition of ff and ~ , straightforward computations show that both sides correspond to maps v: X × A ~ 3 such that, for all x E X, v ( x , - ) : n --+ 3 is aT-algebra morphism, and for all a E A, v ( - , a): X ~ 3 is a continuous function. [3

Up to now we did not apply the hypotheses on 3. The fact that 3 is a regular cogenerator in Top is equivalent to e2 being full and

faithful (see [2, Theorem 9, p. 111]), so we can consider - up to equivalences - Top °p as a full reflective subcategory of Set T, with the evaluation map r/A: A --+ • ~(A) = Top(SetT(A, 3), 3) as unit.

The second assumption that 3 is regular injective is equivalent, by definition, to G preserves regular epimorphisms, and this implies that q~ preserves regu- lar epis, since G T reflects regular epis. Since any T-algebra A is a quotient q: F T ( y ) ~ A of a free one, and free T-algebras are in Top °p, we get the following commutative diagram:

F T ( y ) q • A

¢ ~ ( F T ( y ) ) '~'~(q) • eel(A)

84 MICHAEL BARR AND M. CRISTINA PEDICCHIO

Both • and ~I' preserves regular epis, so O~(q) is a regular epi, hence ~/A is a regular epi for all A. Finally, we have proved the following:

THEOREM 5. Top °p is a quasi-variety.

We remark that Top °p corresponds to the (regular epi)-reflective hull in Set T of the full subcategory given by all spaces {I-Iy 3, Y E Set} . In fact, being Top °p (regular epi)-reflective, it suffices to check it is the smallest possible one containing all {X-[y, 3, Y E Set}; this easily follows, by the characterization of reflective hulls (see [1, Proposition 16.22]) since, for any X E Top, there exists a surjective family of constant morphisms hx: {*} = II¢ 3 --+ X with x E X.

If we replace 3 by any other regular injective, regular cogenerator I for Top (for example I could be the three points space {0, 1,2} with topology f~({0, I, 2}) = {¢, {0}, {0, 1,2}}), we can repeat the same construction we did, by choosing 1 as "schizophrenic object". In this way we define a new triple TI: Set --+ Se t on Set, by T I ( Y ) = T o p ( l l y , I, I).

PROPOSITION 6. The categories Set T and Se t TI are equivalent. Proof. If I is a regular injective, regular cogenerator in Top, then the full

subcategory Z of Top, given by all retracts of powers of I, is the subcategory of all regular injective topological spaces.

By applying ¢x: Top °p -+ Set TI to objects in 77, we get regular projective Tl-algebras since, by definition of the comparison functor ~I , ¢ I (77) corresponds to free Ti-algebras and retracts of free Ti-algebras in Set T~.

Hence ¢I(77) is a projective cover for Set Tr, i.e. not only all A E ~i(77) are regular projective, but also every algebra in Set T~ is a regular epimorphic image of a certain object B E ¢I(Z) . Being Z equivalent to ~I(77) then, for any regular injective cogenerator I, we get ffl(:/?) _~ ~(77), and this equivalence between projective covers can be easily extended to an equivalence between the corresponding categories of T-algebras Set T and Set Tx [12]. D

In [4] the category of T-algebras corresponding to the choice I = 3, has been explicitely constructed as the category of Grids. A grid is defined as a frame (G, V, A) with an additional unary operator ' satisfying, for u, v E G, the fol- lowing axioms (in addition to the frames ones): (1) u" = u; (2) the unary operation u V u' and u A u' are V-homomorphisms; (3) the unary operation u V u' is a A-homomorphism; (4) (u A v) A (u A v) ' = u A v A v'; (5) the interval [u A u ~, u V u'] is a complete atomic boolean algebra with the

operation V a n d ' Moreover in [4] it is shown that Top °p coincides with the full subcategory of

Grids defined by the Horn clause:

(u Vu ' ) V ( t A t ' ) = (v V v') V ( tA t ' ) ~ u V u ' = v V v '

TOPOLOGICAL SPACES AND QUASI-VARIETIES 85

where t denotes the terminal object in G r i d s . In this approach, a topological space X , corresponds to the grid T o p ( X , 3) o f

all pairs (U, W ) with U open in X and W C_ U, with pointwise frame structure and ~ given by the boolean complement on the second component .

R E M A R K 7. The construction given in this note applies to the category C = Sob

of sober spaces and continuous maps, by choosing as regular injective, regular cogenerator the Sierpinski space 2. The triple T: S e t -4 Se t , obtained in this way is the "free f rame" triple defined, for any Y E Se t , by T ( Y ) = Top(I -[y 2, 2) __ f~(I-[y 2), so S e t T is equivalent to the category F r m of frames. The duality between sober spaces and spatial frames is realized by the representable func- tors T o p ( - , 2 ) ~-- f ~ ( - ) : Sob °p -4 F r m and F r m ( - , 2 ) ~- p t ( - ) : F r m -4

Sob °p [7]. Observe that 2 does not maintain the regular cogenerator property if we enlarge Sob to T0-spaces [9].

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