Elements of Topological Algebra. III. The Closed Category of Filters
EELEMENTS OF TOPOLOGICAL ALGEBRAIII. THE CLOSED CATEGORY OF FILTERS
WILLIAM H. ROWAN
Abstract.
We explore the structure of
Fil , the category of filters and germs of admissiblepartial functions. In particular, we show that
Fil is a nonsymmetric closed category, asdefined in [5].
Introduction
In this paper, we study the category of filters , defined almost exactly as defined in [2],the only difference being that we admit as objects in the category of filters,
Fil , filters thatcontain the empty set. This necessitates that we define germs of functions using explicitpartial functions.The point of the paper is not this minor change, but the definition we give of nonsym-metric closed category structure ([5] - see [3, VII.1 and VII.7] for the canonical treatmentof the symmetric case) on the category
Fil . We feel the need for a self-contained definitionand exploration of the properties of this category, to facilitate forthcoming more detailedexplorations of applications, such as [6] and [4], briefly mentioned in Section 10.1.
Filters
In this section, we will begin to define
Fil , the category of filters and germs, with adiscussion of filters . Recall
Definition 1.1. A filter F on a set S is a set of subsets of S such that(1) S ∈ F ;(2) if F ∈ F and F ⊂ F (cid:48) ⊆ S , then F (cid:48) ∈ F ; and(3) if F , F (cid:48) ∈ F then F ∩ F (cid:48) ∈ F .We will denote the set of filters on S by Fil S . Note that some definitions include anothercondition: that F be proper , i.e., that F not contain the empty set. However, when weassume this, we shall explicitly call the filter a proper filter . Remark 1.2.
A filter
F ∈
Fil S uniquely determines S , as S = (cid:83) F . Date : December 14, 2020.2020
Mathematics Subject Classification.
Primary: 08A99.
Key words and phrases. filters, partial functions, germs. a r X i v : . [ m a t h . R A ] D ec WILLIAM H. ROWAN
Ordering the set of filters.
Fil S admits a partial ordering, which we (unlike some authors)take to be reverse inclusion : Proposition 1.3.
Let F , G ∈
Fil S . The following are equivalent (and, if they hold, we willsay F ≤ G ):(1)
G ⊆ F , and(2) For any G ∈ G , there is an F ∈ F with F ⊆ G . Filter bases.Definition 1.4.
We say that a set B of subsets of S is a filter base or base for a filter if F , F (cid:48) ∈ B imply there is a ¯ F ∈ B such that ¯ F ⊆ F ∩ F (cid:48) . Definition 1.5. If B is a filter base (of subsets of S ), then the set of subsets F = { F ⊆ S |∃ B ∈ B such that B ⊆ F } is a filter F , which is the least (in the above ordering) such that B ⊆ F , and which we denote by Fg S B or simply Fg B . In this case, we also say that B is a base for F . If B is not a filter base, then the least filter F such that B ⊆ F , and which westill denote by Fg B , is Fg B (cid:48) , where B (cid:48) is the set of finite intersections of elements of B (andis a filter base). In either case, we say that F = Fg B is the filter generated by B . Subfilters. If F ∈
Fil S , then we say that a filter F (cid:48) ∈ Fil S is a subfilter of F if F (cid:48) ≤ F .We will denote by Fil F the set of subfilters F (cid:48) ≤ F , i.e., the interval sublattice I Fil S [ ⊥ , F ].Fil S is coalgebraic. Recall that a lattice is coalgebraic if its dual is algebraic [4, Defini-tion 15.1].
Proposition 1.6.
Let F be a filter. Then Fil F is a coalgebraic complete lattice, where(1) (cid:87) i F i = (cid:84) i F i and(2) (cid:86) i F i = Fg { (cid:83) i F i } . f ( F ) and f − ( G ) .Definition 1.7. Let S and T be sets, and f : S → T a function. If F ∈
Fil S , then wedefine f ( F ) = Fg { f ( F ) | F ∈ F } ;if G ∈
Fil T , then we define f − ( G ) = Fg { f − ( G ) | G ∈ G } . Proposition 1.8.
We have(1) If f : S → T is a function, F ∈
Fil S , and G ∈
Fil T , then f ( F ) ≤ G ⇐⇒ F ≤ f − ( G ) . (2) If in addition g : T → W is a function and H ∈
Fil W , then g ( f ( F )) = ( gf )( F ) and f − ( g − ( H )) = ( gf ) − ( H ) . II. THE CLOSED CATEGORY OF FILTERS 3 Partial Functions; Restriction If S , T are sets, a partial function f : S → T is a method or rule f which somehow assignsan element f ( s ) ∈ T to s , for some, but not necessarily all, elements s ∈ S . Definition 2.1.
We denote the domain of definition of f , the subset of s ∈ S such that f ( s ) is defined, by dd( f ). We denote the range of f , the set of elements of the form f ( s ) forsome s ∈ S , by r( f ). Definition 2.2. If f : S → T , g : T → W are partial functions, then the composite partialfunction of f and g , denoted g ◦ f , is the partial function that assigns an element s ∈ S to g ( f ( s )), if both s ∈ dd( f ) and f ( s ) ∈ dd( g ). Definition 2.3. If f and g are partial functions from S to T , then we say that f is a restriction of g if dd( f ) ⊆ dd( g ) and f ( s ) = g ( s ) for s ∈ dd( f ). If f is a partial function on S , and D ⊆ S , then we denote by f | D the restriction of f to D , i.e., the rule which assigns f ( s ) to s for s ∈ D ∩ dd( f ) and does not assign anything to elements not in D ∩ dd( f ). f ( D ) and f − ( D ) when f is a Partial Function. If f : S → T is a partial function, and D ⊆ S , we define f ( D ) = { f ( s ) | s ∈ D ∩ dd( f ) } and if D (cid:48) ⊆ T , f − ( D (cid:48) ) = { s ∈ S | s ∈ dd( f ) and f ( s ) ∈ D (cid:48) } . Lemma 2.4.
Let f : S → T , g : T → W be partial functions. Then dd( g ◦ f ) = f − (dd( g )) .Proof. Referring to the definitions, we have s ∈ dd( g ◦ f ) ⇐⇒ s ∈ dd( f ) and f ( s ) ∈ dd( g ) ⇐⇒ s ∈ f − (dd( g )) . (cid:3) Lemma 2.5.
Let f and g be partial functions from S to T . If f = g | ˆ D for some ˆ D ⊆ S then(1) If D ⊆ S , then f ( D ) ⊆ g ( D ) , with f ( D ) = g ( D ) when D ⊆ ˆ D , and(2) if D (cid:48) ⊆ T , then f − ( D (cid:48) ) ⊆ g − ( D (cid:48) ) .Proof. (1): We have t ∈ f ( D ) ⇐⇒ t ∈ g | ˆ D ( D ) ⇐⇒ ∃ s ∈ ˆ D ∩ D ∩ dd( g ) such that t = g ( s )= ⇒ ∃ s ∈ D ∩ dd( g ) such that t = g ( s ) ⇐⇒ s ∈ g ( D ) , with equivalence if D ⊆ ˆ D . WILLIAM H. ROWAN (2): We have s ∈ f − ( D (cid:48) ) ⇐⇒ s ∈ dd( f ) and f ( s ) ∈ D (cid:48) ⇐⇒ s ∈ dd( g | ˆ D ) and g | D (cid:48) ( s ) ∈ D (cid:48) ⇐⇒ s ∈ ˆ D ∩ dd( g ) and g ( s ) ∈ D (cid:48) = ⇒ s ∈ dd( g ) and g ( s ) ∈ D (cid:48) ⇐⇒ s ∈ g − ( D (cid:48) ) . (cid:3) Proposition 2.6.
We have(1) If f : S → T is a partial function, D ⊆ S , and D (cid:48) ⊆ T , then f ( D ) ⊆ D (cid:48) ⇐⇒ D ⊆ ( S − dd( f )) ∪ f − ( D (cid:48) ); (2) if in addition, there is another partial function g : T → W , and D (cid:48)(cid:48) ⊆ W , then g ( f ( D )) = ( g ◦ f )( D ) and f − ( g − ( D (cid:48)(cid:48) )) = ( g ◦ f ) − ( D (cid:48)(cid:48) ) . Proof. (1) We have f ( D ) ⊆ D (cid:48) ⇐⇒ ( s ∈ D and s ∈ dd( f ) = ⇒ f ( s ) ∈ D (cid:48) ) ⇐⇒ ( s / ∈ D or s / ∈ dd( f ) or f ( s ) ∈ D (cid:48) ) ⇐⇒ ( s ∈ D = ⇒ ( s / ∈ dd( f ) or f ( s ) ∈ D (cid:48) )) ⇐⇒ D ⊆ ( S − dd( f )) ∪ f − ( D (cid:48) ); and(2): we have w ∈ g ( f ( D )) ⇐⇒ ∃ s such that s ∈ dd( f ) and f ( s ) ∈ dd( g ) and w = g ( f ( s )) ⇐⇒ ∃ s such that s ∈ dd( g ◦ f ) and ( g ◦ f )( s ) = w ⇐⇒ w ∈ ( g ◦ f )( D ); and s ∈ f − ( g − ( D (cid:48)(cid:48) )) ⇐⇒ s ∈ dd( f ) and f ( s ) ∈ g − ( D (cid:48)(cid:48) ) ⇐⇒ s ∈ dd( f ) and f ( s ) ∈ dd( g ) and g ( f ( s )) ∈ D (cid:48)(cid:48) ⇐⇒ s ∈ dd( g ◦ f ) and ( g ◦ f )( s ) ∈ D (cid:48)(cid:48) ⇐⇒ s ∈ ( g ◦ f ) − ( D (cid:48)(cid:48) ) . (cid:3) II. THE CLOSED CATEGORY OF FILTERS 5 f ( F ) and f − ( G ) when f is a Partial Function. If S and T are sets, f : S → T is apartial function, and F is a filter of subsets of S , then we define f ( F ) = Fg { f ( F ) | F ∈ F } . On the other hand, given f and a filter G of subsets of T , then we define f − ( G ) = Fg { f − ( G ) | G ∈ G } . Remark 2.7.
Note that for a total function f (i.e. if dd( f ) = S ), this definition coincideswith the definition (Definition 1.7) given previously. Also, note that the mappings D (cid:55)→ f ( D )and D (cid:48) (cid:55)→ g − ( D (cid:48) ) are monotone, and, consequently, take filter bases to filter bases. Theorem 2.8.
We have(1) If f is a partial function from S to T , F is a filter of subsets of S such that dd( f ) ∈ F ,and G is a filter of subsets of T , then f ( F ) ≤ G iff F ≤ f − ( G ); (2) and if in addition, we have a partial function g : T → W , and H is a filter of subsetsof W , then g ( f ( F )) = ( g ◦ f )( F ) and f − ( g − ( H )) = ( g ◦ f ) − ( H ) . Proof.
First, we note that by Remark 2.7, if we have f : S → T and F ∈
Fil S , then f ( F ) = Fg { f ( F ) | F ∈ F } = Up { f ( F ) | F ∈ F } where Up B , for a subset B of a lattice (in this case, the lattice of subsets of S ), denotes theset of elements of the lattice greater than or equal to an element in B . Similarly, if G ∈
Fil T ,then f − ( G ) = Fg { f − ( G ) | G ∈ G } = Up { f − ( G ) | G ∈ G } . Then, to prove the statements of the Theorem, we consider that
WILLIAM H. ROWAN (1): by Proposition 2.6, and since dd( f ) ∈ F , f ( F ) ≤ G ⇐⇒ Fg { f ( F ) | F ∈ F } ≤ G⇐⇒ ∀ G ∈ G , ∃ F ∈ F such that f ( F ) ⊆ G ⇐⇒ ∀ G ∈ G , ∃ F ∈ F such that F ⊆ ( S − dd( f )) ∪ f − ( G ) ⇐⇒ ∀ G ∈ G , ∃ F ∈ F such that F ∩ dd( f ) ⊆ f − ( G ) ⇐⇒ ∀ G ∈ G , ∃ F ∈ F such that F ⊆ f − ( G ) ⇐⇒ F ≤ Fg { f − ( G ) | G ∈ G }⇐⇒ F ≤ f − ( G );(2): we also have H ∈ g ( f ( F )) ⇐⇒ H ∈ Fg { g ( G ) | G ∈ Fg { f ( F ) | F ∈ F } }⇐⇒ H ∈ Fg { g ( f ( F )) | F ∈ F }⇐⇒ H ∈ ( g ◦ f )( F ) , and F ∈ f − ( g − ( H )) ⇐⇒ F ∈ Fg { f − ( G ) | G ∈ Fg { g − ( H ) | H ∈ H } }⇐⇒ F ∈ Fg { f − ( g − ( H )) | H ∈ H }⇐⇒ F ∈ ( g ◦ f ) − ( H ) . (cid:3) The Category
LPartialAdmissible domains of definition.
We would like to consider as equivalent, functions(or partial functions) on a set S which have a common restriction, and to work with theresulting equivalence classes of partial functions that we will call germs in Section 4. In thisplain form, the equivalence relation is uninteresting, because any two partial functions on S have a common restriction to the empty set. For this reason, we will limit this relationof having a common restriction to considering two partial functions on S as equivalent, ifand only if they have a common restriction to a subset of S that belongs to a specified setof admissible domains of definition. In order that this result in an equivalence relation, wewill require the specified set of admissible domains of definition be a filter of subsets of S .We call partial functions from S to T , defined on some set in the filter F ∈
Fil S , admissiblepartial functions from F to T . We already saw, in Theorem 2.8(1), a use of the conditionthat a partial function be admissible, although we didn’t yet call it that. Notation 3.1. If F is a filter on some set S , G is a filter on some set T , and G ∈ G ,we denote by Partial ( F , G ) ( LPartial ( F , G )), the set of admissible partial functions from F to G , (respectively the set of local admissible partial functions from F to G ) and by Partial ( F , G , G ) ( LPartial ( F , G , G )) the set of admissible partial functions from S to T II. THE CLOSED CATEGORY OF FILTERS 7 (respectively the set of local admissible partial functions from F to G ), such that for some F ∈ F , f ( F ) ⊆ G . Remark 3.1.
These notations will be useful not only for defining (in this section and thenext) the categories
LPartial and
Fil , but also for defining a nonsymmetric closed structureon
Fil in Sections 8 and 9. Note that elements of
LPartial ( F , G , G ) are not necessarilyarrows in any category, although they play a role in the definitions of categories in the latersections we mentioned. Locality.
In order for admissible partial functions, and germs of admissible partial func-tions, to be the arrows of categories, we will need to impose another condition, locality , thatthe partial functions must satisfy. The problem is that if f and g are partial functions,dd( g ◦ f ) may be so small that g ◦ f is not admissible. Thus, suppose f is an admissiblepartial function from F to T , where F ∈
Fil S , and g is an admissible partial function from G to W , where G ∈
Fil T . As dd( g ◦ f ) = f − (dd( g )) and dd( g ) could be any element of G ,what we want to require of f is that for any G ∈ G , f − ( G ) ∈ F . For, if that is true, thenbecause F is a filter, g ◦ f will defined on dd( f ) ∩ f − ( G ) ∈ F and will be admissible. Definition 3.2.
We say that f is local (with respect to G ) if for each G ∈ G , f − ( G ) ∈ F ,or in other words, there is an F ∈ F such that F ⊆ dd( f ) and f ( F ) ⊆ G . Proposition 3.3. If f : S → T is a partial function, admissible with respect to F and localwith respect to G , and g : T → W is a partial function, admissible with respect to G and localwith respect to H , then g ◦ f : S → W is a partial function, admissible with respect to F andlocal with respect to H . Definition 3.4.
We denote the set of partial functions from S = (cid:83) F to T = (cid:83) G , admissiblewith respect to F and local with respect to G , by LPartial ( F , G ). This defines a category LPartial , where the identity arrow from F to itself is just the identity function on S = (cid:83) F . Proof that
LPartial is a category. If f : F → G and g : G → H , then g ◦ f is defined asthe partial function with domain of definition dd( f ) ∩ f − ( g ), sending s ∈ dd( f ) to g ( f ( s )).That is, it is the partial function corresponding to the relational product of f and g , seen asrelations. The axioms of a category are immediate. (cid:3) Germs and the Category
FilGerms of partial functions.Definition 4.1. If F is a filter of subsets of a set S , then a germ of admissible partialfunctions from F to a set T is an ≡ F -equivalence class of such partial functions, where f ≡ F g iff for some F ∈ F , dd( f ) ∩ F = dd( g ) ∩ F and f ( s ) = g ( s ) for all s ∈ dd( f ) ∩ dd( g ) ∩ F .If f is a partial function, we will denote its germ (the ≡ F -equivalence class containing f ) byΓ f , or by f / F . We will also use Γ as a set-function, so that if Y is a set of admissible partialfunctions from F to T , Γ( Y ) will denote the set of germs of the partial functions in Y . Wewill use Γ in this way particularly in two cases: we will shortly define the hom-set Fil ( G , H ) = WILLIAM H. ROWAN Γ( LPartial ( H , G )) of the category Fil , and we will later define an internal hom-object G H =Fg { Γ( Partial ( H , G , G ) } for the category Fil using the base of sets Γ(
Partial ( H , G , G ). Theorem 4.2.
Let f , g be partial functions from S to T , and F a filter of subsets of S suchthat f ≡ F g . We have(1) If F (cid:48) is a subfilter of F , then f ( F (cid:48) ) = g ( F (cid:48) ) , and(2) if G is a filter of subsets of T , then f − ( G ) ∧ F = g − ( G ) ∧ F Proof.
Let F ∈ F be such that dd( f ) ∩ F = dd( g ) ∩ F and f = g on dd( f ) ∩ dd( g ) ∩ F .Then(1): Since F ∈ F , and F (cid:48) ≤ F , there is an F (cid:48) ∈ F (cid:48) such that F (cid:48) ⊆ F . We havedd( f ) ∩ F (cid:48) = dd( g ) ∩ F (cid:48) and f = g on dd( f ) ∩ dd( g ) ∩ F (cid:48) . Thus, f ( F (cid:48) ) = Fg { f ( F (cid:48) ) | F (cid:48) ∈ F (cid:48) } = Fg { f ( F (cid:48) ) | F (cid:48) ∈ F (cid:48) and F (cid:48) ⊆ F } = Fg { g ( F (cid:48) ) | F (cid:48) ∈ F (cid:48) and F (cid:48) ⊆ F } = Fg { g ( F (cid:48) ) | F (cid:48) ∈ F (cid:48) } = g ( F (cid:48) ) . (2): If F ∈ F is such that dd( f ) ∩ F = dd( g ) ∩ F and f ( s ) = g ( s ) for all s ∈ dd( f ) ∩ dd( g ) ∩ F , then the same statement is true for any smaller F . Consequently, we have f − ( G ) ∧ F = Fg { f − ( G ) ∩ F (cid:48)(cid:48) | G ∈ G , F (cid:48)(cid:48) ∈ F } = Fg { g − ( G ) ∩ F (cid:48)(cid:48) | G ∈ G , F (cid:48)(cid:48) ∈ F } = g − ( G ) ∧ F ; (cid:3) Notation 4.1.
We continue to use roman letters f , g , etc. to denote partial functions, andwill use greek letters ϕ , γ , etc. for germs. ϕ ( F ) and ϕ − ( G ) .Notation 4.2. Let F , F (cid:48) be filters of subsets of S such that F (cid:48) ≤ F , and let G be a filterof subsets of T . If ϕ is a germ of partial functions from S to T and admissible wrt F , thenwe define ϕ ( F (cid:48) ) = f ( F (cid:48) )and ϕ − ( G ) = f − ( G ) ∧ F , where f is any admissible partial function representing ϕ . By Theorem 4.2, these formulaeare independent of the choice of f . Proposition 4.3.
Let
F ∈
Fil S , and G ∈
Fil T . If ϕ is a germ of partial functions admissiblewrt F , then the following are equivalent: II. THE CLOSED CATEGORY OF FILTERS 9 (1) For some admissible partial function f : G → T representing ϕ , f is local with respectto G ;(2) For every admissible partial function f : G → T representing ϕ , f is local with respectto G ;(3) ϕ ( F ) ≤ G .Proof. Certainly (2) = ⇒ (1).(1) = ⇒ (3): Let f : G → T be an admissible partial function representing ϕ , local withrespect to G , and suppose that we are given G ∈ G . Since f is local wrt G , there is an F ∈ F such that f ( F ) ⊆ G . This shows that ϕ ( F ) = f ( F ) ≤ G .(3) = ⇒ (2): Assume that ϕ ( F ) ≤ G , and let f : G → T be an admissible partial functionrepresenting ϕ . Let G be any element of G . Since ϕ ( F ) ≤ G , there is an f ∈ F such that f ( F ) ⊆ G . Since G was any element of G , this shows that f is local wrt G . (cid:3) Theorem 4.4.
Let F be a filter of subsets of a set S , and let G be a filter of subsets ofanother set T . We have(1) ≡ F is an equivalence relation on the set of partial functions from S to T ;(2) if f ≡ F g , then f is admissible wrt F iff g is admissible wrt F ;(3) if f ≡ F g , then f is local wrt G iff g is local wrt G ;(4) ≡ F is an equivalence relation on the set of partial functions from S to T local wrt G ;(5) If f and g are admissible (wrt F ) functions from S to T , and f ≡ F g , then there isan F ∈ F such that F ⊆ dd( f ) ∩ dd( g ) and f | F = g | F .(6) If f , f (cid:48) : S → T are partial functions, admissible with respect to F and local withrespect to G , with f ≡ F f (cid:48) , and g , g (cid:48) ∈ Fil ( G , (cid:83) H ) are admissible with respect to H ,with g ≡ G g (cid:48) , then ( g ◦ f ) ≡ F ( g (cid:48) ◦ f (cid:48) ) .Proof. (1): Let f ≡ F g ≡ F h . Then there is an F ∈ F such that dd( f ) ∩ F = dd( g ) ∩ F and f = g on dd( f ) ∩ dd( g ) ∩ F , and an F (cid:48) ∈ F such that dd( g ) ∩ F (cid:48) = dd( h ) ∩ F (cid:48) and g = h on dd( g ) ∩ dd( h ) ∩ F (cid:48) . Then F ∩ F (cid:48) ∈ F , dd( f ) ∩ F ∩ F (cid:48) = dd( g ) ∩ F ∩ F (cid:48) ,dd( g ) ∩ F ∩ F (cid:48) = dd( h ) ∩ F (cid:48) , and f = h on dd( f ) ∩ dd( g ) ∩ dd( h ) ∩ F ∩ F (cid:48) = dd( f ) ∩ dd( h ) ∩ F ∩ F (cid:48) .Thus, ≡ F is transitive. Reflexivity and symmetricity are obvious.(2): f is admissible wrt F iff dd( f ) ∈ F , and likewise, g is admissible iff dd( f ) ∈ F . If f ≡ F g , then there is an ¯ F ∈ F such that dd( f ) ∩ dd( g ) ∩ ¯ F ; if f is admissible so thatdd( f ) ∈ F , then dd( g ) ∩ ¯ F ∈ F , which implies that dd( g ) ∈ F . Thus, f admissible implies g admissible. The converse follows by symmetry.(3): If f ≡ F g , then there is an ¯ F ∈ F such that f = g on dd( f ) ∩ dd( g ) ∩ ¯ F , and if f islocal wrt G , then for any G ∈ G there is an f ∈ F such that f ( F ) ⊆ G . Then f ( F ∩ ¯ F ) ⊆ G ,which implies that g ( F ∩ ¯ F ) ⊆ G . Thus, g is local wrt G . The converse follows by symmetry.(4): Follows from (1) and (3).(5): We have dd( f ) ∩ dd( g ) ∈ F , and there is an ¯ F ∈ F such that f and g are equal ondd( f ) ∩ ¯ F and dd( g ) ∩ ¯ F . We let F = dd( f ) ∩ dd( g ) ∩ ¯ F . (6): dd( g ◦ f ) = dd( f ) ∩ f − (dd( g )). dd( g (cid:48) ◦ f (cid:48) ) = dd( f (cid:48) ) ∩ ( f (cid:48) ) − (dd( g (cid:48) )). Let F ∈ F besuch that dd( f ) ∩ F = dd( f (cid:48) ) ∩ F and f = f (cid:48) on dd( f ) ∩ dd( f (cid:48) ) ∩ F , and let G ∈ G be suchthat dd( g ) ∩ G = dd( g (cid:48) ) ∩ G and g = g (cid:48) on dd( g ) ∩ dd( g (cid:48) ) ∩ G . We have [perhaps we needto show f − (dd( g (cid:48) ) ∩ G ) ∩ F = ( f (cid:48) ) − (dd( g (cid:48) ) ∩ G ) ∩ F ?]dd( g ◦ f ) ∩ F ∩ f − ( G ) = dd( f ) ∩ f − (dd( g )) ∩ F ∩ f − ( G )= [dd( f ) ∩ F ] ∩ (cid:2) f − (dd( g ) ∩ G ) ∩ F (cid:3) = [dd( f (cid:48) ) ∩ F ] ∩ (cid:2) ( f (cid:48) ) − (dd( g (cid:48) ) ∩ G ) ∩ F (cid:3) = dd( f (cid:48) ) ∩ ( f (cid:48) ) − (dd( g (cid:48) )) ∩ F ∩ ( f (cid:48) ) − ( G )= dd( g (cid:48) ◦ f (cid:48) ) ∩ F ∩ ( f (cid:48) ) − ( G )and if s ∈ F ∩ f − ( G ), then g ( f ( s )) = g (cid:48) ( f ( s )) = g (cid:48) ( f (cid:48) ( s )). (cid:3) Remark 4.5.
Looking at the statements of the Theorem, it makes sense to call the germ f / F of a partial function f : S → T admissible (wrt F ∈
Fil S ) if f is admissible wrt F ,and local (wrt G ∈
Fil T ) if f is local wrt G . Galois Connection.
Now we want to show that like the mappings F (cid:48) (cid:55)→ f ( F (cid:48) ) and G (cid:55)→ f − ( G ) ∧ F , the mappings F (cid:48) (cid:55)→ ϕ ( F (cid:48) ) and G (cid:55)→ ϕ − ( G ) constitute a Galois connection: Theorem 4.6.
Let F and G be filters, on sets S and T , respectively. Let F (cid:48) be a filter suchthat F (cid:48) ≤ F . If ϕ is a germ of partial functions from S to T admissible with respect to F ,then ϕ ( F (cid:48) ) ≤ G ⇐⇒ F (cid:48) ≤ ϕ − ( G ) . Proof.
Let f represent ϕ . Then by Theorem 4.2 and the notation that follows it, ϕ ( F (cid:48) ) ≤ G ⇐⇒ f ( F (cid:48) ) ≤ G⇐⇒ F (cid:48) ≤ f − ( G ) ∧ F⇐⇒ F (cid:48) ≤ f − ( G ) ⇐⇒ F (cid:48) ≤ ϕ − ( G ) . (cid:3) Notation.
Just as we denote by f | F the restriction to F of an admissible partial function f we can restrict a germ ϕ to a smaller subdomain filter. Thus if ϕ = f / F , and ¯ F ≤ F , wecan form ϕ/ ¯ F = ( f / F ) / ¯ F . because since ¯ F ≤ F , there is an F (cid:48) ∈ ¯ F such that F (cid:48) ⊆ dd f . Proposition 4.7.
In this situation, ( f / F ) / ¯ F = f / ¯ F . The category Fil. If S , T are sets, F ∈
Fil S , and G ∈
Fil T , then we denote the set ofgerms of partial functions from S to T , admissible with respect to F and local with respectto G , by Fil ( F , G ). This defines Fil , the category of filters , where the identity arrow from F to F is the germ 1 S / F . We denote by Γ the functor from LPartial to Fil that takes anadmissible, local partial function f ∈ LPartial ( F , G ) to its germ f / F ∈
Fil ( F , G ). II. THE CLOSED CATEGORY OF FILTERS 11
Remark 4.8.
Although we have needed to handle the boundary case that we mentionedpreviously, of filters F such that {} ∈ F , sometimes we know that {} / ∈ F . In this case,proofs can sometimes be simplified by avoiding the need to work with partial functions, for,if γ ∈ Fil ( F , G ), and G is such that {} / ∈ G , then γ has a representative g which is total, i.e.such that dd( g ) = (cid:83) G . 5. Factorization of Arrows in
Fil
In this section, we will define a factorization system (cid:104) E Fil , M Fil (cid:105) for the category
Fil .See [4, Section 2] for a discussion of this concept. We will show that this factorization is aso-called epi, monic factorization system. Finally, we look at the the M Fil -subobject latticeof an object
F ∈
Fil , and show that it can be identified with the lattice of filters F (cid:48) suchthat F (cid:48) ≤ F . The factorization system (cid:104) E Fil , M Fil (cid:105) . We define the subcategory E Fil of Fil to containall germs ϕ : F → G such that ϕ ( F ) = G . We define the subcategory M Fil to contain allgerms ϕ : F → G having the form f / F , where f is an admissible partial function one-oneon its domain of definition. Theorem 5.1. (cid:104) E Fil , M Fil (cid:105) is a factorization system in
Fil , such that M Fil consists ofmonic, and E Fil of epi, arrows.Proof.
Let ϕ : F → G , where
F ∈
Fil S and G ∈
Fil T . If ϕ = f / F , then f is a local partialfunction from F to f ( F ), and if we define (cid:15) : F → f ( F ) by (cid:15) = f / F and µ : f ( F ) → G by µ = 1 T /f ( F ), we have a suitable factorization ϕ = µ ◦ (cid:15) . For, (cid:15) ∈ E Fil ( F , f ( F )) and µ ∈ M Fil ( f ( F ) , G ).If ϕ ∈ E Fil ( F , G ), then let α , β : G → H , and suppose that α ◦ ϕ = β ◦ ϕ . Let a , b ,and f be partial functions representing α , β , and ϕ , where a and b can be taken to havedd( a ) = dd( b ) = G ∈ G . Let F ∈ F be smaller than dd( f ), such that f ( F ) ⊆ G , and suchthat ( a ◦ f ) | F = ( b ◦ f ) | F . Since ( a ◦ f ) | F = ( b ◦ f ) | F , a | f ( F ) = b | f ( F ) , showing that α = β because (remembering that ϕ ( F ) = G ) f ( F ) ∈ G . Thus, ϕ is epi.If ϕ ∈ M Fil ( F , G ), let f be an admissible partial function representing ϕ , and such that f is one-one on dd( f ) = F ∈ F . Let α , β : H → F be such that ϕ ◦ α = ϕ ◦ β . Let a and b berepresentatives of α and β having the same domain of definition H ∈ H , which is such that a ( H ) ⊆ F and b ( H ) ⊆ F , and such that f ◦ a = f ◦ b . (Such representatives can always beconstructed by restriction, since α and β are local and ϕ ◦ α = ϕ ◦ β .) However, f is one-one,implying a = b , which implies that α = β . We have proved that ϕ is monic. Suppose now that germs (cid:15) , α , β , and µ are given, such (cid:15) ∈ E Fil and µ ∈ M Fil , and forminga diagram F (cid:15) (cid:126) (cid:126) α (cid:31) (cid:31) W β (cid:32) (cid:32) G µ (cid:127) (cid:127) H which commutes. Let (cid:15) = e/ F , α = a/ F , β = b/ W , and µ = m/ G , where dd( e ) = dd( a ), r ( e ) ⊆ dd( b ), r ( a ) ⊆ dd( m ), m is one-one, and furthermore b ◦ e = m ◦ a .Let e and a , considered as functions from dd( e ) to dd( b ) and from dd( a ) to dd( m ) re-spectively, be factored as e = m [ e ] ◦ ˜e [ e ] and a = m [ a ] ◦ ˜e [ a ] in the category Set . (Thenotations m [ f ], and ˜e [ f ] for an arrow f , which we employ only in this proof, are defined in[4, Section 2]; e = m [ e ] ◦ ˜e [ e ] is the canonical chosen factorization of the function e in theusual factorization system of the category Set .) We have a commutative diagramdd( e ) = dd( a ) ˜e [ e ] (cid:118) (cid:118) ˜e [ a ] (cid:40) (cid:40) r( e ) b ◦ m [ e ] (cid:40) (cid:40) d (cid:47) (cid:47) r( a ) m ◦ m [ a ] (cid:118) (cid:118) r( b ◦ m [ e ]) = r( m ◦ m [ a ])in the category Set which is uniquely diagonalized as shown by a function we denote by d .For, ˜e [ e ] is an onto function, and m ◦ m [ a ] is a one-one function.The function d is an admissible partial function from W to T because, (cid:15) being in E Fil ,r( e ) ∈ W . d is local, because if G ∈ G , there is an F ∈ F such that a ( F ) ⊆ G , and then d ( ˜e [ e ]( F )) ⊆ G ; however, ˜e [ e ]( F ) ∈ W because (cid:15) ∈ E Fil .Let δ ∈ Fil ( W , G ) be defined by δ = d/ W . Since e , a , b , and m , and d are all admissible,local partial functions, the diagram of germs commutes. The uniqueness of the diagonalarrow δ follows by invoking either the fact that E Fil consists of epi, or the fact that M Fil consists of monic arrows of
Fil . (cid:3) Epi and monic arrows in Fil.Theorem 5.2.
Let ϕ ∈ Fil ( F , G ) , where F ∈
Fil S and G ∈
Fil T . We have(1) If ϕ is epi, then ϕ ∈ E Fil ; and(2) if ϕ is monic, then ϕ ∈ M Fil .Proof. (1): Suppose that ϕ ∈ Fil ( F , G ) but ϕ / ∈ E Fil ( F , G ), i.e. that ϕ ( F ) < G . This meansthat if ϕ = f / F , there is an F ∈ F such that F ⊆ dd( f ) and f ( F ) / ∈ G ; thus, for all G ∈ G , G (cid:54)⊆ f ( F ), so there is an element g G ∈ G such that g G / ∈ f ( F ). Let W = { , } . Let II. THE CLOSED CATEGORY OF FILTERS 13 a : T → W send all elements to 0. Let b : T → W be the same, except for elements of theform g G (for any G ), which it should send to 1. a and b are total functions, so admissible.By construction, a/ G (cid:54) = b/ G . But, we have a ◦ f = b ◦ f , because if x ∈ F , we cannot have f ( x ) = g G for any G . Thus, ϕ is not epi.(2): Suppose ϕ ∈ Fil ( F , G ), but ϕ (cid:54)∈ M Fil ( F , G ). That is, we assume that if ϕ = f / F ,for an admissible partial function f : F → T , then f is not one-one. For every F ∈ F , suchthat F ⊆ dd( f ), there are a F , b F ∈ F such that a F (cid:54) = b F but f ( a F ) = f ( b F ). Let W be theset of F ∈ F such that F ⊆ dd( f ), and define a : W → S , b : W → S by a : F (cid:55)→ a F and b : F (cid:55)→ b F . Let H ∈
Fil W be defined as H = a − ( F ) ∧ b − ( F ). The functions a and b aretotal functions, hence admissible. By monotonicity, we have a ( H ) = a ( a − ( F ) ∧ b − ( F )) ≤ a ( a − ( F )) ≤ F and similarly, b ( H ) ≤ F . Thus, a/ H = α and b/ H = β are local germs, and we have ϕ ◦ α = ϕ ◦ β because f ◦ a = f ◦ b . However, α (cid:54) = β , for, if H ∈ H , then there exist F a , F b ∈ F such that a − ( F a ) ∩ b − ( F b ) ⊆ H , and letting F = F a ∩ F b , we have a − ( F ) ∩ b − ( F ) ⊆ H .Then a F and b F ∈ H , so a | H (cid:54) = b | H . This proves α (cid:54) = β , and the contrapositive, that ϕ isnot monic. (cid:3) Isomorphisms.
As usual with factorization systems, the isomorphisms in
Fil are preciselythose arrows contained both in E Fil and in M Fil . (This follows from axioms [4, Section 2,(F3) and (F4)].) This allows us to characterize them:
Proposition 5.3.
An arrow ϕ ∈ Fil ( F , G ) is an isomorphism in Fil iff there is a partialfunction f representing ϕ such that f is one-one and f ( F ) = G . The partially-ordered sets F / M Fil . Recall [4, Section 2.2] that if we have a factorizationsystem (cid:104) E , M (cid:105) in a category C , c ∈ C , and arrows m and m (cid:48) with common codomain c ,then we say m ≤ m (cid:48) when there is a diagramdom m m (cid:27) (cid:27) f (cid:47) (cid:47) dom m (cid:48) m (cid:48) . (cid:3) (cid:3) c where f is an arrow making the diagram commutative. If f is an isomorphism, so that m ≤ m (cid:48) and m (cid:48) ≤ m , then we say that m and m (cid:48) are equivalent , or m ∼ m (cid:48) . The ≤ relation defines a preorder and the ∼ relation defines an equivalence relation; we denote thecorresponding partially-ordered set of equivalence classes (wrt ∼ ) by c/ M . Theorem 5.4.
Let
F ∈
Fil . Then F / M Fil is isomorphic to the complete lattice
Fil F . Proof.
The elements of F / M Fil are equivalence classes of arrows of
Fil with codomain F .Given a germ µ : G → F , we map µ to µ ( G ) ∈ Fil F . If we have a diagram G µ (cid:24) (cid:24) ϕ (cid:47) (cid:47) G (cid:48) µ (cid:48) . (cid:6) (cid:6) F (5.1)where ϕ is an isomorphism, then because µ = µ (cid:48) ◦ ϕ , µ ( G ) = µ (cid:48) ( ϕ ( G )). Now, ϕ , beingan isomorphism, is an arrow of E Fil , and by definition, this means that ϕ ( G ) = G (cid:48) . Thus, µ ( G ) = µ (cid:48) ( G (cid:48) ). In other words, µ ∼ µ (cid:48) implies µ ( G ) = µ (cid:48) ( G (cid:48) ). Thus we have defined amapping Z : F / M Fil → Fil F , which takes the equivalence class [ µ ] of an arrow µ : G → F to µ ( G ).Suppose now that we have the diagram 5.1, absent ϕ , but knowing that µ ( G ) ≤ µ (cid:48) ( G (cid:48) )(in the lattice Fil F ), and we will construct an arrow ϕ witnessing µ ≤ µ (cid:48) . Let m : G → S , m (cid:48) : G (cid:48) → S be admissible one-one partial functions representing the germs µ and µ (cid:48) ,respectively.Let f be the partial function ( m (cid:48) ) − ◦ m . We will show that f is a local, admissible partialfunction, such that ϕ = f / G completes Diagram 5.1.Let K (cid:48) ∈ G (cid:48) . We have K (cid:48) ∩ G (cid:48) ∈ G (cid:48) since G (cid:48) ∈ G (cid:48) . Then m (cid:48) ( K (cid:48) ∩ G (cid:48) ) ∈ m (cid:48) ( G (cid:48) ). Since m ( G ) ≤ m (cid:48) ( G (cid:48) ) by assumption, there is an L ∈ m ( G ) such that L ⊆ m (cid:48) ( K (cid:48) ∩ G (cid:48) ). There is a K ∈ G such that K ⊆ G = dd( m ) and K ⊆ L . Then, K ⊆ dd( f ) and f ( K ) ⊆ K (cid:48) . So, f islocal.Certainly, m | K = m (cid:48) ◦ f | K , implying that µ = µ (cid:48) ◦ ϕ . (cid:3) Properties of the Category
Fil
In this section, we will verify that the category
Fil satisfies the basic properties in the list[4, 3.1]. (A number of subsequent theorems in [4] will follow.)
Theorem 6.1. M
Fil is well-powered.Proof. If F ∈
Fil S , then by Theorem 5.4, F / M Fil ∼ = Fil F , which is a small set because S is a small set. (cid:3) Theorem 6.2. Fil has limits of all finite diagrams.Proof.
It suffices to show that there are equalizers, and products of finite tuples of objects.(Equalizers): Suppose α , β : F → G , where
F ∈
Fil S and G ∈
Fil T . Let (cid:104)H , µ (cid:105) be theequalizer and its arrow to F , if they exist, in the diagram H µ (cid:47) (cid:47) F α (cid:47) (cid:47) β (cid:47) (cid:47) G , II. THE CLOSED CATEGORY OF FILTERS 15 where we know that µ needs to be monic because that is always the case for an equalizer,and from our analysis of the factorization system (cid:104) E Fil , M Fil (cid:105) , that if the equalizer exists, wecan choose
H ∈
Fil F and µ the germ with respect to H of the identity function 1 S . Also, weknow that if {} ∈ G , then S = {} and µ is an isomorphism, but if not, then by Remark 4.8,there are total functions m , a , and b representing µ , α , and β respectively.We choose H = Fg { [ a = b ] | a/ F = α and b/ F = β } , where [ a = b ] stands for the set of x ∈ S on which the partial functions a and b are definedand a ( x ) = b ( x ). We have H ≤ F , because if F ∈ F , a/ F = α , and b/ F = β , then[ a | F = b | F ] ⊆ F .Suppose now that K is a filter, and γ ∈ Fil ( K , F ) is such that α ◦ γ = β ◦ γ . Let a , b , and g be admissible partial functions representing α , β , and γ respectively. We have [ a ◦ g = b ◦ g ] = K ∈ K . Let h : K → S be defined as h : x (cid:55)→ g ( x ). Clearly α ◦ ( h/ K ) = β ◦ ( h/ K ). We mustshow that h/ K is local. If F ∈ F , then F ⊇ [ a | F = b | F ] ∈ H . Let K (cid:48) = [( a | F ) g = ( b | F ) g ]; wehave K (cid:48) ∈ K and g ( K (cid:48) ) ⊆ F .The germ h/ K is unique, because, µ being one-one on its domain of definition, µ ∈ M Fil ,and by Theorem 5.1, then, µ is monic.(Products of finite tuples of filters): Given a tuple of filters F i , on sets S i , let S = Π i S i ,and let P = (cid:86) i π − i ( F i ), where the π i : S → S i are the projections. We claim that if thetuple F i is finite, then P is the product, with product cone the germs π i / P .Given a filter G (on a set G ), and germs ϕ i : G → F i , let f i : X i → S i be a partial functionrepresenting ϕ i , for each i , where X i ∈ G . We define X = (cid:84) i X i ∈ G (using the fact thatthe index set is finite) and we can define f : X → S , using the universal property of theproduct. It is clear that for each i , ( π i / P ) ◦ ( f / G ) = ϕ i . (cid:3) Theorem 6.3. Fil has pullbacks of small tuples of arrows in M Fil .Proof.
Let
F ∈
Fil S , and for each i in some small index set which we take to be an ordinalwithout loss of generality (as long as we assume the Axiom of Choice), let G i ∈ Fil and γ i ∈ M Fil ( G i , F ). We will prove that the diagram G γ (cid:24) (cid:24) G γ (cid:15) (cid:15) G γ (cid:6) (cid:6) · · ·F (6.1)has a pullback.By Theorem 5.4 and because the lattice Fil F is complete, the M Fil -subobjects γ i / M Fil have a meet, H . Let η : H → F be the germ 1 S / H . By Theorem 5.4, we can then draw the pullback diagram H ϕ (cid:5) (cid:5) ϕ (cid:15) (cid:15) ϕ (cid:25) (cid:25) · · ·G γ (cid:25) (cid:25) G γ (cid:15) (cid:15) G γ (cid:5) (cid:5) · · · (cid:83) F (6.2)where for all i , γ i ◦ ϕ i = 1 S / H , with the ϕ i coming from Theorem 5.4 and the fact that foreach i , H ≤ G i . (cid:3) Theorem 6.4. Fil has coproducts of all small tuples of arrows.Proof.
Let F i ∈ Fil S i , indexed by a small set which, without loss of generality, we take tobe an ordinal number. Let Z be the disjoint union of the sets S i , and for each i , let j i bethe insertion of S i into Z . Then, let (cid:87) i j i ( F i ) be the join over i of the filters j i ( F i ). Thejoin C = (cid:87) i j i ( F ) is the filter of subsets C ⊆ Z such that for all i , there is an F i ∈ F i with j i ( F i ) ⊆ C . Let ι i = j i / F i for each i .Thus we have the diagram F ι (cid:24) (cid:24) F ι (cid:15) (cid:15) F ι (cid:5) (cid:5) · · · ; C in the category Fil , and we will show that C is a coproduct of the tuple of F i . II. THE CLOSED CATEGORY OF FILTERS 17
Suppose we are given a filter
X ∈
Fil W , and a cocone of germs ξ i : F i → X , as shown inthe following diagram, and we will construct and prove uniqueness of the dotted arrow λ F ι (cid:25) (cid:25) ξ (cid:23) (cid:23) F ι (cid:15) (cid:15) ξ (cid:10) (cid:10) F ι (cid:2) (cid:2) ξ (cid:6) (cid:6) C λ (cid:15) (cid:15) · · · ; X such that for all i , ξ i = λ ◦ ι i , proving the universal property.For each i , let x i : F i → W be a partial function representing ξ i . Let C = (cid:83) i F i ⊆ Z . Let (cid:96) : C → W be the arrow given by the universal property of the disjoint union (in Set ). Thenlet λ = (cid:96)/ C . (cid:96) is an admissible partial function; we must show that it is local, that its germ λ satisfies ξ i = λ ◦ ι i for all i , and that if λ (cid:48) is any germ satisfying those equations, λ (cid:48) = λ .Each partial function x i is local, which means that if X ∈ X , there is an F (cid:48) i ∈ F i suchthat x i ( F (cid:48) i ) ⊆ X . It follows that (cid:96) (cid:48) , defined as the restriction of (cid:96) to C (cid:48) = (cid:83) i j i ( F (cid:48) i ), is local.Now suppose that we have any arrow λ (cid:48) : C → X such that for all i , ξ i = λ (cid:48) ◦ ι i . Let (cid:96) (cid:48) : C (cid:48) → W be an admissible, local partial function representing λ (cid:48) . For each i , let F (cid:48)(cid:48) i be the subset of S i such that (cid:96) i ◦ j i = (cid:96) (cid:48) ◦ j i on F (cid:48)(cid:48) i . We know that F (cid:48)(cid:48) i ∈ F i because λ ◦ ι i = ξ i = λ (cid:48) ◦ ι i . Then C (cid:48)(cid:48) ∈ C where C (cid:48)(cid:48) = (cid:83) i j i ( F (cid:48)(cid:48) i ), and (cid:96) = (cid:96) (cid:48) on C (cid:48) . This shows that λ = (cid:96)/ C = (cid:96) (cid:48) / C = λ (cid:48) . (cid:3) Theorem 6.5. M
Fil consists of monic arrows of
Fil .Proof.
More than that, by Theorems 5.1 and 5.2, arrows in
Fil are monic iff they belong to M Fil . (cid:3) Theorem 6.6. E
Fil is stable under pullbacks along arrows of
Fil .Proof.
Let ε ∈ E Fil ( F , G ), and let us pull it back along ϕ : H → G , giving P and ε (cid:48) ∈ Fil ( P , H ) which we want to prove is in E Fil .First, let us deal with the boundary case in which {} ∈ G . Then also {} ∈ F and ε is anisomorphism, whence the pullback ε (cid:48) is too, so that ε (cid:48) ∈ E Fil . Assuming, on the contrary, that {} / ∈ G , we can use Remark 4.8, and drawing the detaileddiagram P ε (cid:48) (cid:127) (cid:127) ψ (cid:32) (cid:32) H ϕ (cid:31) (cid:31) F , ε (cid:127) (cid:127) G we can assume that there are total functions f and e representing ϕ and ε , respectively. Wecan construct our pullback ε (cid:48) by first constructing the product H × F π (cid:123) (cid:123) π (cid:48) (cid:35) (cid:35) H F and then the equalizer of the arrows ϕ ◦ π , ε ◦ π (cid:48) : H × F → G : P λ (cid:47) (cid:47) H × F ϕ ◦ π (cid:47) (cid:47) ε ◦ π (cid:48) (cid:47) (cid:47) G , after which we can set ε (cid:48) = π ◦ λ and ψ = π (cid:48) ◦ λ .To show ε (cid:48) ∈ E Fil , we need to show that H = ε (cid:48) ( P ), or in other words, since we have ε (cid:48) ( P ) ≤ H just because ε (cid:48) is an arrow, that H ≤ ε (cid:48) ( P ). To show this, it will suffice toshow that for some admissible partial function e (cid:48) representing ε (cid:48) , and any P ∈ P , there isan H ∈ H such that H ⊆ e (cid:48) ( P ).Examining the proof of the existence of equalizers, we see that under our current assump-tion that {} / ∈ G , not only ϕ and ε , but also λ can be represented by a total function, andso can π and π (cid:48) from the proof of the existence of finite products. Thus, the partial function e (cid:48) representing ε (cid:48) can be assumed to be a total function.Let P ∈ P . Let F ∈ F , and H ∈ H , be such that f ( H ) ⊆ e ( F ). (This is possible because ε ∈ E Fil and f is local.) At the same time, let H and F be such that π − ( H ) ∩ ( π (cid:48) ) − ( F ) ⊆ P .Consider the diagram in Set , which is a pullback diagram: P (cid:48) = H × e ( F ) F π | P (cid:48) (cid:120) (cid:120) π (cid:48) | P (cid:48) (cid:38) (cid:38) H f | H (cid:38) (cid:38) F e | F (cid:120) (cid:120) e ( F ) II. THE CLOSED CATEGORY OF FILTERS 19 where we know very well that in
Set , since e maps F onto e ( F ), the function π also maps P (cid:48) onto H . But P (cid:48) ⊆ P . Since H can be made as small as desired in the filter H , this provesthat ε (cid:48) = π/ P ∈ E Fil . (cid:3) The Core of a Filter
Definition 7.1.
Let F be a filter of subsets of S . The subset (cid:84) F ∈F F ⊆ S is called the core of F and denoted by core F . If ϕ ∈ Fil ( F , G ), then core ϕ will denote the restriction of f tothe filter Fg { core F } ≤ F . Proposition 7.2.
The mapping S (cid:55)→ { S } , and the functor Γ sending f : S → S (cid:48) to f / { S } ,define a functor L : Set → LPartial . There are adjunctions (cid:104) L, core , η, ε (cid:105) : Set (cid:42)
LPartial where η S = 1 S and ε (cid:104) S, F(cid:105) is the inclusion of core F into S , and (cid:104) Γ ◦ L, core , α (cid:48) (cid:105) : Set (cid:42)
Fil , where Γ :
LPartial → Fil is the functor defined in Subsection 4.Proof.
To have these adjunctions, we must have isomorphisms χ S, G : LPartial ( L ( S ) , G ) ∼ = Set ( S, core G )and χ (cid:48) S, G : Fil (Γ( L ( S )) , G ) ∼ = Set ( S, core G )natural in S and G .Since L ( S ) = { S } , for L ( S ) be admissible, a partial function between values of the functor L must be a total function, and the functor Γ does nothing. Thus, the isomorphisms χ and χ (cid:48) simply relate total functions to total functions. Naturality in S and in G is straightforward. (cid:3) Remark 7.3.
Thus, if we decide to study objects in
Fil that have additional structure, suchas groups or other algebra structures, we have available right adjoint forgetful functors thatwill yield groups or other algebras. However, note that the forgetting that the core functordoes can be very extensive. For example, a filter can have an empty core, or a core withjust one element. Certainly we should expect to have more of interest to study in many suchcases than the trivial group. 8.
Monoidal Products F (cid:3) G . Suppose that F and G are filters of subsets of sets S and T , respectively. If F ∈ F and g : F → G (i.e., g is any function assigning a subset in the filter G to each element of F ), then we define F (cid:3) g = { (cid:104) s, t (cid:105) | s ∈ F, t ∈ g ( s ) } . More generally, if g : F → Sub S is any function such that { s | g ( s ) ∈ G } ∈ F , then we define F (cid:3) g just the same, as { (cid:104) s, t (cid:105) | s ∈ F, t ∈ g ( s ) } . Theorem 8.1.
We have(1) The set { F (cid:3) g | F ∈ F , g : F → G } is a base for a filter F (cid:3) G of subsets of S × T ;(2) the filter F (cid:3) G consists of those subsets H ⊆ S × T such that { s | { t ∈ T | (cid:104) s, t (cid:105) ∈ H } ∈ G } ∈ F ; (3) every subset in F (cid:3) G has the form S (cid:3) g for some g : S → Sub T .Proof. (1): Given F (cid:3) g and F (cid:3) g , we have F (cid:3) g ⊆ ( F (cid:3) g ) ∩ ( F (cid:3) g ), where F = F ∩ F and for s ∈ F , g ( s ) = g ( s ) ∩ g ( s ).(2): Given H satisfying the condition, let F = { s | { t ∈ T | (cid:104) s, t (cid:105) ∈ H } ∈ G } ∈ F . Foreach s ∈ F , let g ( s ) = { t | (cid:104) s, t (cid:105) ∈ H } ∈ G . Then F (cid:3) f ⊆ H , so H ∈ F (cid:3) G .On the other hand, if D (cid:3) f is an element of the base of F (cid:3) G , then it clearly satisfies thecondition. Then we need only see that the set of subsets satisfying the condition is closedupward.(3): Given H ∈ F (cid:3) G , let g ( s ) = { t ∈ T | (cid:104) s, t (cid:105) ∈ H } . Then H = S (cid:3) g . (cid:3) Part (2) of the Theorem suggests some notation we will use later:
Notation 8.1. If F is a filter of subsets of a set S , G is filter of subsets of a set T , and wehave a subset X ∈ F (cid:3) G , then we define(1) F F , G ,X = { s ∈ S | { t ∈ T | (cid:104) s, t (cid:105) ∈ X } ∈ G } and(2) h F , G ,X = [ s ∈ F F , G ,X (cid:55)→ { t ∈ T | (cid:104) s, t (cid:105) ∈ X } ]. Monoidal products in LPartial and Fil.Definition 8.2. If f ∈ LPartial ( F , F (cid:48) ) and g ∈ LPartial ( G , G (cid:48) ), where dd( f ) = F anddd( g ) = G , then we define f (cid:3) p g : F (cid:3) G → F (cid:48) (cid:3) G (cid:48) to be the partial function with domain ofdefinition F × G , sending a pair (cid:104) s, t (cid:105) to (cid:104) f ( s ) , g ( t ) (cid:105) . Theorem 8.3.
We have(1) The foregoing defines a functor (cid:3) p : LPartial × LPartial → LPartial .(2) If f ≡ F f (cid:48) and g ≡ G g (cid:48) , then ( f (cid:3) p g ) ≡ F (cid:3) G ( f (cid:48) (cid:3) p g (cid:48) ) .(3) Setting (cid:3) g = (cid:3) p on objects, and ( f / F ) (cid:3) g ( g/ G ) = ( f (cid:3) p g ) / F (cid:3) G on arrows, definesa functor (cid:3) g : Fil × Fil → Fil .(4) Γ( − (cid:3) p − ) = Γ( − ) (cid:3) g Γ( − ) : LPartial × LPartial → Fil .(5) core( − (cid:3) p − ) = core( − ) × core( − ) : LPartial × LPartial → Set .(6) core( − (cid:3) g − ) = core( − ) × core( − ) : Fil × Fil → Set .Proof. (1): Let f ∈ LPartial ( F , F (cid:48) ), and g ∈ LPartial ( G , G (cid:48) ). F × G ∈ F (cid:3) G , because F × G = F (cid:3) [ s ∈ F (cid:55)→ G ]. Thus, f (cid:3) g is admissible. To show f (cid:3) g is local, consider a baseelement F (cid:48) (cid:3) h (cid:48) ∈ F (cid:48) (cid:3) G (cid:48) , where F (cid:48) ∈ F (cid:48) and h (cid:48) : F (cid:48) → G (cid:48) . We have f − ( F (cid:48) ) ∈ F . Consider II. THE CLOSED CATEGORY OF FILTERS 21 now f − ( F (cid:48) ) (cid:3) h , where h : f − ( F (cid:48) ) → G is defined by setting h ( s ) = g − ( h (cid:48) ( f ( s ))). Then if (cid:104) s, t (cid:105) ∈ f − ( F (cid:48) ) (cid:3) h , we have ( f (cid:3) g ) (cid:104) s, t (cid:105) = (cid:104) f ( s ) , g ( t ) (cid:105) ∈ F (cid:48) (cid:3) h (cid:48) , because f ( s ) ∈ F (cid:48) and g ( t ) ∈ h (cid:48) ( f ( s )).(2): Suppose f | F = f (cid:48) | F and g | G = g (cid:48) | G , where F ∈ F and G ∈ G . Then if (cid:104) s, t (cid:105) ∈ F × G ,we have ( f (cid:3) p g ) (cid:104) s, t (cid:105) = ( f (cid:48) (cid:3) p g (cid:48) ) (cid:104) s, t (cid:105) . But F × G ∈ F (cid:3) G , so ( f (cid:3) p g ) ≡ F (cid:3) p G ( f (cid:48) (cid:3) g (cid:48) ).(3): Follows from (2).(4): Follows by the definition of (cid:3) g in (3).(5), (6): The core functor is a right adjoint functor, after all. (cid:3) Unit object, unit and associativity natural isomorphisms, and coherence. If S , T , and W are sets, let α S,T,W : S × ( T × W ) → ( S × T ) × W be the function defined by (cid:104) s, (cid:104) t, w (cid:105)(cid:105) → (cid:104)(cid:104) s, t (cid:105) , w (cid:105) . If D , D (cid:48) , and D (cid:48)(cid:48) are filters on S , T , and W respectively, we set α p D , D (cid:48) , D (cid:48)(cid:48) = α S,T,W , considered as a partial function. We set α g D , D (cid:48) , D (cid:48)(cid:48) = α S,T,W / ( D (cid:3) ( D (cid:48) (cid:3) D (cid:48)(cid:48) )). Theorem 8.4. α p D , D (cid:48) , D (cid:48)(cid:48) ∈ LPartial ( D (cid:3) ( D (cid:48) (cid:3) D (cid:48)(cid:48) ) , ( D (cid:3) D (cid:48) ) (cid:3) D (cid:48)(cid:48) ) and is an isomorphism.Similarly, its germ in Fil is an isomorphism.Proof. α p D , D (cid:48) , D (cid:48)(cid:48) is admissible because α S,T,W is a total function. To show it is an isomorphism,it suffices to show that both it and its inverse (in
Set ) are local.If X ∈ D (cid:3) ( D (cid:48) (cid:3) D (cid:48)(cid:48) ), then X = S (cid:3) f where f : S → Sub( T × W ). In this proof, we willdenote the subset X = S (cid:3) f of S × ( T × W ) by X [ f ].On the other hand, if Y ∈ ( D (cid:3) D (cid:48) ) (cid:3) D (cid:48)(cid:48) , then Y = Y [ h, k ] = ( S (cid:3) h ) (cid:3) k where h : S → Sub T and k : S × T → Sub W .To see that α − S,T,W (( D (cid:3) D (cid:48) ) (cid:3) D (cid:48)(cid:48) ) = D (cid:3) ( D (cid:48) (cid:3) D (cid:48)(cid:48) ), consider X [ f ], and we will find h and k such that α − S,T,W ( Y [ h, k ]) = X [ f ]. Let h : S → Sub T be defined by h : s (cid:55)→ p ( f ( s )), where p : T (cid:3) g → T is the projection to the first component of a pair. Let k : S (cid:3) h → Sub W be defined by k : (cid:104) s, t (cid:105) (cid:55)→ p ( f ( s )), where p : T (cid:3) g → Sub W projects a pair to its secondcomponent. We have (cid:104) s, (cid:104) t, w (cid:105)(cid:105) ∈ α − S,T,W ( Y [ h, k ]) ⇐⇒ α S,T,W (cid:104) s, (cid:104) t, w (cid:105)(cid:105) ∈ Y [ h, k ] ⇐⇒ (cid:104)(cid:104) s, t (cid:105) , w (cid:105) ∈ Y [ h, k ] ⇐⇒ s ∈ S and t ∈ h ( s ) and w ∈ k ( s, t ) ⇐⇒ s ∈ S and (cid:104) t, w (cid:105) ∈ h ( s ) (cid:3) k ( s, t ) ⇐⇒ s ∈ S and (cid:104) t, w (cid:105) ∈ f ( s ) ⇐⇒ (cid:104) s, (cid:104) t, w (cid:105)(cid:105) ∈ X [ f ];it follows that both α D , D (cid:48) , D (cid:48)(cid:48) and α − D , D (cid:48) , D (cid:48)(cid:48) are local.Applying the functor Γ, we obtain the corresponding statements about α g D , D (cid:48) , D (cid:48)(cid:48) . (cid:3) Notation 8.2.
Let u denote the filter { } = { { } } on the one-element set 1 = { } . If D is a filter on the set S , and λ S : 1 × S → S is the function defined by (cid:104) , s (cid:105) (cid:55)→ s ,then we define λ p D = λ S , considered as a partial function from u (cid:3) p D to D . We also define λ g D = λ p D / D (cid:3) u ∈ Fil ( u (cid:3) D , D ). Similarly, if (cid:37) S : S × → S is the function defined by (cid:104) s, (cid:105) (cid:55)→ s , then we define (cid:37) p (cid:104) S, D(cid:105) = (cid:37) S , considered as a partial function from D (cid:3) u to D , and (cid:37) g D = (cid:37) p D / D (cid:3) u ∈ Fil ( D (cid:3) u, D ).Note that u is a terminal object both in LPartial and in
Fil and that for any filter F ,the set core F is naturally isomorphic both to LPartial ( u, F ) and to Fil ( u, F ). Proposition 8.5.
These definitions yield natural isomorphisms α p : − (cid:3) p ( − (cid:3) p − ) ∼ = ( − (cid:3) p − ) (cid:3) p − , α g : − (cid:3) g ( − (cid:3) g − ) ∼ = ( − (cid:3) g − ) (cid:3) g − , λ p : u (cid:3) p − ∼ = − , λ g : u (cid:3) g − ∼ = − , (cid:37) p : − (cid:3) p u ∼ = − , and (cid:37) g : − (cid:3) g u ∼ = − making (cid:104) LPartial , (cid:3) p , u, α p , λ p , (cid:37) p (cid:105) and (cid:104) Fil , (cid:3) g , u, α g , λ g , (cid:37) g (cid:105) into monoidal categories [3,p. VII.1], [5, Definition 1.1], and the functors Γ :
LPartial → Fil , core p : LPartial → Set , and core :
Fil → Set are strict morphisms of monoidal categories [3, p. VII.1],[5,Definition 1.7]. The Closed Category Structure of
Fil
Let G be a filter of subsets of a set T , H be a filter of subsets of another set W , q ∈ LPartial ( G , G (cid:48) ), and r ∈ LPartial ( H (cid:48) , H ). Since Fil is a category, composition with γ = q/ G on the left induces a function Fil ( H , γ ) : Fil ( H , G ) → Fil ( H (cid:48) , G )and composition with γ (cid:48) = r/ H on the right produces a function Fil ( γ (cid:48) , G ) : Fil ( H , G ) → Fil ( H (cid:48) , G );and we have also Fil ( γ (cid:48) , γ ) = Fil ( γ (cid:48) , G ) ◦ Fil ( H , γ ) = Fil ( H , γ ) ◦ Fil ( γ (cid:48) , G ) : Fil ( H , G ) → Fil ( H (cid:48) , G (cid:48) ) , where hopefully, the reader will recognize easily that these functions are simply forms of theHom functor for the category Fil . We mention them to clarify our notation in what follows.
The internal Hom functor of Fil.
We will be defining the internal Hom functor for
Fil using, among other things, the mapping that takes an admissible (but not necessarily local)partial function to its germ. That is, if we have filters H (of subsets of W ) and G (of subsets of T ) then we can take the germ of an element of Partial ( H , G ) (See Notation 3.1, and Section 4where germs are defined), giving an arrow in Fil ( H , G ). However, formation of germs is usefulmore generally: If G ∈ G , then recall that we denote the set of admissible partial functions f : W → T , such that there is an H ∈ H such that f ( H ) ⊆ G , by Partial ( H , G , G ). If f is such a function, we can form the germ f / H and get an element of the set of germs of II. THE CLOSED CATEGORY OF FILTERS 23 elements of
Partial ( H , G , G ), which set we can denote by Γ( Partial ( H , G , G )) as mentionedin Definition 4.1. Proposition 9.1.
We have(1) The subsets
Fil ( H , G , G ) ⊆ Fil ( H , G , T ) , for G ∈ G , form a base for a filter G H ofsubsets of Fil ( H , G , T ) , and core G H = Fil ( H , G ) .(2) ( − ) ( − ) is a functor from Fil × Fil op to Fil .Proof. (1): clear.(2): Given G and H , other filters G (cid:48) and H (cid:48) , and germs γ ∈ Fil ( G , G (cid:48) ), ρ ∈ Fil ( H (cid:48) , H ),we must show that γ gives rise to an arrow γ H ∈ Fil ( G H , G (cid:48)H ), and ρ gives rise to an arrow G ρ : Fil ( G H , G H (cid:48) ).Let g be a partial function, admissible with respect to G and local with respect to G (cid:48) .(Every γ has such a representative by definition of Fil ( G , G (cid:48) ), so let us say that g is a repre-sentative of γ .) Composition with g on the left is a (total) function from Partial ( H , G , T )to Partial ( H , G (cid:48) , T (cid:48) ) (where G (cid:48) is a filter of subsets of the set T (cid:48) ). The germ of this totalfunction is admissible (the germ of a total function always is); to show it is local, considera basic set Partial ( H , G (cid:48) , G (cid:48) ) where G (cid:48) ∈ G (cid:48) . If G ∈ G is such that g ( G ) ⊆ G (cid:48) (such a G exists because g is local with respect to G (cid:48) ), then composition with g on the left maps thebasic set Partial ( H , G , G ) into Partial ( H , G (cid:48) , G (cid:48) ), as needed to show composition with g is a local function, and so, the germ of the composition function is local. Thus the arrow γ H : G H → G (cid:48)H .Now, let r : (cid:83) H (cid:48) → W be a partial function, admissible with respect to H (cid:48) , local withrespect to H , and representing ρ . Composition with r on the right is once again a totalfunction from Partial ( H , G , T ) to Partial ( H (cid:48) , G , T ), because if f ∈ Partial ( H , G , T ), thenit is admissible, and we have dd( f ) ∈ H . Then since r is local, we have dd( f ◦ r ) ∈H , so that f ◦ r ∈ Partial ( H (cid:48) , G , T ) – i.e., it is an admissible partial function and weconclude that composition with the germ, ρ , is total and admissible. For locality, supposenow that we have G ∈ G ; we want to show that there is a basic set in the filter G H thatwill map into Partial ( H (cid:48) , G , G ), and we will show that Partial ( H , G , G ) will serve. Given f ∈ Partial ( H , G , G ), we have f − ( G ) ∈ H . If we form f ◦ r , then since r is local, we seethat ( f ◦ r ) − ( G ) = r − (dd( r ) ∩ f − ( G ))) ∈ H (cid:48) , so f ◦ r ∈ Partial ( H (cid:48) , G , G ). Since we havenow shown that composition with r on the right is admissible and local, so is its germ; thus,the arrow G ρ : G H → G H (cid:48) .Clearly we have produced a functor ( − ) ( − ) : Fil × Fil op → Fil . (cid:3) For this next definition, we make use of Notation 8.1.
Definition 9.2.
Let H be a filter. For every pair of filters (cid:104)F , G(cid:105) , let χ HF , G : Fil ( F (cid:3) H , G ) → Fil ( F , G H ) be the total function mapping a germ κ to χ HF , G ( κ ) = [ s ∈ F F , G , dd( q ) (cid:55)→ [ w ∈ h F , G , dd( q ) ( s ) (cid:55)→ q ( s, w )] / H ] / F ∈
Fil ( F , G H )where q ∈ LPartial ( F (cid:3) H , G ) is a representative of the arrow (admissible, local germ) κ .Some propositions about this mapping, and Notation 8.1, that we will need later: Lemma 9.3.
Let F , ¯ F , G , ˜ G , H be filters of subsets of sets S , ¯ S , T . ˜ T , and W respectively,and let q ∈ LPartial ( F (cid:3) H , G ) , ¯ q ∈ LPartial ( F , ¯ F ) , ˜ q ∈ LPartial ( G , ˜ G ) , and ˆ q = q ◦ (¯ q (cid:3) p H ) . We have(1) F ¯ F , H , dd(ˆ q ) = ¯ q − ( F F , H , dd( q ) ) ;(2) if s ∈ F ¯ F , H , dd(ˆ q ) , then h ¯ F , H , dd(ˆ q ) ( s ) = h F , H , dd( q ) (¯ q ( s )) ;(3) F F , H , dd(˜ q ◦ q ) ⊆ F F , H , dd( q ) ; and(4) if s ∈ F F , H , dd(˜ q ◦ q ) , then h F , H , dd(˜ q ◦ q ) ( s ) ⊆ h F , H , dd( q ) ( s ) .Proof. (1): F ¯ F , H , dd(ˆ q ) = { s ∈ ¯ S | { w ∈ W | (cid:104) s, w (cid:105) ∈ dd(ˆ q ) } ∈ H } = { s ∈ ¯ S | { w ∈ W | (cid:104) s, w (cid:105) ∈ (dd(¯ q ) × W ) ∩ (¯ q (cid:3) p H ) − (dd( q )) } ∈ H } = { s ∈ dd(¯ q ) | { w ∈ W | (cid:104) s, w (cid:105) ∈ (¯ q (cid:3) p H ) − (dd( q )) } ∈ H } = { s ∈ dd(¯ q ) | { w ∈ W | (cid:104) ¯ q ( s ) , w (cid:105) ∈ dd( q ) } ∈ H } = ¯ q − ( { s ∈ S | { w ∈ W | (cid:104) s, w (cid:105) ∈ dd( q ) } ∈ H } )= ¯ q − ( F F , H , dd( q ) ) . (2): h ¯ F , H , dd(ˆ q ) ( s ) = { w ∈ W | (cid:104) s, w (cid:105) ∈ dd(ˆ q ) } = { w ∈ W | (cid:104) ¯ q ( s ) , w (cid:105) ∈ dd( q ) } = h F , H , dd( q ) (¯ q ( s )) . (3): F F , H , dd(˜ q ◦ q ) = { s ∈ S | { w ∈ W | (cid:104) s, w (cid:105) ∈ dd(˜ q ◦ q ) } ∈ H } = { s ∈ S | { w ∈ W | (cid:104) s, w (cid:105) ∈ dd( q ) ∩ q − (dd ˜ q ) } ∈ H }⊆ { s ∈ S | { w ∈ W | (cid:104) s, w (cid:105) ∈ dd( q ) } ∈ H } = F F , H , dd( q ) . (4): h F , H , dd(˜ q ◦ q ) ( s ) = { w ∈ W | (cid:104) s, w (cid:105) ∈ dd(˜ q ◦ q ) } = { w ∈ W | (cid:104) s, w (cid:105) ∈ dd( q ) ∩ q − (dd ˜ q ) }⊆ { w ∈ W | (cid:104) s, w (cid:105) ∈ dd( q ) } = h F , H , dd( q ) ( s ) . II. THE CLOSED CATEGORY OF FILTERS 25 (cid:3)
Remark 9.4.
Note that just as in our discussion of the natural transformation α , there isnot much mystery about where the partial functions we define send the elements in theirdomains. Theorem 9.5.
We have(1) χ HF , G is a well-defined, one-one, and onto function;(2) χ H is a natural isomorphism from the functor Fil ( − (cid:3) H , − ) : Fil op × Fil → Set tothe functor
Fil ( − , − H ) : Fil op × Fil → Set , resulting in an adjunction (cid:104)− (cid:3) H , − H , χ H (cid:105) : Fil (cid:42)
Fil ; (3) we have a nonsymmetric closed structure (cid:104) Fil , (cid:3) , u, α, λ, (cid:37), { χ H } H∈ Fil (cid:105) on the category
Fil , where u , α = α g , λ = λ g , and (cid:37) = (cid:37) g are defined as in Section 8.Proof. (1): If q : F (cid:3) h → T and q (cid:48) : F (cid:48) (cid:3) h (cid:48) are representatives of κ , then q and q (cid:48) agree onˆ F (cid:3) ˆ h , for some ˆ F ∈ F and ˆ h : ˆ F → H such that ˆ F (cid:3) ˆ h ⊆ F (cid:3) h ∩ F (cid:48) (cid:3) h (cid:48) . Then for every s ∈ ˆ F , h ( s ) ∩ h (cid:48) ( s ) ⊆ ˆ h ( s ), and we have[ w ∈ h ( s ) (cid:55)→ q ( s, w )] / H = [ w ∈ ˆ h ( s ) (cid:55)→ q ( s, w )] / H = [ w ∈ h (cid:48) ( s ) (cid:55)→ q (cid:48) ( s, w )] / H , proving that χ HF , G ( κ ) does not depend on the choice of q .On the other hand, if we have the same q and q (cid:48) , except that q (cid:54)≡ q (cid:48) , then for anyˆ F (cid:3) ˆ h ∈ F (cid:3) H with ˆ F ⊆ F ∩ F (cid:48) and ˆ h : ˆ F → H with ˆ h ( s ) ⊆ h ( s ) ∩ h (cid:48) ( s ) for s ∈ ˆ F , there isan ˆ s ∈ ˆ F such that [ w ∈ ˆ h ( s ) (cid:55)→ q ( s, w )] (cid:54)≡ [ w ∈ ˆ h ( s ) (cid:55)→ q ( s, w )], which means that there issome (cid:104) ˆ s, ˆ w (cid:105) ∈ ˆ F (cid:3) ˆ h such that q (ˆ s, ˆ w ) (cid:54) = q (cid:48) (ˆ s, ˆ w ), and it follows that [ w ∈ ˆ h (ˆ s ) (cid:55)→ q (ˆ s, w )] (cid:54)≡ [ w ∈ ˆ h (ˆ s ) (cid:55)→ q (cid:48) (ˆ s, w )] and χ HF , G ( q/ F ) (cid:54) = χ HF , G ( q (cid:48) / F ). Thus, χ HF , G is one-one. χ HF , G ( κ ) is admissible because it is the germ of a partial function with domain F . It islocal, because if X ∈ G H , there is a G ∈ G such that Γ( Partial ( H , G , G )) ⊆ X , and we willhave χ HF , G ( q | F × G ) ∈ X ;thus, χ XF , G : Fil ( F (cid:3) H , G →
Fil ( F , G H ).To show χ HF , G is onto, let us be given ρ ∈ Fil ( F , G H ), and an admissible, local partialfunction r : F → G H representing ρ , and define¯ χ HF , G ( ρ ) = [ (cid:104) s, w (cid:105) ∈ dd( r ) (cid:3) [ s (cid:55)→ dd( y ( s ))] (cid:55)→ y ( s )( w )] / ( F (cid:3) H )where for every s ∈ dd( r ), y ( s ) is some representative of the germ of admissible partialfunctions r ( s ) ∈ G H . Since we made choices here, this is a one-to-many relation. Ignoring for the moment that this is not a function, and just fixing our choices in defining ¯ χ HF , G ( ρ ),we see that χ HF , G (cid:0) ¯ χ HF , F ( ρ ) (cid:1) = χ HF , G ([ (cid:104) s, w (cid:105) ∈ dd( r ) (cid:3) [ s (cid:55)→ dd( y ( s ))] (cid:55)→ y ( s )( w )] / ( F (cid:3) H ))= [ s ∈ dd( r ) (cid:55)→ [ w ∈ dd( y ( s ))) (cid:55)→ y ( s )( w )] / H ] / F = [ s ∈ dd( r ) (cid:55)→ y ( s ) / H ] / F = [ s ∈ dd( r ) (cid:55)→ r ( s )] / F = r/ F = ρ, showing that χ HF , G is onto.(2): In order to show that χ HF , G is natural in F , it suffices to show that if in addition tohaving κ as we have assumed, we have ¯ κ ∈ Fil ( ¯ F , F ), where ¯ F is a filter of subsets of a set¯ S , then χ H ¯ F , G ( κ ◦ (¯ κ (cid:3) H )) = ( χ HF , G ( κ )) ◦ ¯ κ : ¯ F → G H ; (9.1)indeed, if we set ˆ q = q ◦ (¯ q (cid:3) p H ) : ¯ F (cid:3) H → G , we have χ H ¯ F , G ( κ ◦ (¯ κ (cid:3) H )) = χ H ¯ F , G (cid:0) q/ ( F (cid:3) H ) ◦ (¯ q (cid:3) p H ) / ( ¯ F (cid:3) H ) (cid:1) = χ H ¯ F , G (cid:0) ˆ q/ ¯ F (cid:3) H (cid:1) = [ s ∈ F ¯ F , H , dd(ˆ q ) (cid:55)→ [ w ∈ h ¯ F , H , dd(ˆ q ) ( s ) (cid:55)→ ˆ q ( s, w )] / H ] / ¯ F = [ s ∈ F ¯ F , H , dd(ˆ q ) (cid:55)→ [ w ∈ h ¯ F , H , dd(ˆ q ) ( s ) (cid:55)→ q (¯ q ( s ) , w )] / H ] / ¯ F = [ s ∈ F ¯ F , H , dd(ˆ q ) (cid:55)→ [ w ∈ h F , H , dd( q ) (¯ q ( s )) (cid:55)→ q (¯ q ( s ) , w )] / H ] / ¯ F = [ s ∈ ¯ q − ( F F , H , dd( q ) ) (cid:55)→ [ w ∈ h F , H , dd( q ) (¯ q ( s )) (cid:55)→ q (¯ q ( s ) , w )] / H ] / ¯ F ]= (cid:0) [ s ∈ F F , H , dd( q ) (cid:55)→ [ w ∈ h F , H , dd( q ) ( s ) (cid:55)→ q ( s, w )] / H ] ◦ ¯ q (cid:1) / ¯ F = (cid:0) [ s ∈ F F , H , dd( q ) (cid:55)→ [ w ∈ h F , H , dd( q ) ( s ) (cid:55)→ q ( s, w )] / H ] / F (cid:1) ◦ (¯ q/ ¯ F )= ( χ HF , G ( κ )) ◦ ¯ κ where we use Lemma 9.3(1) and (2).To show that χ HF , G is natural in G , we must show that χ HF , ˜ G (˜ κ ◦ κ ) = ˜ κ H ◦ ( χ HF , G ( κ )) : F → ˜ G H , (9.2)for any ˜ q/ G = ˜ κ : G → ˜ G , where ˜ κ H is the usual shorthand for ˜ κ H . Considering the twosides of Equation 9.2, we have II. THE CLOSED CATEGORY OF FILTERS 27 χ HF , ˜ G (ˆ κ ◦ κ ) = χ HF , G (˜ q/ G ◦ q/ ( F (cid:3) H ))= χ HF , G ((˜ q ◦ q ) / ( F (cid:3) H ))= [ s ∈ F F , H , dd(˜ q ◦ q ) (cid:55)→ [ w ∈ h F , H , dd(˜ q ◦ q ) ( s ) (cid:55)→ ˜ q ( q ( s, w ))] / H ] / F = [ s ∈ F F , H , dd(˜ q ◦ q ) (cid:55)→ [ w ∈ h F , H , dd( q ) ( s ) (cid:55)→ ˜ q ( q ( s, w ))] / H ] / F = [ s ∈ F F , H , dd( q ) (cid:55)→ [ w ∈ h F , H , dd( q ) ( s ) (cid:55)→ ˜ q ( q ( s, w ))] / H ] / F = (˜ q/ G ) H ◦ [ s ∈ F F , H , dd( q ) (cid:55)→ [ w ∈ h F , H , dd( q ) ( s ) (cid:55)→ q ( s, w )] / H ] / F = (˜ q/ G ) H ◦ (cid:0) χ HF , G ( q/ ( F (cid:3) H )) (cid:1) = ˜ κ H ◦ ( χ HF , G ( κ )) , using Lemma 9.3(3) and (4), which force germs with respect to F and H to be the same. (cid:3) Remark 9.6.
Note that each component η HF : F → ( F (cid:3) H ) H of the unit natural transfor-mation η H is the germ of the function sending s ∈ S to the germ of the function sending w ∈ W to (cid:104) s, w (cid:105) . Each component ε HG : G H (cid:3) H → G of the counit natural transformation ε H is the germ of the function sending (cid:104) q/ ( G H (cid:3) H ) , w (cid:105) to q ( w ) for w ∈ dd( q ).10. Applications
Uniform spaces on filters.
In [6], we discuss in detail the theory of uniform spaces withan underlying filter , instead of an underlyiing set . It should not come as a shock that there isa close relationship between
Fil and uniform spaces, when we consider that a uniformity ona set is defined as a filter of entourages.
Fil has a factorization system (Section 5), and weshow in [4] that a certain list of properties of a category with factorization system supporta theory of generalized equivalences, of which uniformities on a set are an example. Theseare the properties of
Fil (and its factorization system) that we proved in Section 6.Thus, we consider in [6] the category of uniform spaces on filters, which we continue todenote by
Unif because it is so natural to do so. A uniformity on a filter is a small gener-alization from a uniformity on a set, but one which manifests interesting new phenomena,especially when completion is considered. If we have a uniformity on a filter, the points ofthe filter that are not in the core play an interesting role. They are not closed points, andcan never be limits of a cauchy filter, but, there can be cauchy filters consisting entirely ofnon-core points. These cauchy filters give rise to new elements when we apply the functorof (hausdorff) completion, C : Unif → Unif .This becomes especially important when we try to define a function space and make thecategory of uniform spaces on filters into a closed category. Indeed this is possible, if we admitthe possibility of a nonsymmetric closed category. The elements of the function space aregerms of admissible partial functions, without regard to being local or uniformly continuous, which properties hold only for germs in the core of the function space object . However, thesenon-core germs can give rise to new arrows when we complete the hom-objects and, usingthe fact[5, Section 1] that completion is a monoidal functor , form the category ˇ C ( Unif ),as defined in [5, Section 3]. The new arrows include an inverse to the unit natural arrowfrom any uniform space into its completion, so that it becomes possible to work with theassumption that all spaces are complete.
Ind [ Fil ] , a base category for Topological Algebra. In [4], we introduce the theoryof
Ind [ Fil ], a cartesian-closed category suitable for the study of Topological Algebra. Inparticular, for varieties of algebras that are congruence-modular,
Ind [ Fil ] has propertiesthat allow us to generalize Day’s Theorem[1], something that is not possible[7] in
Unif . Anyhausdorff, compactly-generated topological algebra can be made into an object of
Ind [ Fil ],and a factorization system (Section 5) in that category allows us to associate with that object,a structure lattice analogous to the congruence lattice or lattice of compatible uniformities onthat algebra, which will be modular if the algebra belongs to a congruence-modular variety.
References [1] Alan Day. “A characterization of modularity for congruence lattices of algebras”. In:
Canad. Math. Bull.
12 (1969), pp. 167–173.[2] V. Koubek and J. Reiterman. “On the category of filters”. In:
Commentationes Mathe-maticae Universitatis Carolinae
11 (1970), pp. 19–29.[3] Saunders MacLane.
Categories for the Working Mathematician, Second Edition . Vol. 5.Graduate Texts in Mathematics. Springer-Verlag, 1998.[4] William H. Rowan.
Elements of Topological Algebra I. Generalized Equivalences . Draft.Nov. 2020. url : .[5] William H. Rowan. Elements of Topological Algebra II. Nonsymmetric Closed Categories .Nov. 2020. url : .[6] William H. Rowan. “Elements of Topological Algebra IV. Uniform Spaces on Filters”.(In preparation).[7] Hans Weber. “On lattices of uniformities”. In: Order
24 (2007), pp. 249–276.
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Email address : [email protected] And, the core of the function space coincides with the hom-set in the category