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Is the category JTF a topos?
Fuzzy Sets and Systems 27 (1988) 11-19 Nogth-HoHand
11
| S T H E C A T E G O R Y ~T~ A T O P O S ? J. COULON and LL. COULON Univevs~ Claude Bernard (Lyon I), O~partemem de Ma~h~nuuiques 2 ° Cycle, 69622 Villeurbanne, France Received June 1.985 G. Blanc [1] has intJn~uced the concept of totally fuzzy set, ~hich has been devek~ed by D. Higgs [4] whose category ~(J) of J-valued sets (J is a complete He~ng algebra) is a topos. D. Ponasse has studied another category of totally fuzzy sets: ~T~ [5]. We have constn~cted an equivalence between ~(J) and a full subcategory of ~ in [6]. T~~ purpose of the foHo~ng work is to show that YrF, which is equivalent to the category of sepa~ted, presheaves introduced in [2], is not a topos, except if J is an anti-ordinal.
Keywords: Totally fuzzy set, J-valued set, Separated presheaf, Topos.
Let J be a complete Heytin$ algebra. JTF is the category defined as fo~ows: * An object is a J-valued set (or totally fuzzy set) (X, o) where: X is a set, o a map from X z to J such that
a(x, y) = a(y, x),
a(x, y) ^ a(y, z) ~ a(x, z);
we denote t~o(x) = o(x, x). . A morphism from (X, o) to (Y, ~) is a relatien R from X to F such that
Vx, Vx' ¢X YY VY° ¢ Y
(x Ry and x' Ry'):~,J(x, x') ~ ~0,, y'), . (x R y and eo(X) ~ ~'(y, y')) =~x R y'.
, Composition of morphisms: if R is a morphism from (X, o) to (Y, ~') and S is a mo~hism from (Y, r) to (Z, p), the molphism SR from (X, o) to (Z, p) is defined by: x SR z ¢:~ :~y ~ Y, 3 z ' e Z such that: x R y, y S z' and or~,(x) ~ p(z, z').
, Identity morphism: x Ix'; ¢~ oo(X) = o ( x , x ' ) Vx c X , Vx' ~X.
1. Cat~o~ D e S e r t . J T ~ is the full subcategory of ~ fuzzy sets (X, o) such that:
who~,e objects are the totally
0165-0114f~/$3.50 ~) 1988, Elsevier Science Publishers B.V. (North-Holland)
J. Coulon, J.L. Coulon:
12
O) o(x, y ) = O~o(X)ffi #o(y)=>x f y; (ii) if i ~< C~o(X), there exists an element y of X such that o(y, x) = ~ro(y) ffi i.
Remark. If (X, o) is an object of TrF °°, the u~ique element y of X satisfying (ii) is denoted by: x/~. ~pm~Wmn 1. The objects of JTF °° are precisely separated presheaves.
~ e f , Let (X, o) be an object of ,rI'F~ . Vx ~ x, put E(x) = C~o(X) and Vx ¢ X, Vi eJ, x I i = x/a^~¢x). Then: x 1 F.(x)-X/.ot~)=x,
E(x) ^ i, x I i 1 j--x 1 i/j^=o(~l~)-x I il~^j^~o(~-xls^.o(.),,^,^~..~ E ( x I ~) = ~ o ( x / , ^ . o ~ )
= i ^ ~o(X) =
but it is easy to see that if j <~i <~OCo(X), (x/~)/i •x/j. So
x I i 1 j = x/,^j^=o<~)= x 1 (i ^ j) ~md (X, o) is a presheaf over J. On the other hand,
o(xl,,
y) ~
ao(xl,) = i,
o(x, y) ~> o(x, xi,) ^ a(xl,, y) = i ^ o(xl,, y) = o(x/,, y), So o(x/,, y ) = i ^ o(x, y) Vx EX, y EX, i ~<=o(X). Then if j <-..=°(x) ^ OLo(y) and ff x/j =yli = z, ~°(z) ---j= j ^ or(x, y) and so j ~< ~r(x, y). Put k = o(x,y), zl = x/k,'z2 = Y/k. Then c , ( ~ , z2) = k ^
o(x,
y ) = k = ~o(Z~) =
~o(Z~)
and (i) :~ z~ = z2. So, we h~ve
o(x, y) = Max{j ~ OCo(X)^ =°(y) such t h a t x / j f y / ~ } and the presheaf (X, o) is separated. Conversely if X is a separated presheaf over J, it is clear that (X, =7) where o(x, y ) = [ x =Yl (see [2]) is an object of JTF °°. .P~pe~fion 2. The categories JTF and J T ~ are equivalent. The proof of this proposition is a consequence of the three following iemmas: L i m a 1. Any obje ~ of JTF is isomorphic (in JTF) to an object of ~ITF sallying (i).
~'eef. We introduce on X the equivaL rice x=- yC~a(x, y ) = oto(x)= ~o(y).
Is the category JTF a topos?
13
Put 5(= X / . and let 2 ~ the class of the element x. We remark that ff x ~ y and x' my ' then o(x, x') = o(y, y ' ) s o we can put @(2, 2') = e(x, x'). It is easy m see that (X; O) is an object of ~ satisfying (i). We define a morphism R from (X, a) to (,~, #) by xR)~ @ ~ # ) =
~(x,y).
We define a morphism S from (~', O) to (X, ~) by:
~Sy ~ ~o(X)= ,~(~, y). It is easy to prove that
xSRx' o x l x ' ¢~ ~o(X)fo(x,x'), ~s~' <~ ~ ' ~ ,~-,,(z) = 0(~, z'). so (X, o) and (X', 0) are isomorphic in 3TF.
l~wma 2. Any object of yrF is isomorphic (in STF) to an object of YFF satis~ing
(ii).
~f.
Let (X', a) be an object of YrF. Put
,~= xeYd' U {(i,x)) and O((i, x), (j, y)) = i ^ j ,', o{,x, y). (,~', O) is an object of 3"FF and =gi, x ) ) = i ^ ¢ o ( X ) = i . If j<~i then ~((j, x)) = j = ~((i, x), q, x)). So (~) is satis~ed. If x e X and (j, y) e X, put x R (j, y) ¢:~ ~o(X) f j A a(x, y). Then R is a morphism from (X, o) to (X, ~).
(~,,(x), x)~ Rx,
~o(X)=j ^ o(x, y),
~(x')=j'
^ ~(x', y')
implies o(x, x ' ) < =o(X) ^ ==(x')=j ^ j' ^ o(x, V) ^ a(x', y'). So
a(x, x') = a(x, x') ^ j ^ j' ^ a(x, y) ^ a(x', y') <~j ^ j' ^ a(y, y') = a((j, y), (j', y')). If =°(x) = j ^ e(x, y) and if COo(X)<~j ^ j' ^ a(y, y'), then =o(X) >~a(x, y')>~j~' ^ a(x, y')~>j ^ j ' ^ e(y, y') ^ e(x, y) >~=o(X) and so ~ro(X)~ j' ^ o(x, y')). If (j, y) ¢ X and if x ~ X, put (j, y) S x <=~ j ~
j ^ j' ^ o~:y,y') <-.a(=, y) ^ o(x', y') ^ =(y, y') < a(x, x'). If j ~ a(x, y) and j ~< e(x, x'), then
j < a(x, y) ^ o(=, =') ~ o(=', y')
J. Coulon, LL, ~ulon
14
so (j, y) Sx' and then S isa morphism from (~', O) to (X, o). x S R z ' ¢~ x l x ' ~ ~ o ( x ) f o(x,x'). For, we have: x SI~ x' ¢~ there exist (L Y) ¢ ~" and x" ¢ X such that Oto(X)ffi j ^ o(x, y), j ~ a(x", y) So
and
¢o(X) <<.a(x', x").
o(x, x') ~ o(x, y) ^ o(y, x") ^ o(x", x') > o(x, y) ^ / ^ ~o(X) ffi ~o(X).
Then x SR x' ~ o(x, x') = O~o(X). Conversely, if o(x,x')=eo(x) put (j, y)=(~Co(X),x) and x"ffix'. We then have x SR x'. (j, y) RS (j', y') ~ (j, y) ~ (j', Y') ~:~ j =j A j' A o(Y, Y') ¢e~ j ~ j ' ^ ¢~(y, y'). For (j, y) RS (j', y') ~ there exist x ¢ X and (k, z) ¢,~" vuch that j ~< a(x, y), to(x) = k ^ o(x, z) and j ~ j' ^ k ^ ~7(y', z). So
j' ^ o(y, y') ~ j ' ^ k ^ or(y, z) ^ ~(z, y') ~ j ^ k ^ o(y, z) ~ . . . ~ j ^/c ^ ~(y, x) ^ ~(x, z) = a~o(X) ^ ~(x, y) ^ j = a(x, y ) ^ j--'j. Conversely if j<<.j' ^ o ( y , y ' ) , . w e put x f z f y , kffi~o(y) and we have (j, y) RS (j', y'). So (X, a) and (X, O) are isomorphic in YIT.
~ma
3. If (X, o) sa~fies (ii) then (~', if) satisfies (ii).
The proof is obvious. We can then conclude that (X, a) is isomorphic (in YFF) to (J~, ~) which satisfies (i) and (ii) and that the categories JTF are ~ are equiva|ent. Notation. If (X, a) is an object of JTF °° (or of ,ITF) we ~hal| write Xj = (x E x i ,~o(~)--- i}.
P~pc~Uon 3. Let (X, a) and (Y, T) be two objects of YTF°°. If R is a morphism from (x, a) to (Y, T) then: (1) R induces a family (Ri)i~j of mappings Ri. Xi--* Yi such th,~t Vx ¢ Xi, Vi ¢ Y, Vj <~i,
Rj(x/,) - Ri(.:)/j.
(N)
(Re)ies is called the natural family of R. (2) If (Ra)iEj b a family oS nuzps from X~ to Y~satisfying (N) there exists a unique morphism R from (X, o) to (]; r) whose natural family is (Ri)iej, which is defined by x R y ~ e'o(x) ffi T[R,(x), y] if x cX,. Proof° Let R be a rnorphism l~rom (X, a) to (Y, ~). If x ¢ Xi, Rx ~ ~. Let y be an element of Rx. Then ~o(X)= i ~<~,(y) =j. (Y, ~') satisfies (ii), so y/i exists. $irlee ~o(x)<~ T(y, Y/i) we have x R y/i. Suppose now that there exist y and z in ~ such that: x R y and x R z. Then • (y, z) ~>~ ( x ) ffi i and T(y, z) <~ o%(y) = i. So i ffi ~(y, z) = o~(y) = ~,(z) and since (Y, T) satisfies, (i), y = z. We have proved that there exists an unique
Is ~ category J T F
a
topos?
15
element of Y~which is in relation with x. We shall denote it by R~(x)and so the maps (R~)~ are defined. Suppose that x ¢ X~ and that ] ~
So
j--" v[R](xb), R~{x)/il--" ~,(Rj(x]j))--- ot,[R,(x)]j]
and (i) implies that R#(x/~)--R~(x)/~. Conversely, suppose that (Ri)~ is a family of maps from X~ to Y~ satisfying(N). W e define R by x R y ¢# ~,,(x) = z [ ~ ( x ) , y].
Then Rx ~ ~ because if x e X , x R R~(x). Suppose that we have x R y and x' R y ' with xeX~ and x ' ¢ X e . o(x, x') = io ~ i ^ i'. Then R,(x)l,o = R,o(xl,,) = R,o(x'l,J =
Put
R,.(x')l,o.
So io < z[R~(x),Rr(x')] and
T(y, y') ~ ~(y, R,(x)) ^ ~(R,(x), R,.(x')) ^ ~(R,.(x'), y') >~~,,(x) ^ o(x, x') ^ oc,,(x') = o(x, x'). Suppose that we have x R y and ¢o(X) ~ ~(y, y') w;th x ~ Xi. Then
So v[Ri(x), y ' ] = Olo(X) and x R y ' . Thez~ R is a morl: ~,,'Ism from (X, _a) tO (Y, ~).
If (Si)~EIis itsnatural family and ifx e."X~, then S~(:~ isthe unique element of Y~ such that x R S~(x). But we have x R R,~(x) so Si(x) = ~'~(x). If S is an other morphism from (3~, o) to (Y, v) associated with the family (Ri)~Ej and if x Sy, then R~(x) = y/~ (ff x ¢ X~). Hence T[R~(x), y] = ac,(y) ^ i = i = ~o(X) and we have x R y. If we have x R y, then x S Rj(x) and Oto(X) = z(y, Ri(x)) so we also have x S y and R = S.
Remark. The morphisms of JTF °° are precisely the morphisms of presheaves defined in [2]. It is a direct consequence of Proposition 3.
2. Prop¢~¢s of $ ~ F~oposifion 4. The category J T ~
is finitely complete.
• Terminal object: (J, ^ ) where ^ (i, j) = i ^ j.
J, Coulon, J.l.,,Coulon
16
* Finite products: (X, o) x (Y, 1:) -- (p, p) where
,((x. y). (x'. y'))= o(x. x') ^ ~(y. y'); the projections from (p, p) to (X, a) and (Y, v) are the morphisms whose natural families are respectively (P~)~o and (q~)~o, P~ and q~ being the projections from X~ × Yj to Xi and Y,. * Finite pullbacks: Consider the following diagram in JTFO°:
(x. o)
(z, e)
..4'.
(Y, ¢) Put
Q , = ( ( x , y ) ~ p , suchthatA,(x)=Bi(y)},
Q=[_JQ,
k = PlQxQ.
It is easy to see that (Q, k) is an object of J T ~ . lfp~ =PdQ, and q~ =qdQ, and if /7' and q' are the morphisms associated with (P~)go and (q[)~,x, then
(x.o)
(Q, x)
(z, e) is a pullback in Yl]~ . (Y, ~)
* Equaliser of a pair {R, S}: Let (.t", o) :~s (Y' ~) be a diagram in JTF °°. Put, for any i e J, Z~ = {x ~ X~ [ R~(x) = Sa(x)}; E~ is the canonical injection from Z~ into X~. If (Z, p) is defined by Z -- U ~ Z~; p = olzxz, it is an object of 3"1"~. The family (Ej)~E~satisfies (N) and if E is the associated morpl~sm, the RE = SE. It is obvious to see that [(Z, p), E] is the equaliser of the pair {R, $}. I[~|fion
5. The category JTF °° has exponentiation.
l ~ b | e m . Let (X, or) and (Y, ~) be two objects of Yl'F°°. We are going to construct an object (Z, p) of YI'F°°, a morphism E from (X, o) x (Z, p ) t o (Y, T) such that, for any object (U, ~) of yFF °° and for any morphism F from (X, a) x (U, p) to (Y, lr) there exists a unique morphism rFI :(U, !~)--~ (Z, p) making the following diagram commutative:
(x, a) x (z, p) ~ l(x.,,)
× rF11! t
(x, ~) x (u, p)
(Y, =)
Is the category J ~
a
~opo~?
17
(a) Construc. on of (g, p). For any i e J, we put Z~ -- {[i, (~)j~i]} where if / ~ i, .~ is a map from Xs to Yj such that if k ~ j andif x eXj then fj(X)/k---fk(X/Ic). Put g---[,.)ie:Zi and define p: Zz---~J by We can see that (Z, p) is an object of YrF °° and we have, if il ~ i,
[i, (~)j.,]/,, (b) Construction
= [i,, (~)j~,].
orE.
Ei'X~ × Zs~ Y~by:' ifx ¢ Xs a n d f = [i, (~)j~t] ¢ Zi then Ei(x, f) =fi(x).
For any i e
J, w e d e f i n e
We can see that (E~)~e: satisfies (N), so there is a morphLsm E from (X, o) × (Z, p) to (Y, ~) whose natural family is (E~)iE:. (c) Let (U, ~) be an object of ~ and let F (whose natural family is (F/)i~s) be a morphism from (X, o) × (U, ~) to (Y, ~). We will construct a morphism rF]: (U, ~u)--*(Z, p): - if u ¢ U~, we put IF l~(u) = [i, (~)j.~,] where j~ is the map from Xj to Yj defined by f~(x) = Fj(x, u/j) if x ¢ Xs and for any j ~
Ee
lXiX FI~,. ! .._
x
commutes, so the diagram (D), (X, c~)x (Z, p) ~
(Y, ~)
!
l(x,o ~ X rfl : !
is also commutative. Finally, [F 1 is the unique morphism: (U, [~)--~ (Z, p) such that D is conunutath,°e. The proof uses the diagrams (D~)~:. 3. Subob]ect d ~ m e r of JTF ~ D e f ~ o n . A complete Heyfing algebra J is an anti-ordinal if any non-empty subset of J has a greatest element. In particular, an anti-ordinal Y is a chain. We have proved in [6] that if J is an anti-ordinal, the categories YrF and ~(J)
18
I. Coulon, J.L. Coulon
(category of 1-sets of Higgs) are equivalent..So, ff J is an anti-ordinal, .r]'F is a topos. It is the same thing to re,~ark that ff I is an anti-ordinal every separated presheaf over J is a sheaf over Jr. ,
But we can now say: ~ r o l m d ~ = 6. If J is not an an~-ord~l, T I ~
is not a topos.
ProM. If I is not an anti-ordinal, there exists a non-empty subset K of J which has no greatest element. Put K = (~t)t=,- and i = Vt~,- ~t, i ~ K. If £rF o° is a topos, there is a subobject classifier (9./, c) in YrF e°. Let T:(J, ^)--~(~[, c) be the tree-morphism. For any I e L , put (~t=[0, ~t] and
¢ = Ut,,. Ct. Put ¢ = [0, i]. ((~, ^) and ((~, ^) are objects of JTFe° and there exists a canonical monomorphism from (~;, A) into ((~, A). So there is a unique morphism Z from ((~, A) to (~, C) such that the following diagram (D) is a pu~back:
,(J, ^) (~,
^)
(a, c)
(~, ^) (D) is commutative. So we have Vl ~ L, X=,,(at) = T=,(ott). Put X,(i) -- a,. Then
a,l~, =
x,(i)/~, = x=,(~t) = r=,(~,) =
So at = c(ad=,, T~(i)/=,)= =t ^ = V,.
c(a,,
~(0/=,
T~(i)) for any l ¢ L and
^
=
^
and then i ~ c(a,, r~(O) -< i, and by (i), a, = r,(O. The diagrm (E)
(1,^) (~, ^)
(~,c) (~ ^)
is commutative and, since (D) is a puUback, there exists a morphism R from (~, ^) to ((~, ^) such that
(~, A) ~
(¢, A)
~ ~ ~,
(fir C)
~(¢, A)/f
Is the category JTF a topos?
19
is commutative. But the existence of R has no sense because ~i = {i} ~ ~ and So J T ~ has no subobject classifier.
References [I] G. Blanc-, Pr~faisceaux et ensembles floes, Unpublished manuscript. [2] M.P. Fourman and D.S. Scott, Sheaves and logic, Lecture Notes in Mathematics No. 753 (Spt~ager, Berlin, 1977) 302-401. [3] R. Goldblatt, Topoi, the Categorical Analysis of Logic, Studies in Logic and the Foundations of Mathematics Vol. 98 (North-Holland, Amsterdam, 1979). [4] D. Higss, [njectivity in the topos of complete Hey~ng algebra valued sets, Canad. J. Math. 36 0984) 550-568. [5] D. Ponasse, Remarques sur les ensembles totalement floes, S6minalre de Math6matique Roue de PUniversit6 de Lyon I (1982-1983). [6] $. Cordon and $.L. Coelon, Remerques set certaines categories d'ensembles totalement floes, BUSEFAL (Hirer 1985) 11-30.
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| S T H E C A T E G O R Y ~T~ A T O P O S ? J. COULON and LL. COULON Univevs~ Claude Bernard (Lyon I), O~partemem de Ma~h~nuuiques 2 ° Cycle, 69622 Villeurbanne, France Received June 1.985 G. Blanc [1] has intJn~uced the concept of totally fuzzy set, ~hich has been devek~ed by D. Higgs [4] whose category ~(J) of J-valued sets (J is a complete He~ng algebra) is a topos. D. Ponasse has studied another category of totally fuzzy sets: ~T~ [5]. We have constn~cted an equivalence between ~(J) and a full subcategory of ~ in [6]. T~~ purpose of the foHo~ng work is to show that YrF, which is equivalent to the category of sepa~ted, presheaves introduced in [2], is not a topos, except if J is an anti-ordinal.
Keywords: Totally fuzzy set, J-valued set, Separated presheaf, Topos.
Let J be a complete Heytin$ algebra. JTF is the category defined as fo~ows: * An object is a J-valued set (or totally fuzzy set) (X, o) where: X is a set, o a map from X z to J such that
a(x, y) = a(y, x),
a(x, y) ^ a(y, z) ~ a(x, z);
we denote t~o(x) = o(x, x). . A morphism from (X, o) to (Y, ~) is a relatien R from X to F such that
Vx, Vx' ¢X YY VY° ¢ Y
(x Ry and x' Ry'):~,J(x, x') ~ ~0,, y'), . (x R y and eo(X) ~ ~'(y, y')) =~x R y'.
, Composition of morphisms: if R is a morphism from (X, o) to (Y, ~') and S is a mo~hism from (Y, r) to (Z, p), the molphism SR from (X, o) to (Z, p) is defined by: x SR z ¢:~ :~y ~ Y, 3 z ' e Z such that: x R y, y S z' and or~,(x) ~ p(z, z').
, Identity morphism: x Ix'; ¢~ oo(X) = o ( x , x ' ) Vx c X , Vx' ~X.
1. Cat~o~ D e S e r t . J T ~ is the full subcategory of ~ fuzzy sets (X, o) such that:
who~,e objects are the totally
0165-0114f~/$3.50 ~) 1988, Elsevier Science Publishers B.V. (North-Holland)
J. Coulon, J.L. Coulon:
12
O) o(x, y ) = O~o(X)ffi #o(y)=>x f y; (ii) if i ~< C~o(X), there exists an element y of X such that o(y, x) = ~ro(y) ffi i.
Remark. If (X, o) is an object of TrF °°, the u~ique element y of X satisfying (ii) is denoted by: x/~. ~pm~Wmn 1. The objects of JTF °° are precisely separated presheaves.
~ e f , Let (X, o) be an object of ,rI'F~ . Vx ~ x, put E(x) = C~o(X) and Vx ¢ X, Vi eJ, x I i = x/a^~¢x). Then: x 1 F.(x)-X/.ot~)=x,
E(x) ^ i, x I i 1 j--x 1 i/j^=o(~l~)-x I il~^j^~o(~-xls^.o(.),,^,^~..~ E ( x I ~) = ~ o ( x / , ^ . o ~ )
= i ^ ~o(X) =
but it is easy to see that if j <~i <~OCo(X), (x/~)/i •x/j. So
x I i 1 j = x/,^j^=o<~)= x 1 (i ^ j) ~md (X, o) is a presheaf over J. On the other hand,
o(xl,,
y) ~
ao(xl,) = i,
o(x, y) ~> o(x, xi,) ^ a(xl,, y) = i ^ o(xl,, y) = o(x/,, y), So o(x/,, y ) = i ^ o(x, y) Vx EX, y EX, i ~<=o(X). Then if j <-..=°(x) ^ OLo(y) and ff x/j =yli = z, ~°(z) ---j= j ^ or(x, y) and so j ~< ~r(x, y). Put k = o(x,y), zl = x/k,'z2 = Y/k. Then c , ( ~ , z2) = k ^
o(x,
y ) = k = ~o(Z~) =
~o(Z~)
and (i) :~ z~ = z2. So, we h~ve
o(x, y) = Max{j ~ OCo(X)^ =°(y) such t h a t x / j f y / ~ } and the presheaf (X, o) is separated. Conversely if X is a separated presheaf over J, it is clear that (X, =7) where o(x, y ) = [ x =Yl (see [2]) is an object of JTF °°. .P~pe~fion 2. The categories JTF and J T ~ are equivalent. The proof of this proposition is a consequence of the three following iemmas: L i m a 1. Any obje ~ of JTF is isomorphic (in JTF) to an object of ~ITF sallying (i).
~'eef. We introduce on X the equivaL rice x=- yC~a(x, y ) = oto(x)= ~o(y).
Is the category JTF a topos?
13
Put 5(= X / . and let 2 ~ the class of the element x. We remark that ff x ~ y and x' my ' then o(x, x') = o(y, y ' ) s o we can put @(2, 2') = e(x, x'). It is easy m see that (X; O) is an object of ~ satisfying (i). We define a morphism R from (X, a) to (,~, #) by xR)~ @ ~ # ) =
~(x,y).
We define a morphism S from (~', O) to (X, ~) by:
~Sy ~ ~o(X)= ,~(~, y). It is easy to prove that
xSRx' o x l x ' ¢~ ~o(X)fo(x,x'), ~s~' <~ ~ ' ~ ,~-,,(z) = 0(~, z'). so (X, o) and (X', 0) are isomorphic in 3TF.
l~wma 2. Any object of yrF is isomorphic (in STF) to an object of YFF satis~ing
(ii).
~f.
Let (X', a) be an object of YrF. Put
,~= xeYd' U {(i,x)) and O((i, x), (j, y)) = i ^ j ,', o{,x, y). (,~', O) is an object of 3"FF and =gi, x ) ) = i ^ ¢ o ( X ) = i . If j<~i then ~((j, x)) = j = ~((i, x), q, x)). So (~) is satis~ed. If x e X and (j, y) e X, put x R (j, y) ¢:~ ~o(X) f j A a(x, y). Then R is a morphism from (X, o) to (X, ~).
(~,,(x), x)~ Rx,
~o(X)=j ^ o(x, y),
~(x')=j'
^ ~(x', y')
implies o(x, x ' ) < =o(X) ^ ==(x')=j ^ j' ^ o(x, V) ^ a(x', y'). So
a(x, x') = a(x, x') ^ j ^ j' ^ a(x, y) ^ a(x', y') <~j ^ j' ^ a(y, y') = a((j, y), (j', y')). If =°(x) = j ^ e(x, y) and if COo(X)<~j ^ j' ^ a(y, y'), then =o(X) >~a(x, y')>~j~' ^ a(x, y')~>j ^ j ' ^ e(y, y') ^ e(x, y) >~=o(X) and so ~ro(X)~ j' ^ o(x, y')). If (j, y) ¢ X and if x ~ X, put (j, y) S x <=~ j ~
j ^ j' ^ o~:y,y') <-.a(=, y) ^ o(x', y') ^ =(y, y') < a(x, x'). If j ~ a(x, y) and j ~< e(x, x'), then
j < a(x, y) ^ o(=, =') ~ o(=', y')
J. Coulon, LL, ~ulon
14
so (j, y) Sx' and then S isa morphism from (~', O) to (X, o). x S R z ' ¢~ x l x ' ~ ~ o ( x ) f o(x,x'). For, we have: x SI~ x' ¢~ there exist (L Y) ¢ ~" and x" ¢ X such that Oto(X)ffi j ^ o(x, y), j ~ a(x", y) So
and
¢o(X) <<.a(x', x").
o(x, x') ~ o(x, y) ^ o(y, x") ^ o(x", x') > o(x, y) ^ / ^ ~o(X) ffi ~o(X).
Then x SR x' ~ o(x, x') = O~o(X). Conversely, if o(x,x')=eo(x) put (j, y)=(~Co(X),x) and x"ffix'. We then have x SR x'. (j, y) RS (j', y') ~ (j, y) ~ (j', Y') ~:~ j =j A j' A o(Y, Y') ¢e~ j ~ j ' ^ ¢~(y, y'). For (j, y) RS (j', y') ~ there exist x ¢ X and (k, z) ¢,~" vuch that j ~< a(x, y), to(x) = k ^ o(x, z) and j ~ j' ^ k ^ ~7(y', z). So
j' ^ o(y, y') ~ j ' ^ k ^ or(y, z) ^ ~(z, y') ~ j ^ k ^ o(y, z) ~ . . . ~ j ^/c ^ ~(y, x) ^ ~(x, z) = a~o(X) ^ ~(x, y) ^ j = a(x, y ) ^ j--'j. Conversely if j<<.j' ^ o ( y , y ' ) , . w e put x f z f y , kffi~o(y) and we have (j, y) RS (j', y'). So (X, a) and (X, O) are isomorphic in YIT.
~ma
3. If (X, o) sa~fies (ii) then (~', if) satisfies (ii).
The proof is obvious. We can then conclude that (X, a) is isomorphic (in YFF) to (J~, ~) which satisfies (i) and (ii) and that the categories JTF are ~ are equiva|ent. Notation. If (X, a) is an object of JTF °° (or of ,ITF) we ~hal| write Xj = (x E x i ,~o(~)--- i}.
P~pc~Uon 3. Let (X, a) and (Y, T) be two objects of YTF°°. If R is a morphism from (x, a) to (Y, T) then: (1) R induces a family (Ri)i~j of mappings Ri. Xi--* Yi such th,~t Vx ¢ Xi, Vi ¢ Y, Vj <~i,
Rj(x/,) - Ri(.:)/j.
(N)
(Re)ies is called the natural family of R. (2) If (Ra)iEj b a family oS nuzps from X~ to Y~satisfying (N) there exists a unique morphism R from (X, o) to (]; r) whose natural family is (Ri)iej, which is defined by x R y ~ e'o(x) ffi T[R,(x), y] if x cX,. Proof° Let R be a rnorphism l~rom (X, a) to (Y, ~). If x ¢ Xi, Rx ~ ~. Let y be an element of Rx. Then ~o(X)= i ~<~,(y) =j. (Y, ~') satisfies (ii), so y/i exists. $irlee ~o(x)<~ T(y, Y/i) we have x R y/i. Suppose now that there exist y and z in ~ such that: x R y and x R z. Then • (y, z) ~>~ ( x ) ffi i and T(y, z) <~ o%(y) = i. So i ffi ~(y, z) = o~(y) = ~,(z) and since (Y, T) satisfies, (i), y = z. We have proved that there exists an unique
Is ~ category J T F
a
topos?
15
element of Y~which is in relation with x. We shall denote it by R~(x)and so the maps (R~)~ are defined. Suppose that x ¢ X~ and that ] ~
So
j--" v[R](xb), R~{x)/il--" ~,(Rj(x]j))--- ot,[R,(x)]j]
and (i) implies that R#(x/~)--R~(x)/~. Conversely, suppose that (Ri)~ is a family of maps from X~ to Y~ satisfying(N). W e define R by x R y ¢# ~,,(x) = z [ ~ ( x ) , y].
Then Rx ~ ~ because if x e X , x R R~(x). Suppose that we have x R y and x' R y ' with xeX~ and x ' ¢ X e . o(x, x') = io ~ i ^ i'. Then R,(x)l,o = R,o(xl,,) = R,o(x'l,J =
Put
R,.(x')l,o.
So io < z[R~(x),Rr(x')] and
T(y, y') ~ ~(y, R,(x)) ^ ~(R,(x), R,.(x')) ^ ~(R,.(x'), y') >~~,,(x) ^ o(x, x') ^ oc,,(x') = o(x, x'). Suppose that we have x R y and ¢o(X) ~ ~(y, y') w;th x ~ Xi. Then
So v[Ri(x), y ' ] = Olo(X) and x R y ' . Thez~ R is a morl: ~,,'Ism from (X, _a) tO (Y, ~).
If (Si)~EIis itsnatural family and ifx e."X~, then S~(:~ isthe unique element of Y~ such that x R S~(x). But we have x R R,~(x) so Si(x) = ~'~(x). If S is an other morphism from (3~, o) to (Y, v) associated with the family (Ri)~Ej and if x Sy, then R~(x) = y/~ (ff x ¢ X~). Hence T[R~(x), y] = ac,(y) ^ i = i = ~o(X) and we have x R y. If we have x R y, then x S Rj(x) and Oto(X) = z(y, Ri(x)) so we also have x S y and R = S.
Remark. The morphisms of JTF °° are precisely the morphisms of presheaves defined in [2]. It is a direct consequence of Proposition 3.
2. Prop¢~¢s of $ ~ F~oposifion 4. The category J T ~
is finitely complete.
• Terminal object: (J, ^ ) where ^ (i, j) = i ^ j.
J, Coulon, J.l.,,Coulon
16
* Finite products: (X, o) x (Y, 1:) -- (p, p) where
,((x. y). (x'. y'))= o(x. x') ^ ~(y. y'); the projections from (p, p) to (X, a) and (Y, v) are the morphisms whose natural families are respectively (P~)~o and (q~)~o, P~ and q~ being the projections from X~ × Yj to Xi and Y,. * Finite pullbacks: Consider the following diagram in JTFO°:
(x. o)
(z, e)
..4'.
(Y, ¢) Put
Q , = ( ( x , y ) ~ p , suchthatA,(x)=Bi(y)},
Q=[_JQ,
k = PlQxQ.
It is easy to see that (Q, k) is an object of J T ~ . lfp~ =PdQ, and q~ =qdQ, and if /7' and q' are the morphisms associated with (P~)go and (q[)~,x, then
(x.o)
(Q, x)
(z, e) is a pullback in Yl]~ . (Y, ~)
* Equaliser of a pair {R, S}: Let (.t", o) :~s (Y' ~) be a diagram in JTF °°. Put, for any i e J, Z~ = {x ~ X~ [ R~(x) = Sa(x)}; E~ is the canonical injection from Z~ into X~. If (Z, p) is defined by Z -- U ~ Z~; p = olzxz, it is an object of 3"1"~. The family (Ej)~E~satisfies (N) and if E is the associated morpl~sm, the RE = SE. It is obvious to see that [(Z, p), E] is the equaliser of the pair {R, $}. I[~|fion
5. The category JTF °° has exponentiation.
l ~ b | e m . Let (X, or) and (Y, ~) be two objects of Yl'F°°. We are going to construct an object (Z, p) of YI'F°°, a morphism E from (X, o) x (Z, p ) t o (Y, T) such that, for any object (U, ~) of yFF °° and for any morphism F from (X, a) x (U, p) to (Y, lr) there exists a unique morphism rFI :(U, !~)--~ (Z, p) making the following diagram commutative:
(x, a) x (z, p) ~ l(x.,,)
× rF11! t
(x, ~) x (u, p)
(Y, =)
Is the category J ~
a
~opo~?
17
(a) Construc. on of (g, p). For any i e J, we put Z~ -- {[i, (~)j~i]} where if / ~ i, .~ is a map from Xs to Yj such that if k ~ j andif x eXj then fj(X)/k---fk(X/Ic). Put g---[,.)ie:Zi and define p: Zz---~J by We can see that (Z, p) is an object of YrF °° and we have, if il ~ i,
[i, (~)j.,]/,, (b) Construction
= [i,, (~)j~,].
orE.
Ei'X~ × Zs~ Y~by:' ifx ¢ Xs a n d f = [i, (~)j~t] ¢ Zi then Ei(x, f) =fi(x).
For any i e
J, w e d e f i n e
We can see that (E~)~e: satisfies (N), so there is a morphLsm E from (X, o) × (Z, p) to (Y, ~) whose natural family is (E~)iE:. (c) Let (U, ~) be an object of ~ and let F (whose natural family is (F/)i~s) be a morphism from (X, o) × (U, ~) to (Y, ~). We will construct a morphism rF]: (U, ~u)--*(Z, p): - if u ¢ U~, we put IF l~(u) = [i, (~)j.~,] where j~ is the map from Xj to Yj defined by f~(x) = Fj(x, u/j) if x ¢ Xs and for any j ~
Ee
lXiX FI~,. ! .._
x
commutes, so the diagram (D), (X, c~)x (Z, p) ~
(Y, ~)
!
l(x,o ~ X rfl : !
is also commutative. Finally, [F 1 is the unique morphism: (U, [~)--~ (Z, p) such that D is conunutath,°e. The proof uses the diagrams (D~)~:. 3. Subob]ect d ~ m e r of JTF ~ D e f ~ o n . A complete Heyfing algebra J is an anti-ordinal if any non-empty subset of J has a greatest element. In particular, an anti-ordinal Y is a chain. We have proved in [6] that if J is an anti-ordinal, the categories YrF and ~(J)
18
I. Coulon, J.L. Coulon
(category of 1-sets of Higgs) are equivalent..So, ff J is an anti-ordinal, .r]'F is a topos. It is the same thing to re,~ark that ff I is an anti-ordinal every separated presheaf over J is a sheaf over Jr. ,
But we can now say: ~ r o l m d ~ = 6. If J is not an an~-ord~l, T I ~
is not a topos.
ProM. If I is not an anti-ordinal, there exists a non-empty subset K of J which has no greatest element. Put K = (~t)t=,- and i = Vt~,- ~t, i ~ K. If £rF o° is a topos, there is a subobject classifier (9./, c) in YrF e°. Let T:(J, ^)--~(~[, c) be the tree-morphism. For any I e L , put (~t=[0, ~t] and
¢ = Ut,,. Ct. Put ¢ = [0, i]. ((~, ^) and ((~, ^) are objects of JTFe° and there exists a canonical monomorphism from (~;, A) into ((~, A). So there is a unique morphism Z from ((~, A) to (~, C) such that the following diagram (D) is a pu~back:
,(J, ^) (~,
^)
(a, c)
(~, ^) (D) is commutative. So we have Vl ~ L, X=,,(at) = T=,(ott). Put X,(i) -- a,. Then
a,l~, =
x,(i)/~, = x=,(~t) = r=,(~,) =
So at = c(ad=,, T~(i)/=,)= =t ^ = V,.
c(a,,
~(0/=,
T~(i)) for any l ¢ L and
^
=
^
and then i ~ c(a,, r~(O) -< i, and by (i), a, = r,(O. The diagrm (E)
(1,^) (~, ^)
(~,c) (~ ^)
is commutative and, since (D) is a puUback, there exists a morphism R from (~, ^) to ((~, ^) such that
(~, A) ~
(¢, A)
~ ~ ~,
(fir C)
~(¢, A)/f
Is the category JTF a topos?
19
is commutative. But the existence of R has no sense because ~i = {i} ~ ~ and So J T ~ has no subobject classifier.
References [I] G. Blanc-, Pr~faisceaux et ensembles floes, Unpublished manuscript. [2] M.P. Fourman and D.S. Scott, Sheaves and logic, Lecture Notes in Mathematics No. 753 (Spt~ager, Berlin, 1977) 302-401. [3] R. Goldblatt, Topoi, the Categorical Analysis of Logic, Studies in Logic and the Foundations of Mathematics Vol. 98 (North-Holland, Amsterdam, 1979). [4] D. Higss, [njectivity in the topos of complete Hey~ng algebra valued sets, Canad. J. Math. 36 0984) 550-568. [5] D. Ponasse, Remarques sur les ensembles totalement floes, S6minalre de Math6matique Roue de PUniversit6 de Lyon I (1982-1983). [6] $. Cordon and $.L. Coelon, Remerques set certaines categories d'ensembles totalement floes, BUSEFAL (Hirer 1985) 11-30.