Elements of monoidal topology Lecture 7: Kleisli monoids
Sergejs Solovjovs
Department of Mathematics, Faculty of Engineering, Czech University of Life Sciences Prague (CZU) Kamýcká 129, 16500 Prague - Suchdol, Czech Republic
Abstract
This lecture considers an alternative representation of the category (T, 2)-Cat as the category T-Mon of monoids in the hom-set of a Kleisli category that avoids explicit use of relations or lax extensions. As a motivating example serves an isomorphism F-Mon = Top, where F is the filter monad on the category Set.
1. A representation of topological spaces through neighborhood filters
1.1. The filter monad on the category Set
Definition 1. The filter monad F = (F, m, e) on the category Set of sets and maps is given by
F F f
(1) a functor Set —> Set, where FX = {y |y is a filter on A} for every set X, and FX —> FY is defined by = {B <ZY\ /_1(-B) G ?} for every map X A Y;
(2) a natural transformation lset      F, where X —^» FX is defined by ex(x) = x = {A C X \ x e A} (principal filter);
(3) a natural transformation FF —> F, where FFX mx> FX is defined by mx(%) = SI (filtered sum or Kowalsky sum), where A e XX iff {y e FX \ A e y} € X. ■
Remark 2. There exists the contravariant powerset functor Setop -> Set defined by P'(X —> Y) =
PY —> FX, where PX and PY are the powersets of the sets X and Y, respectively, and P'(B) = f~x(B) = {x G X | f(x) G B} for every subset B CY. The functor P' is self-adjoint, namely, there exists an adjoint situation (P')op H P* : Setop —> Set. This adjoint situation provides the double-powerset monad P2 = (P'(P')op,m,e) on the category Set. Both the filter monad F and the ultrafilter monad (3 (recall Lecture 1) are restrictions of the above double-powerset monad P2. ■
Definition 3. Given a set X, the set FX of filters on X can be partially ordered by the refinement partial order, i.e., for every y, tj € FX, y ^ t) iff y ~D t) (namely, given a subset A C X, if A e t), then A E f). A filter y is finer than 1) (or t) is coarser than y) provided that y D 1). ■
1.2. The Kleisli category of a monad
Definition 4. Given a monad T = (T,m,e) on a category X, the Kleisli category Xj associated to T is defined as follows. The objects of Xj are those of X. Given two Xj-objects X, Y, the hom-set Xj(A, Y) is
the hom-set X(A, TY) (the elements of which will be denoted X Y). Given two Xj-morphisms A Y, Y ^ Z, their Kleisli composition in Xj is defined via the composition in X as go / = - Tg- f, i.e., as the X-morphism A A TY^TTZ ^ TZ. The identity on an XT-object A is the A-morphism A-^>TA. .
Email address: solovjovs@tf.czu.cz (Sergejs Solovjovs) URL: http://home.czu.cz/solovjovs (Sergejs Solovjovs)
Preprint submitted to the Czech University of Life Sciences Prague (CZU)
February 23, 2022
Example 5. Given the powerset monad P = (P,m,e) (recall Lecture 1), a Setp-morphism X Y is a map X —> PY, which can be considered as a relation X —I—y Y defined by xr y iff y G /(x). Given two
Setp-morphisms X Y and Y Z, the Kleisli composition g o f is the composition s ■ r of the relations corresponding to g and /, respectively. Indeed, if t is the relation corresponding to gof, then for every x G X and every zGZ, xtziffzGgo f(x) iff z G mz ■ Pg ■ /(x) = mz(Pg ■ /(x)) = (J Pg ■ f(x) = (J Pg(f(x)) = [Jy£f(x) g(y) iff there exists y G /(x) such that z G g(y) iff there exists y E Y such that y G /(x) and z G g(y) iff there exists y E Y such that x r y and y s z iff x (s ■ r) z. It follows that Setp = Rel. ■
Remark 6. Given a Kleisli category Xj, there exists a functor Xj X, Gt(X y) = TX my T^> TY = TX       TTY ^ TY. The functor GT has a left adjoint X ^> XT, Fj(X A y) = X Y =
X -A y —^> Ty. The unit lx -^-> GjFt of this adjunction is e, and the co-unit FjGj lxT is given by X-morphisms TX 1tx> TX. The monad associated to this adjunction gives back the original monad T. ■
Remark 7. Given sets X and Y, the hom-set Setp(X,y) is partially ordered by the pointwise refinement partial order of Definition 3, namely, given f,g G Setp(X, y), f ^ g iff /(x) ^ ff(x) for every x G X (recall
/
that both / and g are maps X       ] FY). Therefore, the partially ordered set Setp(X, Y) can be considered
a
as a category, the objects of which are the elements of Setp(X, Y), and, for every two objects / and g, there exists precisely one morphism / —> g provided that / ^ g. ■
Lemma 8. A partially ordered set S, considered as a category as in Remark 1, is a strict monoidal category (recall Lecture 4-) precisely when it has a monoid structure whose multiplication SxS^-S is monotone.
Proposition 9. (Setp(X, X), o, ex) is a strict monoidal category.
Proof. In view of Lemma 8, it will be enough to show that the Kleisli composition preserves the refinement partial order. Thus, given ffi, 32,/l,/2 G Setp(X, X) such that g\ ^ g2 and /1 ^ /2, one has to show that 9i 0 fi ^ 92 0 J2, which is equivalent to mx ■ Fgi ■ /i(x) ^ mx ■ Fg2 ■ /2(x) for every x G X, which, in its turn, is equivalent to m_j • Fgi ■ /i(x) 3 mx ' Fg2 ' f2(x) f°r every x G X.
Take an arbitrary element x G X. Given A G mx ■ Fgi ■ /2(x), by Definition 1(3), it follows that {y G FX I A G y} G Fg2 ■ /2(x) = Fg2(f2(x)), which implies by Definition 1 (1) that g^dt e FX \ A G y}) G /2(x). Since /1 < /2, it follows that /i(x) D /2(x), and then fi^d? "= FX | A G y}) G /i(x). Further, if y G g^dl e FX I A G y}), then ff2(y) G {y G FX | A G y}, namely, A G ff2(y) C ffl(y) since g! ^ g2. Thus, A G ffi(y) implies gi(y) G {y G FX \ A G y}, which gives y G 5l1^1({y G FX \ A G y}). As a consequence, one obtains that ^({y G FX | A G y}) C g~ 1({jc e FX\A e y}). Since ^({y e FX \ A G y}) G /i(x) and /i(x) is a filter, it follows that g± 1({y G FX | A G y}) G /i(x), which implies {y G FX | A G y} G Fgi • /i(x), which finally gives A G mx • Fgi • /i(x). Therefore, mx • Fgi ■ fi(x) D mx ■ Fg2 ■ /2(x) as desired. □
1.3. Kleisli triples
Definition 10. A Kleisli triple on a category X consists of the following data:
T
• a function Ox —> Ox, which sends X to TX;
• an extension operation (—)T, which sends an X-morphism X —> TY to an X-morphism TX —> TY;
• an X-morphism X —^> TX for every X-object X; such that
(ffT-/)T=ffT-/T,   4 = 1tx,   /T-ex = /
for every X-object X and every X-morphisms X ^> Ty, Y A TZ. If one defines g o / = gT • /, then the above conditions are equivalent to this "Kleisli composition" being associative, and ex being its identity,
2
namely, given X-morphisms X A TY, Y A TZ, and Z \ TW, h o (g o f) = h o (gJ ■ f) = hJ ■ gJ ■ f = (hJ ■ g)J ■ f = (hJ ■ g) o / = (h o g) o /, / o ex = fJ ■ ex = f, and eY o / = e\ ■ f = lTY ■ f = f.
Definition 11. A Kleisli triple morphism (S, (—)s,d) —> (T, (—)T,e) is given by a family of X-morphisms SX       TX for every X-object X such that
aY■fs = {ay ■ f)J ■ ax,   ax ■ dx = ex
f
for every X-morphism X —> SY. Observe that a Kleisli triple morphism is a family of X-morphisms preserving the Kleisli composition and its identity, i.e., given X-morphisms X A- SY and Y A SZ, ®z ■ (g °S /) = «z • 9S ■ f = {a-z ■ g)J ■ oly ■ f = (az ■ g) oT (aY • /). ■
Remark 12. A Kleisli triple (T, (—)T, e) on a category X provides a monad T = (T, m, e) on X by setting
Tf = (eY ■ f)J for every X-morphism X —>• Y, and = (1tx)j for every X-object X. A Kleisli triple morphism (S, (—)s, rf)      (T, (—)T, e) provides then a morphism of the corresponding monads S     J. ■
Remark 13. Given a monad T = (T, to, e) on a category X, one gets a Kleisli triple (T, (—)T, e) by setting
/T = TOy • Tf for every X-morphism X ^> TY. A monad morphism S —> J provides then a morphism (S, (—)s, rf) ^> (T, (—)T, e) of the corresponding Kleisli triples. ■
Remark 14. The above passages from a Kleisli triple to a monad and from a monad to a Kleisli triple are inverse to each other, namely, both definitions describe the same structure on a category X (and the respective two definitions of Kleisli composition then correspond). ■
1.4- Monoids in monoidal categories
Definition 15. Let C be a monoidal category (see Lecture 4). A monoid M in C is a C-object together with two C-morphisms M ® M —> M and £AM such that the following two diagrams commute:
M <g> (M (8) M) M <%>M —
(M<E)M)(E)M-
->M <g>M
-e^lM    )M®M>,    lM®e-M®E
A homomorphism of monoids (M, to, e) —> (N, n, d) is a C-morphism M —> N such that the following two diagrams commute:
M <g> M ——-4- N <g> N
E
M ■
-+N
M ■
Monc stands for the category of monoids in C and their homomorphisms. Example 16.
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(1) The category Set is monoidal w.r.t. cartesian product of sets. The category Monset is exactly the category Mon of monoids and their homoniorphisnis in the sense of universal algebra.
(2) The category Sup of \/-seniilattices and \/-preserving maps is monoidal w.r.t. the usual tensor product. The category Quant of quantales and their homomorphisms is exactly the category Monsup-
(3) Given a partially ordered set (S, considered as a monoidal category (S,^,<E), k) as in Lemma 8, a monoid in S is an element s£S such that s<S> s ^ s and k ^ s. Observe that if s is a monoid in S, then k ^ s implies s = s<E>k^s<E>s. Since s <E> s ^ s, it follows that s <E> s = s. ■
1.5. Topological spaces via neighborhood filters
Theorem 17. The category Top of topological spaces and continuous maps is (concretely) isomorphic to the category F-Mon, the objects of which are pairs (X, v) such that X A FX is a monoid in Setf(X, X)
(i.e., vov <i v and ex ^v), and whose morphisms (X, v) —> (Y, /i) are maps X —> Y such that f^ov <i [lof^ where f\Sl = ey ■ f is the image of the map f under the left adjoint functor Set —> Setp of Remark 6.
Proof. Given a topological space (X, t), where t is a topology on a set X, define a map X A- FX by v{x) = {A C X | there exists U E t such that x E U C A} (neighborhood filter of x). It is easy to see that v(x) is contained in the principal filter ex{x) = & for every x G X. Therefore, ex ^ v in the pointwise refinement partial order. To show that v o v ^ v, we will need the following simple lemma.
Lemma 18. For x G X and A C X, A G v(x) iff there exists B G v(x) such that A G v(y) for every y G B.
Proof.
If A G v{x), then there exists U G t such that x G [/ C A.  Put B = U and notice that, first, S G v{x) and, second, A G v(y) for every y E B since U E t.
<=: Given y G B, it follows that A G v{y), i.e., there exists Vy G t such that y E Vy C A. Thus, ■B C UyeB Vy — A, which implies A G v{x), since S G v{x) and        is a filter. □
Given an element x E X and a subset A C X, define a set AF = {y G FX |i e f} (the set of filters containing A). Then 4ei/o iff A G mx • Fi/ • iff A¥ E Fv ■ v(x) = Fv(v(x)) iff v^1(A¥) E v(x) iff there exists B E v{x) such that B C z/_1(J4F) iff there exists S G v{x) such that G A¥ for every y E B iff there exists S G such that A E v(y) for every y E B iff (Lemma 18) A E v(x). As a consequence, one obtains v o        = By Example 16 (3) and the above two properties (v o v <i v and ex ^ v), a
topological space (X,t) provides a monoid v in the monoidal category Setf(X,X).
f
Consider a continuous map (X, t) —> (Y, a), and let v and /i be the monoids corresponding to the spaces (X,t) and (Y,/z), respectively. First, we show that F/ • v ^ M ' /■ Indeed, given x E X and B C Y, B E /J, ■ f(x) = n(f(x)) iff there exists y G a such that /(x) G V C S, which implies (since / is continuous) /_1(y) G t and x G f~x(V) C f~x(B), which results in f~x(B) E v{x), which is equivalent to B E Ff(v(x)) = Ff ■ v(x). As a consequence, one gets Ff ■ v(x) D fi ■ f(x) or F/ • v(x) ^ /i • /(x). Second, since F = (F, to, e) is a monad on Set, the following two diagrams commute:
FY-A FFY Y-^—> FY
FY.
Thus, Ff ■ v = 1 fy ■ Ff ■ v = my ■ Fey ■ Ff ■ v = my ■ F(ey ■ f) ■ v = my ■ Ff§ ■ v = f§ o v by the left-hand side of diagram (1.1), and \i ■ f = Ipy ■ \i ■ f = my ■ F/i • ey ■ f = my ■ F/i • f^ = \i o f^ by the right-hand side of diagram (1.1). As a result, one obtains /joj/^/jo/j.
4
The above constructions define a functor Top F-Mon by G((X, t) -A (Y, cr))) = (X, v) -A (Y, /z). To obtain a functor in the opposite direction, one proceeds as follows.
Given an F-Mon-object (X, v), define t = {U C X | for every x E X, if x E U, then U G ^(x)}- To show that t is a topology on the set X, one notices the following.
• Since the set X is an element of every filter on X, X G t. Since the empty set 0 clearly satisfies the condition on the elements of t, 0 £ t.
• Given [/, V G t, if x G U f] V, then [/ G v{x) and y G v{x), which implies U f] V G t'(x), since is a filter. As a consequence, one obtains that U f] V G
• Given Ui G t for every z G J, if x G IJieJ      then x G £/j0 for some ig G J, which implies Ui0 G Since [/j0 C [JieI Ui and        is a filter, (Jie/ ^« "= vix)- As a result, one obtains that [JieI Ui G v(x).
f f Given an F-Mon-morphism (X,v) —> (Y,/i), to show that the map X —> Y provides a continuous map
(X, t) —> (Y, cr) (where t and a are obtained from v and fi, respectively), notice that given V G cr, for every x G /^(V), e v(x) iff y G Ff{u{x)) = Ff ■ u{x). Since y G a and /(x) G y, it follows that
y G n(f(x)) = (i ■ f(x). Since / is an F-Mon-morphism, Ff ■ v{x) D /i • f(x), and, therefore, y G Ff ■ v(x).
As a consequence, one obtains that /_1(y) G t, i.e., the map X —>• y is continuous.
The above constructions define a functor F-Mon —> Top by H{{X,v) —>• (y/i))) = (X, t) —>• (Y, cr). Straightforward calculations show that the functors G and H are inverse to each other and, moreover, commute with the respective forgetful functors of the constructs (Top, | — |) and (F-Mon, | — |). □
2. Power-enriched monads
Remark 19. Given the powerset monad P on the category Set, the Eilenberg-Moore category Setp of P
(see Lecture 1) is isomorphic to the category Sup. Indeed, given a P-algebra (X, a), one defines an operation
V / FX —> Xby\JS = a(S) providing thus a \/-semilattice (X, \J). A P-homomorphism (X, a) —>• (Y, b) results
then in a \/-preserving map
(X, V) ^ C^V)- Conversely, given a \/-semilattice (X, \/), the map PX ^ X defined by a(S) = \J S provides a P-algebra (X, a). A \/-preserving map (X, \J) —>• (Y, \J) results then in a P-homomorphism (X, a) ^> (Y, b). Altogether, one obtains a concrete isomorphism Setp = Sup. ■
Remark 20. Given the Eilenberg-Moore category XT of a monad T on a category X, there exists a functor XT X, GJ((X,a) A (Y,b)) = X A Y. The functor GT has a left adjoint X ^> XT, FJ(X 4 Y) = (TX,mx) -> (TY,m,Y), where (TX,mx) is the so-called free 1-algebra on a given set X. The
t t
unit lx —> GJFJ of this adjunction is e, and the co-unit FJGJ -^->- 1xt is given by T-homomorphisms
(TX, rax) <X'a)—> (X, a). The monad associated to this adjunction gives back the original monad T. ■
Remark 21. Given a monad T = (T,m,e) on a category X, there exists a full and faithful comparison functor XT A XT defined by K(X ^ Y) = (TX, mx) mY'Tf> (TY, mY).
Proposition 22. Given a monad T = (T,m,e) on Set, there exists a one-to-one correspondence between
(1) monad morphisms P ^> T (recall Lecture 2), where P = (P,n,d) is the powerset monad on Set;
(2) extensions E of the functor Set       Setj along the functor Set ^ ">°> Rel (recall Lecture 1):
G                                                                    | —| (3) liftings L of the functor Setj —^> Set along the forgetful functor Sup-> Set:
Set ......-■■■■> Sup
(4) \j-semilattice structures on the set TX such that the maps TX        TY and TTX mx> TX are \J-
f
preserving for every map X —> Y and every set X.
Proof. In view of Example 5 and Remarks 19, 20, one can identify the category Rel with Setp, the category Sup with Setp, and the forgetful functor Sup -—K Set with Setp     >• Set.
(1) •<=> (2): Given a monad morphism P ^> T, one defines a functor Setp —> Setj by E(X Y) = X Y.   Given now a map X 4 Y, E(-)0(X 4 Y) = X Y, where X A PY is defined
by s(x) = {f(x)}, and Fj(X A Y) = X Y.   For every x e X, ty ■ s(x) = TY({f(x)}) = ty ■
dy(f(x)) TY d= eY ey(f(x)) = ey ■ f(x). Thus, Ty ■ s = ey ■ f, i.e., the required triangle commutes.
Conversely, given an extension Setp Setj, define a monad morphism P ^> T by PX TX = PX Elpxy TX. Diagram chasing shows that t satisfies all the required properties.
(1) •<=> (3): Given a monad morphism P —> T, one defines a functor SetT A Setp by L(X ^ Y) =
{TX, mx ■ ttx) my'T/> (TY, mY ■ tty) (cf. Remark 21). Notice that GPL(X ^ Y) = TX my'T/> TY = f
Gj(X —Y), namely, the required triangle commutes.
Conversely, given a lifting SetT ^> Setp, define a monad morphism P A T by PX ^ TX = PX
PTX A TX, where a is the structure map of the Eilenberg-Moore algebra LX = (TX, a) (recall that GpLX = GjX = TX). Diagram chasing shows that t satisfies all the required properties.
(3) •<=> (4): Given a map x Ay, one obtains a Setj-morphism X ey ^ Y. Since GPL(X ^ Y) =
Gj(X Y) = TX my'T(ey'/}> TY und my-T(eyf) = myTeyTf = (myTey)-Tf = 1TY-Tf = Tf,
it follows that the functor L sends a Setj-morphism X ey ^ Y to a \/-preserving map TX ^4 TY.
Moreover, since TX ±^ X is a SetT-morphism, GPL(TX X) = Gj(TX X) = TTX mx'TlTX> TX and mx ■ Tl^x = mx ■ ^ttx = mx together imply that the functor L sends a Setj-morphism TX X to a V
-preserving map TTX mx> TX. As a consequence, it follows that the conditions of item (4) are just pointwise restatements of the condition of item (3). □
Remark 23.
(1) Given a morphism P ^> T of monads on Set, Proposition 22 (3) equips the underlying set TX of a free T-algebra with a partial order given by
j < t) iff mx -ttx ({?,«J» = >J (2-1)
for every y, tj € TX (cf. Remark 19).
(2) For every set X, the map PX —A TX is monotone, since given A,B <E PX with A C B, the diagram
PPX PTX TTX
mx
PX-yTX
commutes (t is a morphism of monads). As a consequence, mx • ttx({tx(A),tx(E>)}) = mx ■ t^x Ptx({A,B}) = tx-nx({A,B}) = tx(A{JB)= tx(B), namely, tx(A) < tx(B).
6
(3) The hom-sets Setj(X,Y) become partially ordered by the respective pointwise order, i.e., for every
/
X-morphisms X        ) TY, f ^ g iff f(x) ^ g(x) for every x € X. a
(4) Given f,g£ SetT(X, 1") and h e SetT(Y, Z), if / < g, then ho f = mz - Th- f ^ mz - Th-g = hog, since Th, mz are monotone by Proposition 22 (4), i.e., composition on the right is monotone. Composition on
(_)T.j f the left Setj(y, Z) -> Setj(X, Z) for an X-morphism X —>• TY may though fail to be monotone.
(5) To make Setj a partially ordered category (recall Lecture 4), it is enough (—)T to be order-preserving,
/
i.e., / ^ g implies fJ ^ gJ for every X-morphisms X        ] TY If this condition is satisfied, then the
a
functors Rel ^> Setj and Setj Sup of Proposition 22 become functors between partially ordered categories, i.e., preserve the partial order on hom-sets (notice that Lf = fJ for every Setj-morphism
X —Y; and Ef = Ty • f for every Setp-morphism X —Y, where the map ry is monotone). ■
Definition 24. A power-enriched monad is a pair (T,t), where T is a monad on Set and P ^> T is
/
a monad morphism such that / ^ g implies fJ ^ gJ for every Set-morphisms X        ] TY. A morphism
a
(S, u)      (T, t) of power-enriched monads is a monad morphism S     J such that the next triangle commutes
P
S
Example 25.
(1) There exist exactly two trivial monads on Set (admitting only trivial T-algebras), i.e., the monad sending every set to a singleton 1 = {*}, and the monad sending the empty set to itself and all the other sets to 1 (recall Lecture 5). The first one, denoted 1, is clearly power-enriched, where the unique monad
morphism P ^> 1 is given by the unique maps PX —^> 1 for every set X. The second one, say T, is clearly not power-enriched, since there exists no map P0 = 1 —> 0 = T0.
(2) The powerset monad P with the identity monad morphism P P is power-enriched. The partial order on the sets PX induced by condition (2.1) is the usual inclusion of subsets, since V is the union of sets.
(3) The filter monad P is power-enriched, since the principal filter natural transformation t defined on a set X by PX —^> FX, Tx{A) = A = {B C X \ A C B} (principal filter) provides a monad morphism P ^> P. The partial order on FX induced by condition (2.1) is the refinement partial order of Definition 3, and the operation V on FX is given by the intersection of filters. For the latter statement, observe that given a subset {ys | s e S} C FX, V{?s I s <= S1} = rax ■ tfx({Is\s G S}). Therefore, given A c X, A e Vfe I s e S} iff A e mx ■ tfjc({?s | s e S}) iff {3 e FX \ A e 3} e tfjc({?s | s e S}) iff {3 e FX IA e 3} e {B c FX \ {?fl | a e S1} c S} iff {?fl | a e S} c {3 e FX | A e 3} iff A e ?fl for every
s e 5 iff 4 £ Pises?*-- The former statement follows then from the latter, since given y, tj € FX, y ^ ri iff mx -TFxdht)}) = t>iffyn») = t)iff?2t>.
(4) The ultrafilter monad § is not power-enriched, since j30 = 0 (recall from Lecture 1 that an ultrafilter cannot contain the empty set), which is not a Y-semilattice (observe that every Y-semilattice contains a distinguished element \J 0, i.e., the underlying set of every Y-semilattice is non-empty). ■
3. Kleisli monoids
Definition 26. Given a monad T = (T, m, e) on a category X such that the respective Kleisli category Xj is a partially ordered category, T-Mon is the category of J-monoids (or Kleisli monoids), whose objects
7
are pairs (X,v), where X is an X-object, and X X is an Xj-niorphism, which is reflexive (ex ^ v) and transitive (v o v ^ v), where o is the Kleisli composition in the category Xj; and whose morphisnis
(X, v) —>• (Y, fi) are X-morphisnis X —>• y such that T/ • v ^ fj, • f, i.e.,
^y
5= M
->py
Tf
or equivalently /^oj/^^o/j, where     = ey ■ /, i.e.
If T = (T, t) is a power-enriched monad, then the partial order on the hom-sets of Xj depends on t. ■ Remark 27. Given a T-monoid (X, 1/), v = v o ex       o v       implies v ov = v. ■
Remark 28. Given a T-monoid (X, v), the functor Xj-> X has the following property (preservation
of idempotency): z/T = (v o z/)T = (mj • Tz/ • z/)T = mx ■ T(mx ■ Tv ■ v) = mx ■ Tmx ■ TTv ■ Tv ="* rax • Tv ■ mx ■ Tv = vJ ■ vJ, where (f) relies on commutativity of the following diagram
TTX TTTX TTX
TX-
Tu
-4- TTX ■
Example 29.
(1) If T is the trivial monad 1 on the category Set of Example 25 (1), then the respective Kleisli monoids are pairs (X,X      {*}), and the respective morphisms are maps X —>• Y, i.e., 1-Mon = Set.
(2) If T is the powerset monad P with the identity monad morphism P P, then P-Mon is the category Prost of preordered sets and monotone maps that can be seen as follows. First, given a set X, the partial order on PX is the inclusion of sets. Second, a map X A- PX induces a relation ^ on X by x ^ y iff x G v(y). If v is reflexive (ex ^ v), then given x G X, ex(x) = {x} C z/(x) implies x G v(x) implies x ^ x, i.e., ^ is a reflexive relation. If v is transitive (vov ^ z/), then given z£l, 1/01/(2;) ^ z/(z) implies mx • Pi/ • v(z) C z/(z) implies (J Pv(v(z)) C z/(z) implies Uj,eiy(z) ^(2/) — zy(z)- Thus, given x,y,z G X such that x ^ y and y ^ z, x G z/(y) and y G 1/(2;) implies x G Uyeu(z) ^(v) — u(z) implies x G v(z)
f
implies x ^ z, i.e., ^ is a transitive relation. Third, given a P-monoid morphism (X, v) —> (Y, fx), x\ ^ X2 implies X! G i/(x2) implies f(x1) G /(//(x2)) = P/ • v(x2) C /z • /(x2) = ii(f(x2)) implies /(xi) < /(x2), / .
i.e., the map X —> Y is monotone. Fourth, the above-mentioned arguments are reversible.
(3) The filter monad P with the principal filter natural transformation P ^> P provides the category P-Mon, which is isomorphic to the category Top of topological spaces and continuous maps by Theorem 17. ■
Proposition 30. A morphism of power-enriched monads (S = (S,n,d),a)      (J = (T, m, e),r) provides a
p f f
concrete functor S-Mon —^ T-Mon defined by Fa((X, v) —>• (Y, /i)) = (X, ax ■ v) —>• (Y, ay ■ Li).
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Proof. First, observe that there exists a functor Sets Set°> SetT defined by Seta(X     Y) = X "Y ^ Y.
To show that SetQ preserves the Kleisli composition, notice that given Sets-morphisms X     Y and Y Z,
Seta(gof) = az-(gof) = az-nz-Sg-f = mz-Taz-Tg-aY ■ f = mz-T(az ■ g) ■ aY ■ f = («z-g) o (aY ■ f) = Setag o SetQ/, where (f) relies on commutativity of the following diagram
SY ■
Sg
±SSZ-
-tSZ
TY -
Tg
-+TSZ-
Taz
-+TTZ-
-+TZ.
Second, notice that given a set X, the map SX       TX is V
-preserving, which follows from the next
commutative diagram
Pax
^PTX
PSX-
&SX
SSX STX^-^ TTX
SX-
-+TX
and the definition of \J on the sets SX and TX. In particular, it follows that the map ax is monotone.
Third, observe that the functor Fa is correct on objects, since given an S-monoid (X, v), dx ^ v implies ex = ax ■ dx ^ ax ■ v (since a is a monad morphism, whose components are monotone), and v o v ^ v implies v o v = v (by Remark 27) implies (ax • v) o (ax • v) = ax ■ (y ov) = ax ■ v (since SetQ is a functor).
Fourth, notice that the functor Fa is correct on morphisms, since given an S-monoid morphism (X, v) —> (Y, n), Sf ■ v ^ \i ■ f implies aY ■ Sf ■ v ^ aY ■ \i ■ f (since aY is monotone) implies Tf ■ ax ■ v ^ aY ■ \i ■ f by commutativity of the following diagram
SX-^TX
Sf SY-
Tf
^TY.
Fifth, the functor Fa is concrete, since it does not change the underlying sets of Kleisli monoids.
□
4. The Kleisli extension
Definition 31. Define a functor ReFp -^U SetP by (X Yf =Y      X, where the map Y ^ PX
is given by x G r (y) iff xr y (representing the opposite relation Y —I—)• X; cf. Example 5). ■
Definition 32. The functors of Definition 31 and Proposition 22 provide a functor Relop -> Setp =
Relop Setp A SetT A Setp, (X Y)T = TY A TX, where rT = mx ■ T(tx ■ rb) = (tx ■ rb)T. .
Definition 33. Given a power-enriched monad (T, t), the Kleisli extension T of T to Rel (w.r.t. t) is provided by the functions Rel(X, Y)    ~   ' > Rel(TX, TY) (for every pair of sets X and Y) such that
r
for every relation X—I—>Y, and every y G TX, t) G TY, it follows that y (Tr) t) iff y ^ rT(\)), which is
(frf=iTX-rT
equivalently described by a map TY
-> PTX, where iTX (y) = {j G       | 3 < y} (Zower sei). 9
Example 34.
r
(1) Given the terminal power-enriched monad (1,!), the Kleisli extension of a relation X—i—>Y is the
lr
relation {*} —i—>• {*} such that * (lr) *.
(2) Given the powerset monad (P = (P,m, e), lp), the respective Kleisli extension can be described as
r
follows. Given a relation X—i—yY, for every A G PX, B G PY, it follows that A < rip(B) iff A C rlp(B) iff A C mx ■ P(lx ■ rb)(B) iff A C \jPrb(B) = \JyeBr\y) iff for every x G A, there exists y G B such that x G rb(y) iff for every x G A, there exists y G P» such that xry iff A C r°(B), where r°(B) = {x G X | there exists y E B such that xry}. As a consequence, APr B iff A C r°(B), i.e., one obtains the lax extension of the functor P from Lecture 1.
(3) Given the filter monad (F = (F, m, e), t), where P ^> F is the principal filter natural transformation, the
r
respective Kleisli extension can be described as follows. Given a relation X —i—> Y, a subset A C X, and a filter tj G FY, it follows by Definition 1 that A G mx ■ F(tx ■ rb)(tj) iff AF = {y G FX | A G y} G F(rx • rb)(ri) iff fa • rb)-!(AF) G tj iff {y G Y | rx • rb(y) G AF} G tj iff {y G Y | A G rx • rb(y)} G tj iff {y G Y | rb(y) C A} G t} (since tx(B) = {CCI|B C C"}) iff there exists Bet) such that r°(B) C A. As a consequence, rT(ri) = mx • F(tx ■ rb)(t)) =J[px {r°{B) \ B G 1)}, where given a partially ordered set (Z, ^) and a subset S C Z, (S1) = {z £ Z| there exists s E S such that s ^ z}. Thus, given y G FX and t) G FY, it follows that y (Fr) t) iff y < rT(t)) iff y D rT(t)) iff y D r°[tj], where r°[tj] = {r°(B) | S G t)}. Observe that the Kleisli extension of the filter monad coincides with the respective lax extension F. ■
Definition 35. A lax functor C A D of preordered categories (recall Lecture 4) is a pair of maps Oc —^> Cd, -Mc Fm> ■Mo (both denoted F), which satisfy the following axioms:
(1) F(X AY) = FX ^A FY for every C-morphism X A Y;
/
(2) F/ ^ Fg for every C-morphisms X        ] Y such that / ^ g;
a
(3) Fg ■ Ff < F(ff • /) for every C-morphisms X A Y and Y 4 Z;
(4) 1_fc ^ Flc for every C-object C ■ Remark 36.
(1) Recall from Lecture 4 that there is a functor (V-Cat)op ^A- V-Mod defined by ((X, a) A (Y, 6))* =
r (_)-
(Y, 6) —e—)■ (X, a), where f* = f° ■ b. In case of V = 2, this functor induces a functor Prost ->
Modop defined by ((X, <x) 4 (Y, <y))* = (Y, <y)     " o       ; (X,
(2) There exists a lax functor Mod Rel defined by |(X, <x) —«->     <y)U = X —+—> Y, which preserves the composition, but given a preordered set (X, ^x), 1\(x,^x)\l = lx ^ (^x) = ■
Remark 37. In the definition of lax extension of a Set-functor T to the category y-Rel (recall Lecture 1), the following statements are equivalent:
(1) Tf < ff and (Tf)° < f(f°) for every map X A Y;
(2) (T/)° < f(/°) and f(/° • r) = (T/)° • fr for every map X A Y and every relation Z -H—> Y. ■
Proposition 38. Given a power-enriched monad (T,t), the Kleisli extension f of T to Rel provides a lax extension T = (T, to, e) o/ T = (T, to, e) to Rel.
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Proof. To show that Rel —> Rel is a lax functor, one can express it as a composition of lax functors as
r fr (rT)*
follows. Observe first that given a relation X —I—y Y, TX —I—y TY can be expressed as TX —I—y TY
with the help of the functor (—)* of Remark 36 (1). Thus, the Kleisli extension Top can be written as the
composition of functors Relop ^ "* > Sup -—K Prost ^ "* > Modop, where | — | is the forgetful functor. Notice second that the Kleisli extension Top can be expressed as the following composition
Rel°p -LI4 Set,
SetT —Sup -LU Prost Modop Relop,
where all the arrows (except the last one) are functors, and the last arrow is the lax functor of Remark 36 (2).
To show that (Tf)° ^ T(f°) for every map X —>• Y, one can consider the following commutative (except
for the down right part, where one should notice that given a monotone map (X, ^x) (Y, ^y), it follows that f° < f° ■ «y), since lx < «y)) diagram:
rpop
Rel°p Setp —^ SetT —L—t Sup^HH^ Prost -^-U Modop -|-U? Relop
lRel°P
^Relop.
(-)°
(4.1)
The second condition of Definition 37 (2), i.e., T(f° ■ r) = (Tf)° ■ Tr for every map X —>• Y and every
r
relation Z —I—y Y can be shown as follows. Given y e TX and 3 e TZ, y T(/° • r) 3 iff y < (J° • r)T(3) iff
? < rT-(/°)T(3) (since Relop Prost is a functor) iffy < rT-Tf^) (by diagram (4.1)) iff y ((T/)° • fr) 3.
Altogether, it follows that T is a lax extension of the Set-functor T to the category Rel. To show that
X-^-fTX
Y ■
Tr
-fTY
for every relation X —I—y Y, notice first that the following commutative diagram
Pex
PX PPX -^yPTX TTX
PX
-+TX
implies tx = rax ■ t~tx ■ Pex = Vtx 'Pex (recall condition (2.1)). Observe second that given x G X and y <EY such that xry, it follows that (recall Definition 31) ex(x) ^ \fX'erb(y) ex{x') = Vtx Pex(rb (y)) =
rx ■ rb(y) = rax ■ Ttx ■ Trb ■ eY(y) = rax ■ T(tx ■ rb) ■ eY(y) = (tx ■ rb)J ■ eY(y) Deflm^on 32 r^ . eY(y), where
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(f) relies on comniutativity of the following diagram:
Y ■
^TY
PX
Tr"
>PTX
PX
¥TTX
^TX.
As a consequence, one obtains that ex(x) (tr) ey(y). Lastly, to show that
TTX TX
ffr ^
TTY-> TY
Tr
for every relation X—i—vY, observe first that mx ■ ttx' Itx= \Itx ' Itx = Itx, and notice second
,i    ,   / t\T / -7-    1       \T  Definition 32   // k\T    1      \T  Definition 10   / b\T    iT      Definition 32 ^
that (rT)° = (rT ■ Ity) =       ((Tx • r )   • ±ty) =       (Tx • r )   • ^ty       =       r  ' my-
Therefore, given X G TTX and 2) G TTY such that X (ffr) 2), it follows that X < (f r)T(2)), which implies
Proposition 22 (4)
mx(X) < mx((trY(Z)))
. j        / I t\T/<v^ Definition 10
rax ■ (Trym Defl^°n 32 mx . (TTX . (fr)b)T(2j)
lTX ■ (ttx-Itx -rT)'(2)) """" - (llx ■ ttx-Wx -rT)T(2)) = (rax-rTX-Itx tt)t(2)) = (r-)T(2J) = rT ■ ray(2)). As a consequence, one arrives at X (f r) 2), which finishes the proof. □
Definition 33
Proposition 39. Given a monad T = (T, ra, e) on Set, there exists a functor Set —> (T, 2)-Cat defined by t(X -A- Y) = (TX,fax) —^> (TY,fay), where fax = tlx • mx ■ The functor makes the following triangle
Set
-> Set
(T,2)-Cat
commute (\ — \ is the forgetful functor). The preorder on TX induced by fax is given by y ^ t) iff jcTlx t).
Remark 40. Since the Kleisli extension provides a power-enriched monad (T, t) with a lax extension, there exists an induced preorder on TX associated with f as in Proposition 39, i.e., y ^;nd t) iff yf lx f). There also exists a partial order on TX provided by the monad morphism P ^> T as in Remark 23 (1), i.e., y ^T t)
r
iff rax ■ t~tx({z, f)}) = 1)- Following Definition 33, y (tr) t) iff y ^T rT(ri) for every relation X —i—y X. Thus, if r = lx, then y (fix) f) iff y ^T (lx)T(t)) iff y f), since (—)T is a functor. Thus, the induced preorder associated with the lax extension f coincides with the partial order provided by the monad morphism t. Also notice that f fails to preserve identity relations unless T = 1 is the terminal power-enriched monad. ■
Theorem 41. Given a power-enriched monad (T,t) equipped with its Kleisli extension t, there exists a concrete isomorphism (T, 2)-Cat = T-Mon.
Proof. The proof relies on a lax algebraic generalization of the classical correspondence between convergence and neighborhoods in topological spaces. In particular, given a topological space X, a filter y on X converges to some point y G X precisely when y is finer than the neighborhood filter of y. This correspondence can be formalized via maps Set(X, FX) Rel(FX,X) and Rel(FX, X) ^> Set(X,FX), replacing
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the filter monad F with a power-enriched monad (T, t) and identifying Rel(TX, X) with Set(X, PTX), isomorphic as ordered sets. One thus defines conv(z^) =Itx -v and nbhd(r) = \j TX     f°r every map X —> TX
r
and every relation TX —I—y X. In pointwise notation, these maps can be written as y conv(z^) x iff y ^ v(x) and (nbhd(r))(x) = \J{t) G       1t} G rb(x)} = \J{t) G       | t)ri} for every y G TX and every x G X.
Lemma 42. Given a \J-semilattice A, there exists the adjunction \J -\\.\ A —> PA, where PA is the powerset of A ordered by set inclusion, and 4- (a) =i a = {b G A \ b ^ a}, smc/i that \J ■ i = 1a-
Proof. Given a G A and S1 C A, it follows that V S ^ a iff S     a, and, moreover, V I a = a. □
Proposition 43. When Set(X, TX) and Rel(TX, X) are equipped with pointwise partial order, there exists an adjunction (recall Lecture 4-) nbhd H conv : Set(X,TX) —> Rel(TX, X) for every set X. Additionally, the fixpoints of conv-nbhd are precisely the unitary relations (recall Lecture 6), and nbhd- conv = lset(x,tx)> so that the fixpoints of nbhd ■ conv are the maps X A TX.
Proof. Notice that given a map X —>• TX and a relation TX —i—> X, it follows that nbhd(r) ^ v iff (nbhd(r))(x) ^ i/(x) for every x G X iff \/{y G TX|yrx} ^ i/(x) for every x G X iff (Lemma 42) {y G TX | y r x} Cj. v{x) for every x G X iff y r x implies y ^ for every y G TX and every x G X iff y r x implies yconv(z^)x for every y G TX and every x G X iff r C conv(z^) iff r ^ conv(z^). As a consequence, nbhd(r) ^ nbhd(r) implies r ^ conv • nbhd(r), and conv(z^) ^ conv(z^) implies nbhd • conv(z^) ^ v, i.e., lRe\(tx,x) ^ conv-nbhd and nbhd - conv ^ lset(x,tx)- Moreover, both nbhd and conv are monotone maps.
Given a map X A TX, for every x G X, it follows that (nbhd • conv(i/))(x) = (nbhd(conv(i/)))(x) = V{? G TX | y conv(z/) x} = \/{? € TX | y < v{x)} = \J I v{x) Lem=a 42 namely, nbhd • conv(z/) = za As
a result, one obtains that nbhd • conv = lset{x,tx), i-e-, the fixpoints of nbhd • conv are the maps X A TX.
The statement on unitary relations relies on a sequence of technical calculations. □
Moreover, the above adjoint maps nbhd and conv are monoid homomorphisms between Setj(X,X) and
r
(T, 2)-URelop(X, X) (the set of unitary relations TX —I—> X), namely, they satisfy
nbhd(s o r) = nbhd(r) o nbhd(s) conv(/z) o conv(z^) = conv(y o /i)
nbhd(4) = ex conv(ex) = lx
for all unitary relations TX     j   j X, and all maps X        ] TX, where sor = s-Tr-m°x (Kleisli convolution)
s v
and lx = e°x o e°x (properties of power-enriched monads imply that the Kleisli convolution is associative).
a l
Lemma 44. For a set X, a relation TX —I—y X provides a (T, 2)-category (X, a) iff a o a — a and 1 x a. Proof.
=>■: First, notice that given a (T, 2)-category (X, a), it follows that a-Ta ^ a-mx and lx ^ a - ex (recall
mx-mx^lTx ex-ex^lTX
Lecture 1), which implies a o a = a ■ Ta ■ mx ^ a ■ mx ■ rnx ^ a and e°x ^ a • ex ■ e°x ^ a. Second, observe that the operation o is monotone in both arguments by its very definition. Third, notice that given a lax extension T = (T,m,e) to y-Rel of a monad T = (T, to, e) on Set, it follows that Tlx = Te°x ■ m°x, which implies a o e°x = a ■ Te°x ■ m°x = a ■ Tlx ^ a ■ Tlx = a ■ Itx = a, since T is a lax extension of T. Thus, a^aoe^ ^ a o a ^ a (i.e., a o a = a) and lx = exoex^aoa^a (i.e., lx ^ a).
lTTX^mx-mx
<^=: Observe that, first, aoa = a implies a-Ta-m°x = aoa ^ a implies a-Ta ^ a-Ta-mx-mx ^
„ i T is a lax extension of T
a • mx (i-e., a • Ta ^ a • mx), and, second, lx ^ a implies      = e°x ■ Itx = e°x ' Tlx ^
• Tlx = e°x ' Te°x ' mx =4°4^a implies lx < ex • ex < a • ex (i.e., lx < a • ex). □
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Given a (T, 2)-algebra (X,r), it follows that (X, nbhd(r)) is a T-monoid, and conversely if {X,v) is a T-monoid, then (X, conv(i/)) is a (T, 2)-algebra. Moreover, this one-to-one correspondence is functorial. □
Corollary 45. The category Top is concretely isomorphic to the category (F, 2)-Cat, whose objects are
a
pairs (X,a), where FX—i—> X is a relation, which represents convergence and which satisfies (denoting "a" and "Fa" by "—>") X —> t) and t) —> z imply X —> z, and x —> x for every X G FFX, t) G FX, x, z G X (notice that X —> t) iff X ~D a°[t)] as in Example 34 (3)); and whose morphisms (X, a) —> (Y, b) are convergence-preserving maps X —> Y, namely, y —> z implies Ff(j) —> f(z) for every y G FX, z G X.
Proof. The statement follows from Theorems 17, 41. □
Corollary 46. Given the up-set monad U (for a set X, UX = {jC PX \ 'fpx a = a}) equipped with the Kleisli extension associated with the principal filter natural transformation, there is a concrete isomorphism Cls = (U,2)-Cat, where Cls is the category of closure spaces and continuous maps (cf. Lecture 1).
References
[1] J. Adámek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories: the Joy of Cats, Repr. Theory Appl. Categ. 17 (2006), 1-507.
[2] D. Hofmann and G. J. Seal, A cottage industry of lax extensions, Categ. Gen. Algebr. Struct. Appl. 3 (2015), no. 1, 113—151. [3] D. Hofmann, G. J. Seal, and W. Tliolen (eds.), Monoidal Topology: A Categorical Approach to Order, Metric and Topology,
Cambridge University Press, 2014. [4] S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer-Verlag, 1998.
[5] J. MacDonald and M. Sobral, Aspects of Monads, Categorical Foundations: Special Topics in Order, Topology, Algebra,
and Sheaf Theory (M. C. Pedicchio and W. Tholen, eds.), Cambridge University Press, 2004, pp. 213—268. [6] G. J. Seal, A Kleisli-based approach to lax algebras, Appl. Categ. Structures 17 (2009), no. 1, 75—89. [7] G. J. Seal, Tensors, monads and actions, Theory Appl. Categ. 28 (2013), 403-434.
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