Download Aspects of categorical algebra in initialstructure categories

C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE
CATÉGORIQUES
M ANFRED B ERND W ISCHNEWSKY
Aspects of categorical algebra in initialstructure
categories
Cahiers de topologie et géométrie différentielle catégoriques, tome
15, no 4 (1974), p. 419-444
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Vol. XV-4
CAHI ERS DE TOPOLOGIE
ET GEOMETRIE DIFFERENTIELLE
ASPECTS OF CATEGORICAL ALGEBRA
IN INITIALSTRUCTURE
by Manfred
an
Bernd WISCHNEWSKY
F: K - L, the
Initialstruc ture functors
of BOURBAKI’s notion of
CATEGORIES*
« initial
categorical generalization
objects [3], equivalent
to
Kenni-
pullback stripping functors, which Wyler calls Top-functors, reflect
almost all categorical properties from the base category L to the initial
structure category K briefly called INS-category [1,4,5,6,8,11,12,13,
13,16,18,19,20,21,22,37]. So for instance if L is complete, cocomplete, wellpowered, cowellpowered, if L has generators, cogenerators,
proj ectives, injectives, or ( coequalizer, mono )-bicategory structures ( ==&#x3E;
homomorphism theorem ) , then the same is valid for any INS-category over
L . Finally all important theorems of equationally defined universal algebra ( e.g. existence of free K-algebras, adjointness of algebraic functors... )
can be proved for algebras in INS-categories, if they hold for algebras in
L . The most well known INS-categories over Ens are the categories of
topological, measurable, limit, locally path-connected, uniform, compactly
son’s
generated,
or zero
dimensional spaces.
This survey article deals with the
braic
-
-
three types of
over
of PFENDER
algebraic categories
sense
of
alge-
with
INS-categories, namely
algebraic categories of Z-continuous functors [29,30,33] ,
monoidal algebraic categories over monoidal base categories
categories
sense
-
following
in the
[35],
in the
sense
of
T H I E B A U D
[36]
resp. in the
EILENBERG-MOORE.
These three types of
«algebraic categories»
include comma-catego-
categories of finite
algebras, locally presentable categories, ElLENBERG-MOORE: categories,
categories of monoids in monoidal categories.
ries, algebraic categories in the
* Conference
donn6e
au
sense
of LAWVERE,
Colloque d’Amiens ( 1973)
419
In this paper it is shown that each of the above
ries
is
algebraic
catego-
Hence the whole theo-
INS-category again
INS-category.
here
the first part, can again be applied
in
INS-categories, presented
to algebraic categories over INS-categories. In particular it is shown that
this implies that adjointness of « algebraic» functors over L induces
adjointness of « algebraic » functors over an INS-category K . Since furthermore together with K also K°P is an INS-category, and since the algebras
in K°P are just the coalgebras in K , the theory is also valid for coalgebras in INS-categories. Moreover I will show that all the basic results on
algebraic categories even hold for reflective or coreflective subcategories
of INS-categories if they hold for the corresponding INS-categories. So
over an
an
ry of
for instance let U be
a
coreflective
or even an
epireflective
of
a
coreflec-
subcategory of the category of topological spaces and continuous
mappings like the categories of compactly generated, locally path connec-
tive
ted,
7-spaces,
or
i = 0, 1, 2, 3, and
let Z C [C ,S ]
be
a
set
of functo-
morphisms in [C ,S]. Then the algebraic categories Z ( C , U )
Z -continuous functors with values in U are complete, cocomplete,
rial
powered,
and
cowellpowered.
adjoint. The
gebras in U .
are
1.
same
of all
well-
Furthermore all inclusion functors
holds for the
category Z (C, U°p )
for
all Z-coal-
INS-functors, INS-categories, and INS-morphisms.
Let F: K - L be
a
functot. Let I be
an
arbitrary
necessarily
resp. by AL :
not
discrete, small category, and denote by Ak : K-&#x3E; [I,K]
L -&#x3E; [I, L ] the corresponding diagonal functors, and for T E [I, K ] the
comma
categories
If there is
no
(AK, T)
resp.
confusion, I’ll write
of all functorial
for A K resp. A L
simply A.
weakly isomorphisms. With
transportabl e if F
give the following
creates
1.1.
Let F:K - L be
D E F I N I T IO N .
(AL , FT)
morphisms
F is called
this notation
we can
faithful, fibre-small functor. F i s called
420
INS-functor,
an
and K and
T
1, and for all functors
has
an
is
over
L , if for all small categories
the functor
with counit
cone, then the cone
a
GT qj : Al* -&#x3E; T
is called
an
INS -
INS-object generated by qj - If t,: I - F k is a monoGO: l* -&#x3E; k is called an embedding, and l* an INS-subob-
and l* E K
an
morphism, then
ject of k .
Dually one
1.2.
E [1, K J
adjoint ( = right adjoint) GT
If Y : Al -&#x3E; F T
cone,
INS-category
EXAMPLES:
category of
sets
defines coinitialstructure
The
following categories
functor,.....
with obvious INS-functors: the
uniform, measurable,
or
completely regular,
Let
us now
of
topological,
principal limit
compactly generated spaces,
categories
based spaces, of Borel spaces, of
spaces, of spaces with bounded structure, of
of limit,
over S the
INS-categories
are
or zero
dimensional spaces....
recall G R O T H E N D I E C K’ s construction of
a
split
cate-
completely ordered sets with suprema
preserving mappings. Then there corresponds by GROTHENDIECK
( [8]) to each functor P : L°p -&#x3E; Ord (V) a split category Fp : Kp - L : the
objects of Kp are the pairs (l, k) with k E P l and /6L. The Kp-morphisms f:(l,k)-&#x3E; (I’,k’) are exactly those L-morphisms f:1-1’ with
P f k k’ . The functor Fp : K p - L is the projection (l,k) l-&#x3E;H .
gory. Let Ord (V) be the category of
Herewith
we can
give
the
following
[1,8,11,15,20,23]. Let F: Ksmall functor. Regard the following assertions :
( i ) F is an INS-functor.
( ii ) F°p is an INS-functor.
1.3.
THEOREM
( iii ) The category ( K, F)
over
L is
L
-L-equivalent
be a faithful, fibre-
to a
split category
( Kp , Fp ).
( iv ) F preserves limits and has
a
full
421
and
faith ful adjoint J .
Then
(i)
( ii ) ==&#x3E; (iii) ==&#x3E;(iv). 1 f
plete, then (iv) ==&#x3E;(i).
==&#x3E;
The
ANTOINE
equival’ence ( i )
[1]
==&#x3E;
(ii)
ROBERTS [15],
and
L and K
moreover
==&#x3E;
( iii )
are com-
proved by
straightforward from
was
but follows
first
categories [8]. The characterization ( iv ) ==&#x3E;(i) was first given by HOFFMANN [11]. A proof of ( iv )
==&#x3E; (i), in some way simpler than the original one, can immediately be
obtained from both the following statements :
GROTHENDIECK’s
1) A
cone
an
fibied
is
INS-cone if and only
an
cone
INS-cone,
2 ) Let K,
L be
preserving
tion
on
( l* , Y : Al* -&#x3E; T), TE [I, K],
if the induced
is
paper
categories
functor with
(cf. [14]
with
Let F: K - L be
an
inverse. Then F is
pullbacks.
adjoint right
a
pullback
a
fibra-
Corollaire 3.7) (cf. Theorem 1.9a).
Hence, if F fulfills (iv), F is
by 2 and thus together with 1
an INS-functor provided K and L are complete.
The full and faithful right adjoint of an INS-functor is obtained
from definition 1.1 by regarding the void category as index category. The
objects of the image category of this functor are called codiscre te Kob je cts.
1.4.
we
[18,20,22].
following assertions :
PROPOSITION
have the
a
1 ) K-limits ( K-colimits )
fibration
Let F:K-L be
an
INS-functor.
Then
limits ( colimits ) in L
supplied with the
«initialstructure» ( coinitialstructure ) generated by the projections ( injections ).
2 ) F preserves and reflects monos and epis.
3 ) K is wellpowered (cowellpowered) if and only if L has this proare
perty.
4 ) K has generators, cogenerators, projectives,
ly if
K has such
objects.
422
or
injectives if
and
on-
5 ) K is a ( coequalizer, mono )-bicategory if and only if L is such
a
category.
Let
K -&#x3E; K’
1.5.
as
now
well
F:K-L and F’:K’-L’ be INS-functors, and let M :
N:L-L’ be
as
[22,23,37].
DEFINITION
called
an
arbitrary
INS-morphism
The
if both of the
functors.
pair (NI, N): (K, F)-&#x3E;(K’, F’) is
following conditions hold :
1) NF=F’M,
2 ) for all small categories I and all functors 7B’/-’K
right
inverses
G’MT
gram commutative up
resp.
to an
GT
of ’
F’MT
resp.
F?..
make the
adjoint
following diathe
isomorphism
pair (M, N) preserves INS-cones.
In WYLER’s language of Top-categories, i.e. reduced INS-categories
the INS-morphisms correspond to his taut liftings [22,23] .
In this
case we
say that the
[19]
1.6. REMARKS.
1) The category Initial of all INS-functors and INS-morphisms is
ble category in the
sense
of
a
dou-
EHRESMANN.
2) The category Initial (L) of all INS-categories and INS-morphisms
over
a
constant
base category L
is
canonically isomorphic
to
the full
functor category
(L°p Ord (V] ( cf. [19]).
1.7. THEOREM
[22 , 23]. Suppose that (M, N) : (K,F) -&#x3E; ( K’ F’) be an
INS-morphism. Then
( i ) Af is adjoint.
( ii ) N is adjoint.
we
have the
following equivalent
The
statements :
implication (i) ==&#x3E; (ii) is trivial by theorem 1.3 ( iv ) . In
case
categories involved are complete, it suffices to find a soluset
for
tion
M, since M preserves obviously limits by proposition 1.4
and definition 1.5 . If R : L’ - L denotes a coadjoint of N, then for all
that all
423
k’ E K’ the fibre
F-1 (RF’(k’))
is
a
solution
for k’,
set
as
one sees
immediately by factorizing F’f : F’k’- N F k, f : k’-&#x3E;M k E K’, through the
unit of (R -l N), and then by supplying R F’ k’ with the INS structure...
In the general case one takes the infimum of all objects in this fibre appearing in any of the above factorizations.
For the
INS-categories
1.8.
rest
are
DEFINITION.
assume now
that all base
of
categories
complete.
Let F:K-L, resp. F’:K’-L’, be INS-functors. De-
J’ ,
adjoint right inverse functor of F , resp. F’ , and
by O : Id -&#x3E; J F, resp. O’: Id -&#x3E; J’ F’, the corresponding units. Let furtherN F . The
more M : K - K’ and N : L - L’ be a pair of functors with F’ M
pair (M, N) preserves codiscrete objects if
note
by J ,
of this paper
resp.
an
=
pair o f functors betINS-functors. 1 f ( M, N ) preserves codiscrete objects, the following
1.9. THEOREM. Let ( M, N ) : ( K, F ) -&#x3E; ( K’, F’ ) be
ween
statements are
a
equival ent :
(i) (M, N): (K, F)-&#x3E;(K’, F’) is an INS-morphism.
( ii ) M and N preserve limits.
The
cones
and
implication (i) ==&#x3E;(ii) is obvious, since
hence in particular limit-cones. Thus one
(ii) ===&#x3E;(i) .
M preserves INS-
has
only
From the remark 1 of theorem 1.3 it follows that
to
prove
one
has
only to show that ( M , N ) preserves INS-cones of the form (l*, Y: 1* - k )
generated by F Y : l -&#x3E; Fk , i.e. ( M , N ) is a morphism of fibration in the
sense of GROTHENDIECK. But this follows immediately from the fact,
that
a) the
INS-cone Y : l* -&#x3E; k
where O : Id -&#x3E; J F
is
a
projection
denotes the unit of the
424
of
adjoint
a
pullback, namely
functor
pair ( F, J ),
and
ia’* denotes the
b) ( M , N ) preserves
1.10.
induced
by idl : l -&#x3E; l .
codiscrete objects, and
K-morphism
EXAMPLE [18,20,cp.22,36,37].
in the
M preserves
Let A be
of L A W V E R E and let F : K -&#x3E; L be
sense
an
pullbacks.,
algebraic theory
INS-functor
an
over
a
complete category L . Then we obtain the following commutative diagram
of limit preserving functors between complete categories
denote the
obviously
particular :
all
corresponding forgetful
assumptions of theorem 1.9.
VK
is
adj oint
VK
resp.
VL
fulfills
I.II.
COROLLARY.
1.12.
EXAMPLE.
Each
Let A be
Then the functor
if and
morphism
an
Alg ( A , - )
only
if
in Initial (
algebraic theory
defines in
an
VL
is
functors.
Hence
L ) is adjoint.
in the
sense
obvious way
Alg ( A , H ) is an INS-morphism, it is coadjoint by
lary. A typical example for this situation is the functor
induced by the
1.13. DEFINITION
a
strictly
U is closed
20132013201320132013&#x3E;
topological
-
topological
a
of
LAWVERE.
functor [19] :
the above Corol-
groups
INS-morphism
uniform spaces
be
obtain in
adjoint.
Since
uniform groups
we
(VK, V L)
[19,23].
Let F: K - L be
subcategory of K
under INS-objects, i.e.
full
an
INS-functor, and let
. U is called
if for all small
425
spaces.
U
INS-subcategory if
categories I and all
an
T E [I, U]
functors
lies
the
INS-object
again in U .
INS-subcategories
1.14.
[19].
can
l*
generated by
be characterized in the
Let F: K - L
be
(Y : Al -&#x3E; FT)
a cone
following
way:
INS-functor. A strictly full
subcategory U of K is an I NS-subcategory of K if and only if
1 ) U is closed under products and INS-subobjects,
2 ) U contains all codiscrete K-objects.
THEOREM
an
EXAMPLES:
INS-category over S . Since K is complete, wellcowellpowered, a stricly full subcategory U of K is epi-
1 ) Let K be
powered, and
an
reflective in K if and
only if U
objects [9,18 19] . Hence
we
is closed under
products
and INS-sub-
get the
COROLLARY [19]. Let K be an INS-category over S, and le t U
strictly full subcategory of K containing all codiscrete K-objects.
there are equivalent :
( i ) U is an INS-subcategory of K .
( ii ) U is epire f lective i n K .
be
a
Then
K-object of an INS-category over S
are generators in K , we obtain by dualizing and applying the results of
HERRLICH -STRECKER [101 the following
2 ) Since all but
one
discrete
COROLLARY [19,23]. Let K be an INS-category over S, and let
a strictly full subcategory of K containing all discrete K-objects.
there
equivalent :
( i ) U is a COINS- su bcatego ry o f
(ii) U is coreflective in K .
U be
Then
are
In
of
case
these corollaries
Finally
K =r x’op
one
K.
can
find
a
whole host of
examples
for
in [9] .
we
1.16. DEFINITION
will need the
[18].
following
Let F:K- L be
an
INS-functor.
subcategory U of K is called a PIS-subcategory
under products and INS-subobjects.
426
of
K,
A strictly full
if U is closed
Recall that each extremal
K-monomorphism
in
an
INS-category K
embedding [18, 19 , 20]. Furthermore if K is a complete, wellpowered and cowellpowered category, then a strictly full subcategory is epireflective in K if and only if it is closed unde r products and extremal sub-
is
an
objects [9] . Hence we
1.17.
PROPOSITION.
and
wellpowered,
is epireflective
obtain the
Let F:K-L
an
INS-functor
over
a
category L . Then each
cowellpowered
I f moreover in
PIS-subcategory and
in K .
then the notions
complete,
PIS-subcategory
L each monomorphism is a kernel,
epire flective subcategory are equi-
be
[18].
valent
strictly full subcategory of the category of locally convex spaces is epireflective if and only if it is a PIS-subcategory,
i.e. closed under products and subspaces, since the category of locally
convex spaces is an INS-category over the category of complex-valued
So for instance
vector
a
spaces.
Algebraic categories of Z-continuous
2.
functors with values in INS-
categor ies .
Let
tors
us
is
or :
A
a - b the
is
morphisms,
a
2-continuous funcand 2 C Mor K a class of
standard notions
a
category,
on
called 2-bijective, or Z-continuous,
shall denote the full
in K . If C is
of functorial
some
Let K be
K-object k
mapping
bijective. 2 K
objects
tor
recall
[28,29,30,31].
K-morphisms .
all
briefly
subcategory
of all
2-bijective
and Z C Mor [ C , S ] is
pair (C, L) is called a theory.
small category
then the
A : C - K, where K is
an
arbitrary
if for
category, is called
a
K-
class
A func-
a 5: -algebra
if all functors
2-bijective.
theory (C, 2) is called algebraic if the inclusion
functor Z ( C , S )-&#x3E;. [C, S] is adjoint. Let 2* denote an arbitrary class
of theories. A category K is called 2* -algebraic if for all theories (C, L)
are
A
427
in Z* the
inclusion
Herewith
we
functor Z ( C , K) -&#x3E; [C, K] is adj oint, where Z ( C , K )
denotes the full subcategory of all 5i -algebras in K .
Let now (C, Z) be a theory, and let F: K - L be an INS-functor
with coadjoint right inverse D . Let furthermore
obtain the
Hence if A is
the
same
a 5i-algebra
method
one
commutative
following
in
diagram [37] :
K , then F C A
shows that the functor
is
a 5i-algebra
in L.
F Z = FC Z ( C , K)
has
By
an
adj oint right inverse induced pointwise by that of F . For the rest of this
chapter assume that the base category L is complete, in order to be able
to apply characterizations of INS-functors and INS-morphisms given in the
preceeding chapter, although the following theorems are valid without
any restriction.
2.1.
THEOREM ( cp.
plete category
1 ) The
[37]). Let
L . Then
we
F: K - L be
obtain the
an
following
INS-functor
over a com-
assertions :
is
functor
an
INS-functor.
Denote by
the evaluation functors for
c
E C , and by
(C, ¿) and (D, BII )
be theories. A functor f : D - C is called a morphism of theories if the
induced functor [f,S] : [C,S I -&#x3E; [D, S] preserves algebras. If f :
(D ,Y) -&#x3E;( C , Z) is a theory morphism, and K an arbitrary category, then
the canonical functor [ f, K ] : [C, K ] -&#x3E; [D, K] preserves algebras
the
corresponding
inclusion functors. Let furthermore
428
[7.
The restriction
[37]
functor
this notation
and denoted by
if
only if
3 ) The
class 2*
(vK, vL )
vc
is
an
if there is
no
an
INS-morphism.
In
particular vK
is
adjoint
is
adjoint.
pair ( E, E ) of
K is
particular
fK
algebraic
misunderstanding. With
is called
obtain
we
2 ) The pairand
of [ f, K] on Z ( C , K )
inclusion
Z*-algebraic if
and
functors
only if L
is
is
INS-morphism. In
L* -algebraic for any
an
of theories.
f : ( D , Y) --&#x3E; ( C, Z)
morphism of theories. Then the pair
(fK, f L) of algebraic functors is an INS-morphism. In particular an algebraic functor over K is adjoint if and only if the corresponding algebraic functor over L is adjoint.
4) Let
As
2.2.
a
an
immediate
THEOREM.
set, and
be a
application
Let P* be
let F : K- L be
obtain the
following
the class o f all theories (C, 2), where I is
an INS-functor over a locally presentable cawe
tegory L . Th en
1 ) K is P* -algebraic.
2 ) Each P* -algebraic functor over K is adjoint.
3) Each evaluation funct or vc: L (C , K) - K is adjoint.
4 ) Z( C , K) is complete, cocomplete, wellpowered
and
cowellpower-
ed.
5) Z( C, K ) is again
an
INS-category
over a
locally presentable
tegory.
6) Each theory ( C, Z) defines
a
functor
Z(C,-):Initial(L)-·Initial(Z(C,L))
In
particular Z (C ,H)
is
adjoint
for all
429
INS-morphisms
H.
ca.
2.3.
Theorem 2.2 allows
REMARK.
us
to construct
in
a
simple
way
a
lot
INS-categories over locally presentable categories. So for instance
start with an INS-category over the locally presentable category S , the
category of sets: F:K-S. Then take any locally presentable theory
(C,¿), as e.g. an algebraic theory in the sense of LAWVERE, a GROTHENDIECK-topology, or more general a limit-cone bearing category in
the sense of BASTIANI-EHRESMANN [24]. Then the functor
of
FZ: Z(C, K)-&#x3E;Z(C, S)
locally presentable category 5i ( C , S ) . Now one
can continue with this procedure applying theorem 2.2.5. Thus one obtains that the categories of topological, measurable, compactly generated,
locally convex, bornological or zero dimensional spaces, groups, rings,
sheaves...are P* -algebraic, bicomplete, biwellpowered...
is
an
INS-functor
over
the
Since with K also KOP is
locally presentable
we get the following
category is
2.4.
K be an
a
THEOREM.
category. Then
Let
we
obtain the
INS-category over
following statements :
1) K°p is P* -algebraic.
2 ) Each P* -algebraic functor
3 ) Each evaluation functor
2.5.
a
locally presentable
KOP is adjoint.
( C, K°P ) - KOP is adjoint.
over
vc:¿
regard algebraic categories of Z-continuous
subcategories of INS-categories.
Let
over
INS-category, and since each dual of
P* -algebraic ( B A ST I A NI unpublished),
an
now
us
THEOREM
[37].
Let F:K-L be
an
INS-functor
functors
and UC K be
a
PIS-subcategory. Then the following assertions are valid:
1) 1f (C, 2) is a theory, then 2 (C, U) is a PIS-subcategory o f,
Z (C , K) . In particular i f Z (C , K) is complete and biwell powered then
Z (C , U) is an epire f l ective subcategory of Z ( C , K ) .
2 ) 1 f L is ( C, Z )-algebraic and if Z (C, L) is cowellpowered, then
U is again (C , Z) -algebraic.
From
this theorem follows for instance that each
430
epireflective
subcategory of an INS-category over S
reflective subcategory of the category
algebraic.
3. Monoidal
algebraic categories
over
is P*
of
-algebraic, or that each epilocally convex spaces is p*-
monoidal INS-categories.
theory of monoidal universal algebra over monoidal categories,
exactly over S-monoidal categories, is in some way a generaliza-
The
or
more
equationally defined universal algebra in the classical sense.
The S-monoidal categories, defined by M. PFENDER [35] using ideas of
BUDACH-HOEHNKE, are a generalization of monoidal or symmetric monoidal categories in the sense of EILENBERG-KELLY. A S-monoidal
theory (C, 0, can) is a small category C equipped with a S-monoidal
structure. The S-monoidal algebras are functors from a S-monoidal theory
into a S-monoidal category preserving the given S-monoidal structure.
Standard examples for this procedure are the monoids in monoidal categories as e.g. the monads over a fixed category, Hopf-algebras, resp. coalgebras in the sense of SWEEDLER, or monoids in the classical sense. In
order not to complicate the presentation here by lengthy technical details,
I will regard here only monoids over monoidal categories in the usual sense.
Everything, which is stated here in the following for these special
monoidal algebraic categories, is also valid for arbitrary S-monoidal algebraic categories.
By a monoidal functor I always mean a strict monoidal functor.
tion of the
Let
now
F : K -&#x3E; L be
has
an
3.1.
LEMMA.
an
INS-functor
adjoint right-inverse,
we
F: K - L
be
Let
over
a
monoidal category. Since
F
get the
an
L = ( L , 0, can). Th en there exists
INS-functor
at
least
one
over
a
monoidal category
monoidal
structure
on
K,
such that
is a mono’idal
In
functor.
general
F becomes
a
there
are a
lot of monoidal
monoidal functor,
as
the
431
structures on
following examples
K,
show.
such that
by ( S, X , can ) the category of sets with the cartesian closed structure defined by the product X . Let now F : K - S be an
INS-functor over S . Then the most important monoidal structures on K
are the following:
1) K = ( K, II , can ) with the product-monoidal structure lifted by the
INS-functor F . In general ( K , rl , can ) is no more cartesian closed as the
Denote
EXAMPLES.
cases
K = Top
K = Uni f show.(*)
or
2 ) Denote by K = ( K, D X , can ) the category
structure
defined
K-obj ect functor
by
the
D : S -&#x3E;
product-monoidal
K,
K
structure on
with the monoidal
S , and
the discrete
i.e.
Then each functor
adjoint, but ( K, D X , can ) is in general not closed monoidal.
3 ) Denote by K = ( K D ,can) the category K together with the «inductive» cartesian product-structure, i.e. k D k’ is the cartesian product
on S supplied with the finest K-structure, such that id Fk x Fk’ is an Fmorphism (continuous, uniformly continuous, measurable...( [20])) in
each argument. The canonical functorial morphisms « can» are defined as
has
an
in S . Then (K, 0, can) is
each coreflective
In
subcategory
it is
general
tary property, i.e. if
again
a
an
not
closed monoidal category. So for instance
of
Top
or
Unif
known if closed
INS-category
over a
monoidality
INS-categories
over
is
an
INS-heredi-
closed monoidal category, is
closed monoidal. Furthermore there do
rizations of those
is closed monoidal.
not
exist internal characte-
cartesian closed
categories
which
again cartesian closed. The most well-known cartesian closed INScategories over S are the categories of compactly generated and of quasiare
topological
spaces.
In the
following
we assume
that K carries any monoidal structure,
such that
(*) K =( K, il,
can.)
is cartesian closed iff k
sp ecial adjoint functor theorem.)
432
II-,
k EK ,
preserves colimits.
( Apply
monoidal functor. Denote by Mon K resp. Mon L the categories
of monoids over K resp. over L . With this notation we obtain the followbecomes
a
ing
3.3.
F : ( K , D, can ) -&#x3E; ( L , D, can )
be
a
monoidal INS-
Then the
functor.
1) The
2 ) The
defines
In
Let
THEOREM.
an
following assertions are valid:
induced functor Mon F : Mon K -&#x3E; Mon L
pair o f forget ful functors VK : Mon K
again
-K
and
an
INS- functor.
VL : Mon L -&#x3E; L
1 NS-morphi sm :
is
particular VK
In
is
is
adjoint if and only if VL
particular
the
adj oint right-inverse
adjoint.
of Mon F is
given by
Mon J : Mon L - Mon K :
with
In the
prove the
one can
3.4.
as
way
for
equationally defined algebraic categories
following
Let K be
arbitrary monoidal ,categorw,
corresponding forgetful functor. Then
LE M MA .
be the
1) V
2)
same
creates
an
limits.
V creates absolute
Since
coequalizers.
lute
adjoint functor is monadic
coequalizers, we obtain the following
3.5.
COROLLARY.
an
functor . Then
(i)
and V : Mon K -&#x3E; K
the
if and
only
if it
creates
abso-
Let F: (K, D, can)-&#x3E;(L, D can) be a monoidal INS-
following
assertions
are
equivalent :
V : Mon K - K is monadic.
( i i ) V : Mon L - L is monadic.
3.6.
COROLLARY.
functor
over a
Assume
that
Let F: (K,D, can)-&#x3E;(L, D, can) be
a
monoidal INS-
monoidal category (L, 0, can) with countable
1 0 L - and - 0 L 1
preserve these
433
coproducts.
coproducts. Then V :
Mon K- K is monadic.
3.7.
The
forgetful functor V:Top(R-mod)-R-mod from
topological R-modules over a topological ring R into the
EXAMPLE.
category of
tegory of R-modules is
R-mod is defined
a
monoidal INS-functor. The monoidal
the
ca-
structure on
product and on Top(R-Mod) by the inductive topology on the tensor product. Since R-mod is closed monoidal, all
assumptions of the corollary 3.6 are fulfilled. Hence the forgetful functor
by
the
tensor
topological R-algebras into Top (R-mod ) is monadic.
the functor « topological tensor algebra» .
(k, J.L ’ e) E Mon K. Denote by Lact ( k , K) the category
from the category of
The
coadj oint
Let
is
now
of
K-objects,
be
a
on
which k
the left. Let F : ( K , D, ca.) --&#x3E;(L , 0, can )
monoidal INS-functor. Then F induces
With this notation
3.8.
acts on
THEOREM.
Then the induced
we
a
functor
obtain the
Let F : (K, can) --&#x3E; (L , can) be a monoidal
functor
INS- functor.
again an
Lact F : Lact ( k , K) - Lact ( F k , L ) is
INS-functor.
3.9.
Let K be
INS-category over S , e.g. the category of
topological, measurable, compactly generated, zero dimensional or uniform
EXAMPLE.
an
spaces. The category K(Ab) of all abelian groups in K is
a
closed
mo-
INS-category over Ab , the category of abelian groups. The monoidal structure is given by the « inductive tensorproduct». The monoids in
in K ( Ab ) are just the rings in K . The category Lact ( r, K ( Ab ) ) is the
category of all K-modules over the K-ring r. Hence the forgetful functor
noidal
K(r-mod)-r-mod into the category of all r-modules in Ens is an INSfunctor. Hence the category K (r-mod) is complete, cocomplete, well-
powered, cowellpowered, has generators, cogenerators, projectives, injectives and a canonically defined (coequalizer, mono) bicategory structure. But K ( r-mod ) is in general not abelian, since bimorphisms need not
to be isomorphisms.
434
4.
Algebraic Categories ( in
the
T H IE B A UD )
of
sense
over
INS-categories.
algebraic category [36] includes E ILENBERG-MOORE categories, categories of finite algebras, comma-categor ies... It is defined in a completely natural way by BEN A B 0 u’ s profunctors. I regard here algebraic categories over arbitrary base categories,
but restrict myself in the case of the underlying algebraic types to types
which are induced by Ens-valued algebraic functors.
THIEBAUD’s notion of
an
4.1. ALGEBRAIC FUNCTORS AND CATEGORIES
defined in his thesis
the
StrA
structure
gories
over an
( unpublished )
and the semantics
arbitrary category A
to
pair
a
Sem A
of
[36].
adjoint
the category of comonoids
A . This
has
functors
from the category of all
in the monoidal category Dist ( A ) of all distributors
BENABOU’s
THIEBAUD
cate-
Comon (A)
( = profunctors
in
adjoint functors induces
a monad on (Cat,A). The algebras in the corresponding EILENBERGMOORE category are called algebraic functors, resp. algebraic categories.
In the presence of an adjoint (or a coadjoint) the notions of categories
algebraic over A and categories monadic (resp. comonadic ) over A coincide. In other words this pair of adjoint functors allows us to associate a
category of algebras to an arbitrary functor, in such a way that, if this
functor has an adj oint or a coadj oint, then we obtain the category of algebras or coalgebras in the sense of EILENBERG-MOORE.
Let us now briefly recall the basic definitions of algebraic functors as well as some of their properties, in particular the stability under
pullbacks. We assume that the reader is familiar with bicategories in the
terminology)
sense
of B E N A B O U
[25] .
4.1.1.
DE FINITION
[251.
Ens )
-
is
the
over
The
bicategory
given by the following data:
set of objects of Dist is the
( sm all rel .
to
pair
of
Dist of all distributors (over
category Cat
Ens);
435
of «all»
categories
- for
A , B E Cat , the category Dist (A , B) of all distributors
A -P B
category [AoP B , Ens I ;
Iet O : A U- B and 9 : B l- C be two distributors. The composition
O O Y : A -+-&#x3E; C is defined pointwise
coequalizer in Ens of the followis defined
to
X
be the functor
-
as
ing pair (d0 , d1)
of
morphisms :
These attachments define
-
for
bifunctor
A E Cat , the Hom-functor A ( - , - ) is defined
distributor
-
a
abbreviation
isomorphisms
we
REMARK
are
write 00qj
OOY (a , c) generated by
4. 1.2.
be the
identity-
1A : A + A .
The coherent natural
As
to
defined in
for the
an
obvious way.
equivalence
class in
with
[36].
in
I
iff there exists
a
finite chain
in
B , and elements
and
such
that for all i ,
visualized by the
following
commutative
436
diagram :
The category Dist (A , A), A E Cat , is
the above defined
functor
« O»
the category of all comonoids in
adj oint
functors
functorial
Dist (A , A) . We define
now
the
pair
of
( StrA , SemA ) .
Let f (G, e, d) be
are
as
monoidal category with
multiplication. Denote by Comon ( A )
a
morphisms.
a
comonoid on A . In
A
G- algebra is
a
particular
pair (a, x),
x E
G ( a, a), such that
A
morphism f : (a , x) -&#x3E; (a’ , x’) of G-algebras
is
a
where a E A and
A-morphism f : a -&#x3E; a ’,
such that
is commutative, i.e.
We shall denote
underlying forgetful
more
each
by Alg (A , G)
functor is denoted
comonoid-morphism O: (G ,
canonical way
a
functor
437
the category of
G-algebras. The
by U ( G ) : Alg ( A , G ) -&#x3E; A . FurtherE,
d) -&#x3E; (G’, E ’ , 8’) defines in
a
over A . The assignment
defines
a
futictor
SemA : Comon (A) -&#x3E; ( Cat, A).
The functor
is defined in the
fines in
following
obvious way
an
two
way: Let U : B -&#x3E; A be
functor. Then U de-
a
distributors :
Let O E Dist (A, B ) and 9 E Dist ( B , A ) be
two
arbitrary 1-cells,
i.e. dis-
tributors of Dist . Recall that 4Y is
if there exist
satisfying
2-cells
the relations
One of the
important properties of the bicategory Dist is the fact
each functor has a coadjoint, i.e. for any functor U : B -&#x3E; A
most
that in Dist
the distributor
pair
of
define
by
the
.
Dist-coadjoint (left-adjoint) to Y,
( functorial morphisms )
OU : B -l-&#x3E; A
is
Dist-caoadjoint
Dist-adjoint
functors defines
StrA (U)
the comonoid
pair (O U,
as
OU)
of
The
of functors
adjoint pair
monad denoted
a
8u )
8U
comonad, we can
on A generated
functors.
Structure is
adjoint
( StrA , Sem A)
to
induces
semantics.
on
(Cat, A)
a
by AIgA .
4.1.4. D E F I NI TI ON . A category algebraic
pbisms of algebraic categories
4.1.5.
OU : B -l-&#x3E; A . Since each
monad resp.
(OUOOU,
Dist-adj oint
PROPOSITION ( THIEBAUD ) .
a
to
over
A
are
A is
AlgA-algebra.
the AlgA-morphisms.
over
an
Mor-
EXAMPLES.
1) Let
G-algebras
is
G be
a
algebraic
comonoid
over
on
A . Then the category
A.
438
Alg (A, G )
of
2 ) Let F : A - C and G : B - C be functors. Then the comma category ( F, G ) is algebraic over A X B . In particular the category (A, a)
of objects over a and (a, A) of objects under a, a E A , are algebraic
over A .
4.1.6.
is monadic
.
be a
[36].
only if U
PROPOSITION
if
and
[36 .
PROPOSITION
pullbacck·diagram in
4.1-8.
is
algebraic
a
category
and has
a
over
A. Then U
coadjoint.
Let
Cat . Then
[36].
PROPOSITION
Let U:B -A be
if
U is
algebra!ic
Let u : B - A be
those limits and colimits which
are
so
is U’.
Then U
algebraic.
creates
absolute.
4.2. ALGEBRAIC CATEGORIES OVER INS-CATEGORIES.
4.2.1 .
A-obj ects
gory of
defined
as a
or
category
[17]).
Let T : A -&#x3E; Ens be
functor. The
cate-
T-objects T-obj (K) in an arbitrary category
over K by the pullback in Cat :
K is
(cf.
DE FI NITION
a
where Y is the
Since
4.2.2.
PROPOSITION.
algebraic
4.2.3.
Yoneda-embedding.
with T also [KoP, T]
over
algebraic
if T-obj ( K ) - K
One
is
algebraic
algebraic,
over
we
get from 4.1.7 :
Ens, then
T-obj(K)
is
K.
COROLLARY.
category
IfA
is
can
is
Let T:A - Ens be monadic. Then
over
K , and in
T-obj (K) is a
particular again monadic, if and only
adjoint.
easily prove
the
following
439
4.2.4.
PROPOSITION.
T:A-Ens
Let
be
an
algebraic functor.
Then
1 ) T-obj (K) is complete, if K is complete.
2 ) The forgetful functor T-obj ( K ) - K creates all colimits which
the
preserved by
Let
induces
4.2.5.
a
Yoneda-embedding
now
F: K - L
be
an
Y:
are
K -&#x3E;[K°p , Ens I -
INS-functor with
coadjoint
D . Then F
functor
THEOREM.
Let F:K-L be
1) T-obj F : T-obj (K) -&#x3E; T-obj (L)
an
INS-functor. Then
is an INS-functor iff T-obj F
is
fibresmall. This is equivalent to the condition, that there exists up to
isomorphisms only a set of structures Ak, k E K, such that (k, Ak ) is
a T-obj ( K )-algebra. This for instance is always the case, when K is an
INS-category over L and T-obj ( L ) - L is monadic. In the following we
assume that T-obj F is always fibresmall.
2 ) The pair of forgetful functors
forms
an
INS-morphism
visualized
In
by
particular
6.
over
UK
is monadic
if
and
only if UL
COROLLARY ( ERTEL-SHUBERT
[6]).
is monadic.
Let K be
Ens , and let T : A - Ens be monadic. Then
INS-category
T-obj (K)-&#x3E; K is again
an
monadic.
4.2.7. COROLLARY [18]. Let K be
category L , and let A be an
Then the forgetful functor
the
an
INS-category
category in the
over an
arbitrary
algebraic
of LAWVERE .
K
is
monadic
if and only if
UK : Alg ( A , K) forgetful functor UL : A lg ( A , L ) - L i s monadic.
440
sense
4.3. CONNECTION BETWEEN THE CATEGORIES OF T-OBJECTS IN K
AND PRE-T-OBJECTS IN K.
4.3.1.
Let F:K-L be
DE FINITION.
algebraic functor. The
following pullback in Cat :
be
an
i.e. the
objects
arbitrary
« K
The
4.3.2.
is
of
-structure » on
are
the
objects
of
by
the
with
an
is defined
T-obj (L)
it.
following propositions
PROPOSITION.
Pre-T-obj (K)
category
pre-T-obj (K)
INS-functor, and let T:A-Ens
an
The
are
categorical
routine.
forgetful functor
INS-functor.
an
4.3.3.
UPK : Pre-T-obj (K) -&#x3E; K is algebraic. Up
is monadic. The pair (UP, UL ) defines
if and only if UL
ph i sm.
4.3.4.
is monadic
PROPOSITION.
an
INS-mor-
Assume that T: A - Ens is monadic. Then
T-obj(K)
isomorphisms
B IR KH O F F-subcategory o f Pre-T-obj (K), i. e.
1) T-obj ( K ) is closed under products in Pre-T»obj ( K).
2) T-obj (K) is closed under extremal monos.
3 ) T-obj ( K) is closed under retracts in Pre-T-obj( K).
PROPOSITION.
is up to
4.3.5.
a
Assumption as above. T-obj (K)
category o f Pre-T-obj (K) . In particular T-obj (K) is
subcategory o f Pre-T-obj (K) for any category K.
PROPOSITION.
is
an
an
INS-sub-
epire flective
4.4. CATEGORIES OF FINITE ALGEBRAS OVER INS-CATEGORIES.
this
paragraph
we
have
since otherwise the
to
restrict ourselves
following
the category of finite
sets
notions
give
in Ens .
441
to
INS-categories
no sense.
over
In
Ens ,
Denote by Fin (Ens)
4.4.1.
F: K - Ens
Let
DE FINITION.
be
an
INS-functor. The category
Fin ( K ) of finite K-objects is defined by the following pullback in Cat : .
4.4.2.
PROPOSITION.
1 ) Fin (K) -&#x3E; Fin (Ens) is
an
INS- functor.
2 ) Fin (K) -&#x3E; K is algebraic.
The
proof
of assertion 1 is trivial, whereas 2 follows from the
fact that Fin(Ens)-Ens is
algebraic [36]
and that (*) is
a
pullback
diagram.
4.4.3.
ry
is
is
Let T:A-Ens be
algebraic functor. The categoT-obj (Fin (K)) is called the category of finite T-objects in K and
denoted by Fin (T-obj (K)). In particular Fin (T-ob j (K)) -&#x3E; Fin (K)
an algebraic functor.
DEFINITION.
4.4.4.
THEOREM.
Let T : A -&#x3E; Ens be
be
an
INS-functor.
Then the
is
again
4.4.5.
an
an
alge,braic functor
forgetful functor
an
and F : K - Ens
INS-functor.
TH EO RE M.
Let T : A - Ens be
an
algebraic functor.
Then the
as-
signement
defines a functor.
4.4.6.
EXAMPLE.
Let
H:Unif-Top
category of uniform spaces
H is
is
an
again
INS-morphism,
an
to
be the canonical functor from the
the category of
topological
the induced functor
INS-morphism,
and hence in
442
particular adjoint.
spaces. Since
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BRÜMMER,
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