Because of the symmetry inherent in the Cartesian monoidal
structure of the category of sets, there is no difficulty
in treating on an equal footing a category and its opposite.
Consequently, once one understands what to mean by a (covariant)
functor from one category to another, contravariant functors
can be explained with the greatest of ease as either covariant
functors from the first category to the opposite of the second,
or, equally effectively, as covariant functors from the
opposite of the first category to the second itself. The only
small snag here is that it is the orientation of the maps in
the target category that determines the natural direction of
the maps from one functor to another in the functor category.
But this problem already rears its head in the identification
of covariant functors from
A to
B with covariant
functors from
Aop to
Bop ,
and is dealt with simply by making a well-motivated choice.
Symmetry comes into play again in defining the product of two
(or more) categories, and, hence, in explaining what to mean by
functors of two (or more) variables, of fixed, or mixed, variance:
one takes them, for example, to be (covariant) functors from
a product
Aop × B to
C .
A much larger and less easily surmountable snag arises once one
wants to generalize the observations above to the setting of
categories enriched in a closed, or monoidal, or, most generally,
multilinear category
V whose structure (like that of
almost any endofunctor category
AA
under the monoidal operation of composition) manifests no
inherent symmetry whatsoever. Here, given a
V-category
A , there may well be
no appropriate understanding
at all for what the
V-category opposite of
A
should be. Hand in hand with this difficulty comes the further
annoyance that, despite the easy availability of a notion of
covariant
V-functor from one
V-category to another,
absence of symmetry totally stymies any attempt to elucidate
what a
contravariant V-functor between two
V-categories should be. By the same token, absence of
symmetry makes it impossible to explain what to mean by a
V-functor of several variables, of fixed, or mixed,
variance.
On the other hand, even when the multilinear base category
V is not itself a
V-category (i.e., is not closed)
– even if, say,
V is non-symmetrically monoidal, like our
AA example earlier – it miraculously
remains possible to explain what to understand by
V-valued
V-functors – of
either variance –
defined on an arbitrary
V-category A . What is more,
even a notion of
V-valued V-functor of mixed variance,
defined on a pair of
V-categories A and
B ,
contravariant in
A and covariant in
B, is readily
available: this comes down to a system
F consisting of
a rule assigning to each object
A of
A and each
object
B of
B a value-object
F(A, B)
of
V , along with further information providing
V-multilinear maps
F:
< B(B, B'),
F(A, B),
A(A', A) >
F(A', B')
all satisfying conditions analogous to the usual associativity and unit laws.
Such an
F would be termed a
V-valued
V-functor of the
two variables
A and
B , contravariant in
A and
covariant in
B . A prime example of such a functor is the actual
V-valued hom-functor of the
V-category A itself.
What is more, despite the likely absence of any
V-category on which
such a
V-valued V-functor of two variables might be defined,
there is available a “law of exponents,” of sorts, identifying
V-valued
V-functors, of mixed variance, contravariant in
A and covariant
in
B, with actual (covariant)
V-functors on
B –
taking values (when
A is small enough, and
V complete enough)
in a sort of presheaf
V-category V(Aop)
of contravariant
V-valued V-functors on
A . Applied to
A’s own
V-valued hom-functor, this identification yields a
Yoneda representation
A
V(Aop) ,
which (sure enough) turns out to be
V-full-and-faithful. Even more
surprisingly, there is a twist on this “law of exponents” permitting
identification of such
V-valued V-functors of mixed variance with,
instead, single covariant
V-functors from
A to a
V-category
best described, intuitively, as the
V-opposite
(VB)op
of the category of covariant
V-valued V-functors on
B .
Note:
NOT defined on
Aop – there
may well not
BE any
V-category serving as opposite for
A ; and
NOT taking values in
VB
– there may well not be even a reasonable candidate (no matter how small
B or how complete
V may be) for a “covariant
V-valued
V-functors on
B”
V-category
VB ;
BUT defined on
A and taking values in
(VB)op .
This identification, applied to the
V-valued hom functor of
A ,
yields the “other” Yoneda representation,
A
(VA)op ,
which, like the former one, is again
V-full-and-faithful.
An amusing corollary of these considerations and others like them is the
following moral regarding the direction of morphisms in Kleisli categories,
and the variance of Lawvere-style algebras as (roughly speaking)
“representable” functors on such Kleisli categories.
One way to concoct the Kleisli category of a
V-adjoint pair
F
|– U: X A
(with
F: A X left adjoint
to
U: X A)
is to form the
full image of
F, that is, the
V-category with same objects as
A
but
V-objects-of-morphisms
[A, B] = X(F(A), F(B)).
Another way, even in the conceivable absence of
F, is to use as
morphisms from
A to
B a suitable
V-object
of
V-natural transformations between
UA and
UB
(here
UA: X V
is the
V-functor
given by
UA(X) = A(A, U(X)) ).
Now while it has traditionally been thought that it should be the transformations
from
UA to
UB – mimicking
the direction of
A-ary operations – that ought to serve as the
Kleisli-maps from
A to
B, we see here that, it being not
VX but
(VX)op
that stands any chance of being a
V-category, we really need to use
(VX)op(UA, UB) , whose “elements”
are the
V-natural transformations from
UB to
UA (!), as the Kleisli maps from
A to
B.
Fortunately, this observation is consistent with (rather than opposite to) the one
provided by the full image of
F, for, by Yoneda and adjointness,
[A, B] =
X(F(A), F(B)) ≈
(VX)op(Y(F A), Y(F B)) ≈
(VX)op(UA, UB) .
And the
V-category of Lawvere-style algebras “over”
the Kleisli
V-category K can
then be taken as the full
V-subcategory of the
V-category
V(Kop)
of contravariant
V-valued V-functors on
K whose
compositions with
A
K are explicitly representable
(which is to say: as the pullback of the diagram
| | | | | | V(Kop) | | |
| | | | | | ↓ | | |
| | A | | | | V(Aop) | | |
of
V-categories and
V-functors).
Historical remarks. Much of what is to be told here has appeared in
the author’s ancient articles in Springer LNM ## 99 and 195, or was promulgated
orally on the occasion of the esteemed Professor Charles Ehresmann's 70
th birthday
celebration at Chantilly/Fontainebleau in the summer of 1975. But the oral
seeds seem never to have taken root, having instead been simply scattered by
the winds of time, so it has seemed worthwhile to broadcast them once again,
hoping that this time they will fall on more fertile soil.
Technical remarks. It is of course a gross abuse of language to speak
of either of these functor categories
V(Aop) ,
(VA)op as
V-categories. The actual structure available is, at best, for each
pair of
V-functors F ,
G , a “job-description” for the
desired
V-ish hom-object
n.t.(F, G) , in the form of a
suitable
Sets-valued functor on
Vop – or better, on
(M0(V))op , the (opposite of the)
strictly associative monoidal category
M0(V)
(of finite strings of objects of
V)
that serves (in SLNM 195) as the
multilinear structure of
V –
with
Sets any available category of “sufficiently large” sets.
Any object of
V representing this “job-description” functor
will serve as the desired
V-natural transformations object.
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