Variance Vs. Volition

F.E.J. Linton

Wesleyan Univ., Middletown, CT 06459 USA

    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) > long right arrow 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 long right arrow 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 BV-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 long right arrow (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 long right arrow A (with F: A long right arrow X left adjoint to U: X long right arrow 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 long right arrow 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 long right arrow K are explicitly representable (which is to say: as the pullback of the diagram
    A   long right arrow  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 70th 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.

27 May 2005, New Haven, CT (USA). All rights reserved.