The term heteromorphism (Ellerman 06, Ellerman 07) is used for the concept of a morphism not (necessarily) between two objects in the same category, but between objects in two different categories that are related by a functor, and typically by an adjoint functor. Indeed, sets of heteromorphism may be used to characterize adjunctions. Generally, the set of heteromorphisms is that assigned by the corresponding profunctor to the pair of objects.

The concept is also known as the cograph of a functor. While in traditional category theory literature the concept is maybe somewhat neglected, it serves for instance as the very definition of adjoint (infinity,1)-functors in the context of quasi-categories in (Lurie 06) (without using the term β€œheteromorphism” there).


Given two categories π’ž\mathcal{C} and π’Ÿ\mathcal{D} and a functor L:π’žβ†’π’ŸL \colon \mathcal{C} \to \mathcal{D}, then for cβˆˆπ’žc\in \mathcal{C} and dβˆˆπ’Ÿd \in \mathcal{D} two objects, the set of heteromorphisms between them is the hom set

Het(c,d)β‰”π’Ÿ(L(c),d). Het(c,d) \coloneqq \mathcal{D}(L(c),d) \,.

When LL has a right adjoint R:π’Ÿβ†’π’žR \colon \mathcal{D}\to \mathcal{C} then this is of course equivalent to

Het(c,d)β‰…π’ž(c,R(d)). Het(c,d) \cong \mathcal{C}(c,R(d)) \,.

More generally, for Het:π’ž opΓ—π’Ÿβ†’SetHet \colon \mathcal{C}^{op}\times \mathcal{D}\to Set a profunctor from π’ž\mathcal{C} to π’Ÿ\mathcal{D}, then Het(c,d)Het(c,d) may be called its set of heteromorphisms from cc to dd.

The general heteromorphic treatment of adjunctions is due to Pareigis 1970. A pair of adjoint functors arise when a Het profunctor is a representable functor on both the left and right. The left adjoint is representing functor on the left:

π’Ÿ(L(c),d)β‰…Het(c,d) \mathcal{D}(L(c),d) \cong Het(c,d) \,

and symmetrically the right adjoint is the representing functor on the right:

Het(c,d)β‰…π’ž(c,R(d)). Het(c,d) \cong \mathcal{C}(c,R(d)) \,.

Putting the two representations together gives the usual natural isomorphism characterization of a adjunction but with the het middle term:

π’Ÿ(L(c),d)β‰…Het(c,d)β‰…π’ž(c,R(d)). \mathcal{D}(L(c),d) \cong Het(c,d) \cong \mathcal{C}(c,R(d)) \,.


Characterization of adjunctions

Heteromorphisms may be used to express/characterize adjunctions. For more on this see at


Revised on June 24, 2017 08:38:52 by Thomas Holder (