nLab heteromorphism

Idea

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 “heteromorphism” there).

Definition

Given two categories $\mathcal{C}$ and $\mathcal{D}$ and a functor $L \colon \mathcal{C} \to \mathcal{D}$, then for $c\in \mathcal{C}$ and $d \in \mathcal{D}$ two objects, the set of heteromorphisms between them is the hom set

$$Het(c,d) \coloneqq \mathcal{D}(L(c),d) \,.$$

When $L$ has a right adjoint $R \colon \mathcal{D}\to \mathcal{C}$ then this is of course equivalent to

$$Het(c,d) \cong \mathcal{C}(c,R(d)) \,.$$

More generally, for $Het \colon \mathcal{C}^{op}\times \mathcal{D}\to Set$ a profunctor from $\mathcal{C}$ to $\mathcal{D}$, then $Het(c,d)$ may be called its set of heteromorphisms from $c$ to $d$.

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:

$$\mathcal{D}(L(c),d) \cong Het(c,d) \,$$

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

$$Het(c,d) \cong \mathcal{C}(c,R(d)) \,.$$

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

$$\mathcal{D}(L(c),d) \cong Het(c,d) \cong \mathcal{C}(c,R(d)) \,.$$

Properties

Characterization of adjunctions

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

• adjoint functor – In terms of cographs/heteromorphisms

• adjoint (infinity,1)-functor – In terms of cographs/Heteromorphisms

References

• Bodo Pareigis, Categories and Functors , New York: Academic Press 1970. (section 2.2; link)

• David Ellerman, A Theory of Adjoint Functors — with Some Thoughts on Their Philosophical Significance, in What Is Category Theory?, edited by Giandomenico Sica, 127–83. Milan: Polimetrica. (2006)

• David Ellerman, Adjoint Functors and Heteromorphisms (arXiv:0704.2207)

• David Ellerman, Mac Lane, Bourbaki, and Adjoints: A Heteromorphic Retrospective (pdf)

• Jacob Lurie, Higher Topos Theory (arXiv:0608040)