nLab heteromorphism

Idea

The term heteromorphism (Ellerman 2006, Ellerman 2007) refers to 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, in which case the notion is first made explicit in Pareigis 1970, §2.2. Indeed, sets of heteromorphism may be used to characterize adjoint functors. 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 this is maybe somewhat neglected, it serves for instance as the very definition of adjoint $(\infty,1)$-functor in the context of quasi-categories in Lurie 2009) (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

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

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

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

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 $d$.

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:

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

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

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

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

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

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, section 2.2 in: Categories and Functors, Pure and Applied Mathematics 39, Academic Press (1970) [doi:10.5282/ubm/epub.7244]

• David Ellerman, A Theory of Adjoint Functors — with Some Thoughts on Their Philosophical Significance, in: G. Sica (e.) What Is Category Theory?, 12783. Milan: Polimetrica. (2006)

• David Ellerman, Adjoint Functors and Heteromorphisms [arXiv:0704.2207]

• David Ellerman, Mac Lane, Bourbaki, and Adjoints: A Heteromorphic Retrospective (2015) [pdf, doi:10.2139/ssrn.2620333]

• Jacob Lurie, Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press (2009) [pup:8957, arXiv:0608040&rbrack