Contents

category theory

# Contents

## Idea

A retrofunctor (also called a cofunctor) is a kind of morphism between categories. In contrast to a functor, the assignment on objects of a retrofunctor goes in the opposite direction to the assignment on morphisms.

Retrofunctors generalise both bijective-on-objects functors and discrete opfibrations.

Retrofunctors arise naturally in the study non-cartesian internal categories.

In the literature, the terminology “cofunctor” is common. However, terminology is misleading, as a cofunctor is not the opposite of a functor. We use the terminology “retrofunctor”, as this avoids confusion, and aligns with the concept of monad retromorphism (of which retrofunctors are a motivating example).

## Definition

A retrofunctor $\varphi : A \nrightarrow B$ from a category $A$ to a category $B$ consists of a map sending each object $a \in A$ to an object $\varphi_{0}a \in B$ and a map sending each pair $(a \in A, u : \varphi_{0}a \to b \in B)$ to a morphism $\varphi_{1}(a, u) : a \to a'$ in $A$, where $a' = cod(\varphi_{1}(a, u))$, such that

• $\varphi_{1}$ respects codomains: $\varphi_{0}a' = cod(u)$ where $a' = cod(\varphi_{1}(a, u))$,

• $\varphi_{1}$ preserves identity morphisms: $\varphi_{1}(a, 1_{\varphi_{0}a}) = 1_{a}$,

• $\varphi_{1}$ preserves composition: $\varphi_{1}(a, v \circ u) = \varphi_{1}(a', v) \circ \varphi_{1}(a, u)$ where $a' = cod(\varphi_{1}(a, u))$.

Given a pair of retrofunctors $\varphi : A \nrightarrow B$ and $\gamma : B \nrightarrow C$, their composite retrofunctor $\gamma \circ \varphi \colon A \nrightarrow C$ sends each object $a \in A$ to an object $\gamma_{0}\varphi_{0}a \in C$ and each pair $(a \in A, u : \gamma_{0}\varphi_{0}a \to c \in C)$ to a morphism $\varphi_{1}(a, \gamma_{1}(\varphi_{0}a, u))$ in $A$. This defines a category $\mathbf{Cof}$ whose objects are small categories, and whose morphisms are retrofunctors.

## Properties

###### Proposition

The category $\mathbf{Cof}$ of small categories and retrofunctors has an orthogonal factorization system $(Bij^{op}, DOpf)$ which factors each retrofunctor $\varphi : A \nrightarrow B$ into a bijective on objects functor $A \leftarrow I$ followed by a discrete opfibration $I \to B$.

###### Proposition

Let $\mathbf{Poly}(1, 1)$ be the monoidal category arising from the bicategory of polynomials on the singleton set. Then $\mathbf{Cof}$ is isomorphic to the category of comonoids in $\mathbf{Poly}(1, 1)$.

###### Proof

Originally proven in (Ahman-Uustalu 2016). See (Spivak-Niu 2021, Theorem 6.26) for details.

## Examples

• Every function $A \to B$ yields a retrofunctor $disc(A) \nrightarrow disc(B)$ between discrete categories. This defines a fully faithful functor $\mathbf{Set} \to \mathbf{Cof}$.

• Every monoid homomorphism $A \to B$ yields a retrofunctor $B \nrightarrow A$. This defines a fully faithful functor $\mathbf{Mon} \to \mathbf{Cof}^{op}$.

• Every bijective on objects functor $A \to B$ yields a retrofunctor $B \nrightarrow A$.

• Every discrete opfibration $A \to B$ yields a retrofunctor $A \nrightarrow B$.

• Every split Grothendieck opfibration has an underlying retrofunctor given by the splitting.

• More generally, every delta lens has an underlying retrofunctor.

• Let $\mathbb{N}$ denote the monoid of natural numbers under addition. A retrofunctor $\tau : A \to \mathbb{N}$ is the same as a choice of morphism $\tau(a, 1)$ out of every object in $a \in A$.

• A comorphism of Lie groupoids is an internal retrofunctor in the category Diff of smooth manifolds; see (Higgins-Mackenzie 1993, definition 5.12).

## References

The notion of retrofunctor first appeared under the name comorphism in the paper:

The definition of internal retrofunctor between non-cartesian internal categories was introduced in Section 4.4 of the thesis:

• Marcelo Aguiar, Internal categories and quantum groups, PhD thesis, Cornell University, 1997 (pdf)

The characterization of retrofunctors as morphisms between directed containers is developed in the papers:

The link between retrofunctors, delta lenses, and split Grothendieck opfibrations is developed in the papers:

Retrofunctors between groupoids and the link with inner automorphisms of groupoids is explored in the paper:

The notion of retrofunctor between partite internal categories is introduced in Definition 5.5 of the paper:

A detailed account of the relationship between retrofunctors and polynomials appears in Chapter 6 of the draft textbook:

• David Spivak, Nelson Niu, Polynomial Functors: A General Theory of Interaction, 2021 (pdf)

Enrichment in the category of $\mathbf{Cof}$ of categories and retrofunctors is considered in the paper:

The terminology retrofunctor was introduced in:

• Matthew Di Meglio, The category of asymmetric lenses and its proxy pullbacks. Master’s thesis. Macquarie University, 2021. doi: 10.25949/20236449, pdf

The terminology is justified by the fact that retrofunctors are monad retromorphisms in the double category $Span(Set)$.

Last revised on November 3, 2023 at 09:24:39. See the history of this page for a list of all contributions to it.