nLab poly-morphism

This page is about the notion of poly-morphism considered by Shinichi Mochizuki. For the concept in computer science see polymorphism. Or see universe polymorphism for the concept in type theory.

Idea

For a category $C$, a poly-morphism is a collection of morphisms with common source and target, considered as a morphism in a new category $C^{poly}$ with the same objects, but with hom-sets $C^{poly}(a,b) \coloneqq P(C(a,b))$, the power set of the original hom-sets.

These play a substantive rôle in Shinichi Mochizuki‘s inter-universal Teichmüller theory (Mochizuki 12, section 0). Here we put these into the broader context of change of enriching categories along lax monoidal functors $F$ as the special case that $F$ is a power set-functor, or an equivariant version thereof.

Abstract approach

In the following, fix a monoidal category $V$ and a $V$-enriched category

$$C \in V Cat \,.$$

Examples to keep in mind for $V$ are the category Set of sets considered as a cartesian monoidal category, or the category G-set for a given group $G$, again with the cartesian monoidal structure.

Recall:

Definition

(change of enriching category from $V$ to itself)

For $F\colon V\to V$ a lax monoidal functor from $V$ to itself, the image $F_\ast(C)$ of $C$ under the corresponding change of enriching category is the $V$-category with the same objects as $C$, and with $V$-hom objects given by

$$F_\ast(C)(a,b) \coloneqq F(C(a,b)) \,.$$

The composition-, unit-, associator- and unitor-morphisms of $F_\ast(C)$ are the images of those of $C$ under $P$ suitably composed with the structure morphisms of the lax monoidal functor $P$.

For the case of poly-morphisms the relevant lax monoidal functors are power set-functors

Examples

(lax monoidal power set-functors)

• For $V=$Set equipped with its cartesian monoidal category structure, $C$ an arbitrary locally small category, take $F \coloneqq P \;\colon\; Set \to Set$ to be the covariant power set functor. This is lax monoidal since every pair of subsets $U \subseteq S$ and $V\subseteq T$ gives a subset $U\times V \subseteq S\times T$.

• For $G$ a group, $V=$GSet with the cartesian monoidal category structure, $C$ a category enriched in G-sets, let $F \coloneqq P\;\colon\; GSet \to GSet$ also be the covariant power set functor, but for $S$ a $G$-set, equip $P(S)$ with the $G$-action

  $$U\mapsto g\cdot U = \{g\cdot x \in S \mid x\in U\} \,.$$

Definition

(poly-morphisms)

For $C$ a plain category and $F \coloneqq P \colon Set \to Set$ the power set-functor (Def. ), we write

$$C^{poly} \;\coloneqq\; P_\ast(C)$$

for the image of $C$ under the change of enriching category (Def. ) along $P$.

We say that the morphisms of $C^{poly}$ are the poly-morphisms of $C$.

Explicitly, this means composition is defined to be

$$C^{poly}(a,b) \otimes C^{poly}(b,c) = P(C(a,b)) \otimes P(C(b,c)) \to P(C(a,b)\otimes B(b,c)) \to P(C(a,c)) = C^{poly}(a,c)$$

and the unit map is

$$I \to P(I) \to P(C(a,a)) = C^{poly}(a,a) \,.$$

A poly-isomorphism of $C$ is defined to be a poly-morphism of the core of $C$, hence a morphism of $(Core(C))^{poly}$. Hence a poly-isomorphism is a collection of invertible morphisms of $C$. (Note that these are not in general the isomorphisms in $C^{poly}$: all the isomorphisms in $C^{poly}$ are actual isomorphisms in $C$.)

References

The above construction for enrichment in plain sets is considered (without the above category-theoretic formulation) in:

• Shinichi Mochizuki, section 0 of Inter-universal Teichmüller theory I, Construction of Hodge theaters (2012) (pdf)

The construction for enrichment in posets is considered in

• Alveen Chand, Ittay Weiss, An ordered framework for partial multivalued functors, Computer Science and Engineering (APWC on CSE), 2015 2nd Asia-Pacific World Congress on Computer Science (arXiv:1511.00746)