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.
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.
In the following, fix a monoidal category $V$ and a $V$-enriched category
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:
(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
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
(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
(poly-morphisms)
For $C$ a plain category and $F \coloneqq P \colon Set \to Set$ the power set-functor (Def. ), we write
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
and the unit map is
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 $C$.)
The above construction for enrichment in plain sets is considered (without the above category-theoretic formulation) in:
The construction for enrichment in posets is considered in
Last revised on April 7, 2020 at 04:10:41. See the history of this page for a list of all contributions to it.