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Created a page for the poly-morphisms of Mochizuki. For a while I laboured under a misconception as to what they are, and I suspect so did others, despite the definition being deceptively simple. To put the construction in perspective, I generalised the definition to take as input a -category and a lax monoidal endofunctor of , rather than just and the endofunctor the (covariant) power set functor.
Taylor Dupuy is interested in a model theoretic view of how can think of interpretations of the categories resulting from this construction, we are discussing this privately.
Something looks a bit odd. Do you want to be a lax monoidal functor ? That way you can apply to hom-objects valued in .
Yes, thanks for picking that up. Stupidly, I’ve been thinking for ages that poly-morphisms are somehow morphisms given by an equivalence class of morphisms of the original category, and so really just a morphism in a quotient category. This was what made Mochizuki’s “full poly-morphisms” so odd: they are the arrows in given by the whole set of arrows between a fixed source and target. Mochizuki’s description is, technically, flawless, but he nowhere actually specifies that he’s making a new category, with such and such properties. And he does little to dispel the confusion when Scholze and Stix take a representative in an equivalence class and use that in their simplified picture.
Nice! Just an observation that this notion of polymorphism is not far from the notion of a hyperstructure.
In the general case of a lax monoidal functor, I don’t think the terminology “poly-morphism” is appropriate. I would just call that category , of course it’s a special case of the fact that lax monoidal functors induce operations between enriched category theories.
Yes, it’s true, but in highlighting this special case I hoped to end some mystique around the concept. Scholze-Stix for instance, couldn’t be convinced over the course of a week as to their utility. The original source offers no help. There’s more I plan to put in.
I (or someone else) can change the notation if the current one is really too terrible.
What about making a separate page about the general construction (if we don’t have one already), and just referring to it from the poly-morphism page?
That’s probably a better idea.
Not sure what that other page should be called though. “induced enriched category”? “pushforward of an enriched category”?
Induced enriched category sounds good
Base-changed category?
It’s called “change of base” in Emily Riehl’s Categorical Homotopy Theory in Lemma 3.4.3.
… and that’s also what it’s called at enriched category in the section Change of enriching category
Our page base change is just about pullback…
“Change of enrichment base”? And “change of base” when the context is clear?
The beginning of
suggests that using “change of base” here is due to Eilenberg-Kelly’s “Closed categories”. But I can’t check while away from the institute, since it’s behind a paywall and “libgen.io‘ is momentarily unavailable again.
added pointer to change of enriching category.
Also slightly adjusted the definition clause to make it clearer that the induced enriched structure is not something one just checks, but that one needs to define first.
I took the liberty of editing the entry a bit more.
The very good point David is making is that Mochizuki’s “poly-morphisms” are a special case of the general and well-known concept of change of enriching base – but this good point is being diluted if one renames the general concept by this special case. So I tried to edit to make the situation more explicit.
Also I changed the wording in the References-section from claiming that Mochizuki “introduced” poly-morphisms from saying he “considers” them, since the point of the whole entry now is to explain that Mochizuki has really been re-discovering here a concept that has been well known.
To amplify this further, we should add other references that consider enrichment in power sets. I think these are plentiful. But for the moment I am out of time now.
Is there a reason we talk about base? Why not something shorter like “change of enrichment”?
Offhand I can’t think of any other uses of the powerset functor as a change of enriching category.
I suspect the original intent was that “base” by itself is even shorter than “enrichment”.
added pointer to
which discusses essentially this construction, for the case of enrichment in posets and considering also powersets of objects.
Currently this entry attributes, in the References-section, its content as follows:
The above construction for enrichment in plain sets is considered (without the above category-theoretic formulation) in:
(Where I just added the two copies of “above” and the hyperlink. in order to clarify what this sentence is saying.)
But is this accurate? Is it true that it’s clear that the formulation given on this page is implicit in SM’s work, and that this entry did nothing but spell out the evident formalization?
From discussion elsewhere I am getting the impression that this is putting words in SM’s mouth for which there is no evidence that he would agree with or even recognize.
If this is the case, I suggest the entry should trade its politeness for clarity and instead say something as follows:
The above definition was motivated as an attempt to make precise sense of the informal notion of “poly-isomorphism” introduced in (IUT…). It may or may not agree with what (IUT) has in mind.
?
I suggest the entry should trade its politeness for clarity and instead say something as follows:
I concur.
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