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Stream: learning: questions

Topic: monoid-annotated morphisms

Asad Saeeduddin (Sep 19 2021 at 22:04):

Hello there. I am working with a construction in which I have a category of "measured" arrows, such that every arrow in a hom collection is annotated not merely with a source and target object, but also a "measurement" that is valued in some monoid. the measurements of an identity arrow and a composition of arrows work as you might expect.

i am not sure what to call this kind of thing, and i can't seem to neatly reconcile it with existing concepts i'm (admittedly tenuously) familiar with, such as enrichment. as near as i can see, if a category is a "monad in Prof", this thing is like some kind of 2-categorical analogue of a "graded monad" or "lax monoidal pseudofunctor into Prof".

is there already a name for this concept, and is there some less awkward way of thinking about this than a "graded monad in Prof"?

Nathanael Arkor (Sep 19 2021 at 22:10):

If you consider the monoid as a one-object category, this is just a functor from your category to the monoid: each arrow gets sent to a monoid element (viewed as an endomorphism), in such a way that composition respects the monoidal structure.

Asad Saeeduddin (Sep 19 2021 at 22:29):

Ah, very nice; i didn't see that coming. Thank you!

Asad Saeeduddin (Sep 19 2021 at 22:52):

It's kind of interesting. On paper, these seem like entirely equivalent representations of the concept. But computationally, they seem to be very different. In one approach it is possible to map a triple of a monoid element, source object and target object to an appropriate type of arrows. But when the functor is from the category to the monoid, the type of morphisms of a given measurement involves a comprehension over the collection of all the arrows in a given hom.

Matteo Capucci (he/him) (Sep 20 2021 at 06:59):

The Para construction is quite similar: if you have a monoid acting on a category, it produces a category with the same objects but where an arrow $X \to Y$ is given by a choice of an element of the monoid, say $p$, and an arrow $p \bullet X \to Y$

Max New (Sep 30 2021 at 13:49):

So yes, these are "just" functors to a monoid, but there are usually two ways to look at indexing: a fibration can be viewed as a functor satisfying certain lifting properties, or its corresponding functor to Cat. In general a functor $X \to Y$ is the same as a lax functor from $Y$ to Span (or normal lax functor to Prof). So in your case, your monoid-indexed category is the same as a lax functor from the monoid to Span, which probably reads more closely to the original definition you gave. Your intuition that the latter presentation is nicer "computationally" has been argued here: https://arxiv.org/abs/1705.04296

Mike Shulman (Sep 30 2021 at 17:55):

I think this is also the same as a category enriched over the monoidal category $Set^M$, with Day convolution monoidal structure induced from your monoid $M$ regarded as a discrete category.

Dylan McDermott (Oct 05 2021 at 10:36):

Richard Wood's thesis (https://dalspace.library.dal.ca/handle/10222/55465) is about these. His large V-categories (Def 1.1) are like categories, but where each morphism is annotated with an object of $V$ (where $V$ is a monoidal category, such as a monoid viewed as a discrete category). Wood shows (Thm 1.6) that large $V$-categories are categories enriched in $\mathrm{Set}^{V^{\mathrm{op}}}$ with Day convolution.