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David Corfield (Dec 01 2024 at 08:53):I was thinking about a particular situation where there's an indexing collection equipped with some kind of partial ordering, which is indexing some collection of quantities. What then is the advantage to packaging things in a cumulative way?
For instance, in probability theory one can work with either the probability density function or the cumulative density function. I know here there are representational benefits to the latter when combining discrete and continuous distributions, but I'm looking for a category-theoretic understanding of the difference, if one is possible.
It seems one gains in the representational theory of the symmetric groups by looking at FI-representations of the category of finite sets and injections. This allows for the study of certain 'stability' phenomena.
One might imagine that the benefits of cumulativity lie in the use of information relating to inclusion and colimits. Is there anything sufficiently generic to say that would speak also to the probability theory case?
Any other examples in this ballpark?
John Baez (Dec 01 2024 at 17:31):Before I saw your "for instance", I thought you were talking about 'cumulative hierarchies' in set theory, like the von Neumann hierarchy! Those must be an example of what you're talking about.
Structural set theorists may mock the 'cumulative' approach to the universe of sets, but most working set theorists cling to it with a passion that is surely not purely wrongheaded. Somehow the cumulative approach lets us get a tight grip on things in certain ways....
I had never thought of 'cumulative density functions' and the 'cumulative hierarchy' at the same time before!
David Corfield (Dec 01 2024 at 19:08):Good example. Presumably the gain there is through certain colimits.
I was thinking what makes for the least likely advantage. A totally ordered indexing set for starters, maybe. In the FI-representation case, the multiplicity of inclusions/endomorphisms is key.
Maybe the way to think about it is whether it's possible to pass back and forth between an assignment and its 'integral'. I wonder if the categorical probabilists have anything to say about cdf's and pdf's.
David Egolf (Dec 01 2024 at 23:14):I had never heard of FI-representations before! Interesting!
To try and get more examples of attaching data to ordered/nested structures, perhaps one could consider functors from categories of the form , where is a category and refers to the subcategory of obtained by keeping only monomorphisms. I think we recover the notion of FI-representation by letting , and letting our functor map to some category of vector spaces or modules.
Any presheaf or sheaf on a topological space can be viewed as a functor from a thin category. In a thin category, every morphism is a monomorphism, so a presheaf or a sheaf on a topological space is a functor from a category of the form .
Maybe this level of generality is not specific enough to the cases of interest, though.
David Corfield (Dec 02 2024 at 08:01):I see just today there's further work categorifying integration:
John Baez (Dec 02 2024 at 17:08):"Categorizing". :unamused: