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John Onstead (Oct 22 2025 at 20:23):Hi! My name is John Onstead. This will be an extremely long post here, but I wanted to start a page for myself to help keep track of my ongoing goals and explorations (currently 6 of them) from my time here on this server. Think of it as a summary or recap. It's also a good resource for me to refer anyone to when I start a new discussion under one of these goals so I don't have to re-explain where I currently am in my understanding. It also acts as an index where I can easily find past conversations on here. So without further ado here are all my goals, listed in no particular order!
John Onstead (Oct 22 2025 at 20:23):Goal:
Mathematical objects can be viewed in three ways: Materialism, explicit constructions from primitive objects; Structuralism, sets with specified structures and properties; and Category Theory, abstraction of structure itself. These form a spectrum from explicit (material) to implicit (categorical). This project aims to relate the implicit notion of structure in category theory to the more explicit formulations of structuralism.
Progress:
I initially assumed “stuff, structure, property” in category theory (which I will call cSSP) mirrored the same “stuff, structure, property” as in structuralism (which I will call sSSP). I later learned they differ fundamentally:
In sSSP, we begin with sets/sorts (stuff), define relations/functions (structure), and impose equations or constraints on those relations (property). In cSSP, we instead examine functors and classify what exactly is “forgotten” by them: fully faithful (property), faithful (structure), or non-faithful (stuff). However, identifying a functor alone doesn’t specify the explicit form of the structure that is forgotten— multiple explicit constructions can yield the same functor forgetting them. Thus, explicit constructions should be seen as different presentations of the forgetful functor, akin to how distinct model categories present the same infinity category, or as a direct generalization of how in categorical logic multiple different sketches can present/possess equivalent categories of Set-models.
Finally, while sSSP aligns with taking limits in Cat (building categories whose objects are explicit tuples subject to explicit restrictions), cSSP aligns with colimits in Cat (using the 2-notion of image factorization in Cat). The goal of this project is to bridge these—linking the explicitness of sSSP with the implicitness of cSSP within category theory.
Next Steps:
The main question to answer here is the following: "Every explicit construction gives rise to an appropriate forgetful functor. Does every forgetful functor admit at least one presentation in terms of explicit constructions? If so, how about non-trivial presentations, and if not, then when do non-trivial presentations exist?"
Another question is how the different kinds of 2-limit in Cat: product, equalizer, and equifier, might correspond to the equipment of stuff, structure, property respectively, and how this association might generalize to higher versions of Cat with higher notions of limit and "n-stuff".
Relevant Posts:
1 Connection between limits in Cat and stuff, structure, property. Through this was when I first learned that there was a difference between cSSP and sSSP.
2 Related discussion trying to tie together the idea of objects, morphisms, and diagrams (a structuralist way of thinking) with the approach given by forgetful functors
3 A discussion on syntax-semantics duality and categorical logic. First attempt to try and find a way of making the implicit structure of a concrete category explicit
John Onstead (Oct 22 2025 at 20:24):Goal:
Category theory offers a common framework in which a repeating pattern across math can be "externalized" into a single category theoretic concept, of which every occurrence is then just an "internalization" of that concept into the appropriate category. For instance, the idea of cartesian product of sets, group product, and product space are all cases of the categorical product's universal property. The goal of this project is to find such a common externalization of every instance of "locality" and "local-global correspondence" in math.
Progress:
I've actually made quite a bit of progress on this one. Thanks to discussions involving generators and covers, it seems the best candidate for an internal notion of "locality" is the idea of a joint regular/effective epimorphism (regular/effective epi sink). These sinks seem to allow one to define a good notion of "cover" such that both the stuff and structure inside the object is appropriately "covered", allowing one to interpret each morphism in the sink as a "local part" of the object, and hence interpret the sink itself as a "local decomposition" or "local resolution" of an object. It also seems that one can characterize the idea of a "local property" in terms of these sinks, by simply defining an object to locally satisfy a property if there exists an effective epi sink where the domains all exist in some subcategory of the target category (interpreted as all objects that "globally" satisfy the property). Lastly, it seems these sinks can also aid in understanding local-global correspondences given how one can go from local to global by "gluing" these local components together over overlaps via the definition of effective epi.
One major source of local-global correspondences are sheaves. I've recently made great strides in understanding them, as well as in relating them back to effective epi sinks. Indeed, the defining property of a sheaf is that it is a local object with respect to a choice of sieves, and localizing at a sieve is essentially turning some selected sink into an effective epi sink. So a sheaf is basically a presheaf that "thinks" some sink is effective epic!
Next Steps:
While there is finally a good candidate for an externalization we can work with, the point of the project is to devise an externalization that works in all cases. So confirming effective epi sinks are a good notion of "locality" means considering notions of locality and local-global correspondences across all of math and ensuring they can all be thought of as special cases of this concept.
That's not to say there aren't other candidates either. One of them is the idea of a "(semi) final sink" which seems to be a more "relative" version of (effective) epi sinks. It's not clear which concept is the better fit- effective epi sinks work without specifying any additional information to the category, but there might be cases where the final sinks work better, especially if one is working with a category concrete over Set.
Another component to this project is in trying to better understand calculus and analysis from the category theoretic point of view by approaching these concepts from the local-global point of view. My goal here is to fully understand why there's so many notions of differentiation and integration, even just within the context of standard differential geometry (and the problem seems to get worse, not better, when generalizing beyond).
Still another component is finalizing my understanding of sheaves. While sheaves to Set make sense from the above point of view, more work needs to be done to connect the above point of view to sheaves targeted in other categories, as well as sheaves for Lawvere-Tierney topologies.
Relevant Posts:
1 This establishes joint effective epi sinks as the best candidate to define local properties with respect to, since they are the sinks that actually do "cover" all the structure on an object such that the object.
2 Shows that effective epi sinks are a good candidate to use to define a notion of "local property" internal to an arbitrary category, since they include all examples of local topological properties.
3 Preliminary discussion on generators, which later contributed to the discussion on effective epi sinks.
4 A first attempt to trying to understand integrals from the local-global point of view, by trying to relate them to sheaves.
5 When I was first able to finally understand sheaves and what they are doing from the perspective of "gluing".
6 Preliminary discussion on sheaves in the context of locality.
7 Understanding local objects to warm up to sheaf theory
8 and 9 Understanding resolutions, the bar construction, and monadic descent
10 First post on the topic
John Onstead (Oct 22 2025 at 20:24):Goal:
The idea of "space" is not well defined mathematically, and there are many competing descriptions. The aim of the project is to establish a minimal definition of what it means for something to be a "space", and build on top of that a hierarchy that includes increasingly more intricate and rich notions of space.
Progress:
There is significant overlap between the previous goal and this one, as often times spaces are thought of as abstracting notions of "locality". So progress there might inform progress here and vice versa.
In any case, there doesn't seem to be a unified picture of what a "space" is, either from the perspective of sets with structure, or the more abstract perspective of defining the term synthetically using categories with extra structure. In the former case, a good candidate is the idea of a (T,V) algebra, and in the latter case one has concepts like topologically concrete categories, model categories, infinity categories, extensive categories, and certain kinds of site.
I realized the best approach might then be to embrace the idea there is no one unified notion of space by instead trying to come up with a "space hierarchy"- a systematic way of looking at a category (or category-like structure) and determining how "space-like" the objects of that category are. So instead of declaring one kind of object "a space" and another "not a space", instead it's more of a spectrum of "space-ness".
Next Steps:
The obvious next step is to define this hierarchy, and try to make it as "narrow" as possible (IE, minimizing cases where notions of "space" merely converge/overlap rather than one being a strict generalization of the other).
As part of this, one of my goals is to determine a "root" to this hierarchy: a minimum criteria that something must have to be "space-like" in any capacity. One candidate I'd like to look into more is the notion of a "connected component". It seems intuitive that something cannot be space-like at all if there's no notion of a connected component. However, just like with the idea of "space", there's many kinds of "connected component" and so this line of questioning would involve needing to find the most general kind of structure possible we can equip onto a category that gives a good notion of "connected component" to its objects.
Relevant Posts:
1 The main thread on defining categories of spaces.
2 The main thread on defining spaces using sets with structure.
John Onstead (Oct 22 2025 at 20:25):Goal:
One of my initial goals in understanding category theory was to find the most general setting to do mathematical things. At first, category theory itself seemed a good place, but even category theory itself was generalized by enriched category theory, and that itself is generalized by formal category theory in a virtual equipment or even augmented VDC. The goal is to fully understand the mechanisms of passing from categories to VDCs and more general structures like VTCs and so on. There are both exploratory and learning elements to this project.
Progress:
I've learned a great deal about higher dimensional structures like double categories, VDCs, and similar concepts. I understand the idea of a "generalized multicategory" as being a category-like structure, and the idea that one can do formal category theory on objects if they form a virtual equipment (and perhaps more generally, an augmented virtual double category).
Interestingly, it seems a notion of "formalization" exists for an arbitrary generalized multicategory (or at least well behaving ones). That is, one can start with some monad T on a VDC and find another monad T' whose generalized multicategories can be thought of as "formalizing" those of T (as first covered in Leinster's generalized enrichment paper). For instance, if we let T be the identity monad on Span, the T-multicategories are plain categories (since a category is just a monad in Span). T' is then the fc monad, whose generalized multicategories are fc-multicategories, also known as VDCs, which are indeed settings for formal category theory (at least, some of them are, those being the virtual equipments).
Next Steps:
A large amount of these concepts can be taken further in a vast number of different directions. For instance, for the last point, if we let T be the fc monad whose multicategories are VDCs, do T'-multicategories correspond to VTCs, or does the formalization from VDCs to VTCs require something beyond the framework of generalized multicategories? What does formal category theory looks like in a VTC? Is there a triple version of a virtual equipment, such that one might study a "formal VDC theory" within them, like how one studies formal category theory in a virtual equipment? Might this even mind-bendingly lead to a "meta formal category theory", IE, a "formal formal category theory"?
We might also ask: How might generalized polycategories work, and how might they contribute to this framework and line of inquiry?
Going another way, one can realize augmented VDCs as a special case of poly-VDC, where cells consist of chains of horizontal morphisms in both the source AND the target. Is there any benefit in formal category theory to exploring these objects (which also happen to include co-VDCs like Span for categories that don't have pullbacks)? Is there a notion of a "poly-generalized multicategory" (not to be confused with the aforementioned generalized polycategory!!!) which are based on monads on poly-VDCs? If so, maybe the "poly-generalized multicategories" that formalize normal categories are poly-VDCs, in the same way as the generalized multicategories that formalize normal categories are the VDCs?
Relevant Posts:
1 Covers interesting facts about VDCs and hints that functors between non-exponentiable VDCs might form VTCs and so on.
2 Search for a generalized notion of enrichment
3 Preliminary discussion on VDCs and intro to VTCs
John Onstead (Oct 22 2025 at 20:26):Goal:
This one is less an exploratory project and more of a learning project. I really want to understand gauge theory, but even after reading books about it like John Baez' book, it's still hard to comprehend.
Progress:
I've gotten to understanding principal bundles, connections, and associated bundles (though I'll probably need to do a quick refresher on those before I jump back into this topic).
Relevant Posts:
1 Main thread
John Onstead (Oct 22 2025 at 20:26):Set theory:
1 Godel's second incompleteness theorem in CT
2 Connection between Set theory and category theory
3 Comparing the ability to define mathematical objects internal to categories with defining them internal to material foundations like set theories or second order arithmetic
New mathematical structures:
1 What IS a base representation?
2 What category do mathematical models of physical systems form? First discussion
3 Considering what an "analytic structure" might be like. Might tie into the discussion on locality with defining the different notions of calculus
Misc:
1 Understanding typing in category theory and how to handle functions between unlike objects
2 An exercise in the relation between proving something and axiomatically defining something
Elisha Goldman (Oct 22 2025 at 23:31):Cool! For locality, it might be helpful to use Lie theory as a main source of questions and examples, since it's the other main definition of local/global correspondences. I believe categorical Lie theory is less developed than other forms of category theory so there's a lot more to explore, but that might just be my ignorance
John Onstead (Oct 23 2025 at 19:21):Elisha Goldman said:
Cool!
Thanks!
Elisha Goldman said:
For locality, it might be helpful to use Lie theory as a main source of questions and examples, since it's the other main definition of local/global correspondences. I believe categorical Lie theory is less developed than other forms of category theory so there's a lot more to explore, but that might just be my ignorance
Interesting, I will have to look into that!
David Corfield (Oct 24 2025 at 09:26):John Onstead said:
In sSSP, we begin with sets/sorts (stuff), define relations/functions (structure), and impose equations or constraints on those relations (property). In cSSP, we instead examine functors and classify what exactly is “forgotten” by them: fully faithful (property), faithful (structure), or non-faithful (stuff). However, identifying a functor alone doesn’t specify the explicit form of the structure that is forgotten— multiple explicit constructions can yield the same functor forgetting them. Thus, explicit constructions should be seen as different presentations of the forgetful functor, akin to how distinct model categories present the same infinity category, or as a direct generalization of how in categorical logic multiple different sketches can present/possess equivalent categories of Set-models.
I'm not convinced these are so different. The factorization of the forgetful functor from say to gives you a way to carry out your sSSP construction. Start with the stuff of sets. For a given set consider all the group structures which may be placed on it. For any group, see if it has the property of being abelian.
Look at David Jaz Myers doing just this composition with an extra layer (2-stuff) in his recent lecture on systems. The 2-stuff of modules of systems, the 1-stuff of systems and interactions, the structure of algebraic laws, and property of algebraic law laws (around 24:00 explicitly on stuff, property, structure).
As to the difference between a presentation and the thing presented, try Mike Shulman's What is an n-theory? account of syntactic and semantic -theories.