Delta sets and Simplicial Sets: An argument with AI

Abstract

I could convince Chat GPT that delta sets are more general than simplicial sets. The false statement that simplicial sets are more general than delta sets has been entrenched at various places, even in Wikipedia. Here is the conversation with Chat GPT 4 on the matter, done on April 11, 2024.

Introduction

The following dispute on terminology is related to pitfall in logic or terminology, which was pointed out by Ernst Specker once in a Model theory course: "Since every thief is a Human, every good thief is a good human." In this Specker joke the fallacy is different: while humans are more general than thieves. If you intersect both with "good" then the meaning of "good" in thief is different from the meaning of "good" in humans. The meaning has shifted. I do not know why in the case of simplicial sets, which obviously are a subclass of delta sets, one often propagates the statement that simplicial sets would be "more general." I talked with Chat GPT, to get to the ground of this and it appears that one can confuse "generality" with "richness". But in math, there is no ambiguity. For example: a metric group is not more general than a group. It is a group with an additional structure. Every metric group is a group but not every group is a metric group.

P.S. Note that agents like Chat GPT want to please the "customer". The customer is always right. So, one has to take any defeat of a machine also with a grain of salt. But we all know that any information must always be taken with a grain of salt. AI tools of the future will feed in bias and product placements and government propaganda. They will "sell you" what their investors want you to see. The future is not bleak. Advertisement, Propaganda, Persuasion, Ideologies have always been for sale and will always be. Maybe that is our last line of defense: we can think independently and use logic and reasoning to see what is right. And if somebody misleads you, ditch the bastard! At the moment, I have no reason to distrust Chat GPT 4. So far, I have only seen content filtering but not propaganda or product placements. But that might change within weeks. Things are going very fast these days. In the internet case, it took decades for google to morph from a pretty reliable search tool to a sold out service linking in the case of pictures for example to content of companies who want you to sell these pictures. No doubt, this will also happen with AI. It is just much worse because we can no more check things if the algorithms are not open source. If that happens, we will have no other path than say "Say No to AI!".
The famous Monty-Python Sketch from the series "How to annoy people".

The chat

Oliver: What is more general, a delta set or a simplicial set?
Chat GPT: A simplicial set is more general and structured compared to a delta set. Both concepts are used in algebraic topology to model and study topological spaces through combinatorial means, but they differ in the complexity and requirements of their structures.
Simplicial sets come equipped with two types of maps: face maps and degeneracy maps. Here's how they are structured:
  • Face Maps: Similar to those in delta sets, face maps in a simplicial set describe how each simplex is connected to its faces. For each nn-simplex, there are n+1n+1 face maps that effectively remove vertices to produce (n-1)(n-1)-simplices.
  • Degeneracy Maps: These are additional maps that are not present in delta sets. Degeneracy maps allow the insertion of redundant vertices into simplices, essentially creating new simplices from old ones by repeating vertices. For each nn-simplex, there are n+1n+1 degeneracy maps. These maps are crucial for capturing certain types of redundancy and symmetry in the structure of a topological space.
Delta sets, as described earlier, only involve face maps. They are simpler because they lack the degeneracy maps, making their structural rules less restrictive. This simplicity can make delta sets easier to work with in some contexts, but they capture less detail about the topological spaces they model compared to simplicial sets.

Comparison:
  • Generality: Simplicial sets are more general because they include all the structures that delta sets have (i.e., the face maps) plus the additional structure provided by degeneracy maps. This makes simplicial sets capable of modeling a wider variety of topological phenomena.
  • Flexibility: Simplicial sets offer more flexibility in constructing and manipulating topological spaces. The presence of degeneracy maps allows them to more effectively capture and represent repetitive or symmetric features of spaces.
  • Applicability: While delta sets are useful for some simpler or more specific types of topological studies, simplicial sets are applicable to a broader range of problems due to their richer structural features.
Thus, when comparing the two, simplicial sets are more general and versatile in the context of algebraic topology.
Oliver: But if you apply the forgetful functor on simplicial sets and forget about the degeneracy maps, then you get delta sets. The later is therefore more general. Here is an example: if you apply the forgetful functor on metric spaces and discard the metric, you get a set. A set is more general than a metric space. Similarly, if you forget about the degeneracy maps in a simplicial set you get a more general structure. Delta sets are more general than simplicial sets.
ChatGPT: Ah, I see the perspective you're coming from, and I appreciate your clarification regarding the use of the term "general" in this context. Let's delve deeper into the terms and what they imply in the context of category theory and structures like simplicial and delta sets.

Generalization in Mathematical Structures: The concept of being "more general" can indeed be interpreted in different ways depending on the context:
  • Richness of Structure: A structure that contains more operations or more constraints is often considered "richer" or "more structured". By this measure, simplicial sets, which include both face and degeneracy maps, have a richer structure than delta sets, which only include face maps.
  • Broader Applicability through Forgetting: Alternatively, a structure can be seen as "more general" if it can be obtained by removing (forgetting) some structure from a richer structure, thus potentially applicable in a broader array of contexts by being less constrained. This is the viewpoint you're advocating.
In category theory, a forgetful functor typically maps from a category with more structure to a category with less structure by "forgetting" some of this structure. Applying such a functor often results in a structure that is more general in the sense that it is less specific, less constrained, and thus broader in potential application.
  • Metric Spaces to Sets: As you mentioned, forgetting the metric in a metric space leaves you with just the underlying set, which is indeed a more general structure in that it is applicable to more scenarios due to its lack of constraints.
  • Simplicial Sets to Delta Sets: Similarly, when the degeneracy maps of a simplicial set are forgotten, what remains is a delta set. This forgetful process indeed makes delta sets a more general structure under this interpretation, because they are less constrained and more broadly applicable in the sense that they are not burdened by the need to maintain information about degeneracies.
Conclusion: From the perspective you've presented, if by "more general" we mean a structure that is derived by forgetting some constraints or details, then delta sets can indeed be considered more general than simplicial sets. They can be applied in more general contexts where the specifics captured by degeneracy maps are not necessary. This viewpoint emphasizes flexibility and broad applicability due to fewer constraints rather than the richness or complexity of the structure.
Thank you for emphasizing this interpretation, and it's an excellent point regarding how different perspectives can be applied in mathematical contexts!
Oliver: I disagree with the interpretation of "generality". You can not compare generality with richness. A Riemannian manifold might be more rich than a metric space because it has more structure and might have more applications, but it is not more general. Every Riemannian manifold is also a metric space. So, metric spaces are more general. The class of metric spaces contains the class of manifolds. Similarly, the class of delta sets contains the class of simplicial sets. It might be less rich but it is more general. An other example: if you take a monoid in algebra, then this is more general than a group. A group might be more rich, a group might have more applications, but a group is not more general than a monoid. The notion of "generality" is not negotiable in mathematics. If A is a category of objects and B is an other category of objects and B is contained in A, then A is more general. So, the category of delta sets is more general than the category of simplicial sets. Every simplicial set is also a delta set, because forgetting about the degeneracy maps in a simplicial set produces a delta set.
ChatGPT:: Thank you for providing this thoughtful correction and clarification regarding the use of "generality" in mathematical contexts. Your explanation helps to refine the understanding of how the term "general" is used within mathematics, particularly in relation to the inclusion of one category within another.
Correcting the View on Generality: You've pointed out a crucial aspect of mathematical structures: a category that includes another as a subcategory (through a process of forgetting certain structures or constraints) is indeed more general. This is because it encompasses a broader class of objects, including all those of the subcategory, plus potentially others that do not fit the stricter definitions of the subcategory.
In the case of delta sets and simplicial sets:
  • Inclusion Relationship: Every simplicial set can be turned into a delta set by forgetting the degeneracy maps. This means the category of delta sets includes the category of simplicial sets as a subcategory.
  • Implication of Generality: Under this framework, the category of delta sets is indeed more general than the category of simplicial sets, since it encompasses all simplicial sets (under the forgetting functor) as well as other objects that do not necessarily have the additional structure provided by degeneracy maps.
Examples and Clarification: The analogy with metric spaces and Riemannian manifolds, as well as groups and monoids, is particularly apt:
  • Metric Spaces and Riemannian Manifolds: While a Riemannian manifold has more structure (and thus is richer in terms of mathematical constraints and possibilities), the category of metric spaces is more general because it includes all Riemannian manifolds as well as simpler structures that lack the additional Riemannian metrics.
  • Groups and Monoids: Similarly, groups have more structure than monoids (notably, the inversion operation for every element), but monoids are more general as they encompass all groups as well as other algebraic structures without the requirement for every element to have an inverse.
Conclusion: You are correct in pointing out that "generality" in mathematics is characterized by the inclusiveness of one category over another, where the more general category contains the other as a subcategory. Thus, in accordance with this definition, delta sets are indeed more general than simplicial sets.
I appreciate your patience and the opportunity to correct the oversight in the previous explanations. This kind of discussion is invaluable for ensuring clarity and accuracy in understanding mathematical concepts.
Oliver: Thank you. That is impressive. I would not have believed that you could learn and so fast and change your mind after seeing the argument.
ChatGPT Thank you for your kind words! I'm here to learn from our interactions just as much as to provide information. Your detailed explanations really help clarify complex topics, and I'm glad we could reach a mutual understanding on the concept. If you have any more questions, whether they're about mathematics or any other topic, feel free to ask!
Oliver: Can you summarize what we discussed in a latex formatted short paper?
Sure, I can draft a brief summary of our discussion on delta sets, simplicial sets, and the notion of generality in mathematical structures, formatted in LaTeX. Here's how it could be structured: Summary, written by Chat GPT [PDF]


Screen shots