Skeletal Subcategories: Tensor Structure Preservation

Hey everyone! Today, let's dive into a fascinating question in category theory that touches on the interplay between skeletal subcategories and tensor structures. This is a bit of a niche topic, but it has some really cool implications for how we think about the structure of categories, especially those with some kind of product operation defined on them. So, grab your metaphorical thinking caps, and let's get started!

The Central Question: Preserving Tensor Structure in Skeletal Subcategories

Category theory, at its heart, is about abstracting away the specific details of mathematical objects and focusing on the relationships between them. One of the key tools in this abstraction is the notion of a category itself: a collection of objects and morphisms (or arrows) between them, satisfying certain composition laws. But sometimes, categories can be quite large and unwieldy. That's where the idea of a skeleton comes in handy. A skeleton of a category is a smaller, equivalent category that contains exactly one object from each isomorphism class of the original category. Think of it as a minimal representative for each “kind” of object in your category.

Now, let's throw in another layer of structure. A tensor category is a category equipped with a bifunctor, often denoted by ⊗ (like a product), which allows us to “multiply” objects within the category. This tensor product gives the category a kind of algebraic structure. Common examples include the category of vector spaces with the usual tensor product, or the category of sets with the Cartesian product. The main question we're tackling today is this: Suppose we have a (small) category C equipped with a bifunctor ⊗ : C × C → C. Does there always exist a skeleton C’ ⊆ C such that whenever c and d are objects in C’, their tensor product c ⊗ d is also in C’? In simpler terms, can we find a skeleton that “respects” the tensor structure, meaning that the tensor product of any two objects in the skeleton remains within the skeleton?

This question is significant because it asks whether we can simplify a category while preserving its essential multiplicative structure. If such a skeleton exists, it would allow us to work with a smaller, more manageable category without losing the information encoded in the tensor product. This has implications for various areas of mathematics where tensor categories appear, such as representation theory, quantum field theory, and topological quantum computation. Imagine trying to understand a complex system with many interacting components. A skeleton that preserves the tensor structure would allow us to focus on the essential building blocks and their interactions, simplifying the analysis.

Diving Deeper: Why This Question Matters

To really appreciate the question, let's break down why it's not immediately obvious that such a skeleton should always exist. The process of forming a skeleton involves choosing a representative object from each isomorphism class. This choice, while seemingly innocuous, can have subtle consequences when we introduce a tensor product. The tensor product c ⊗ d might be isomorphic to an object that is not in our chosen skeleton, even if c and d are. This is where the challenge lies: we need to make our choices of representatives in a way that ensures the tensor product stays within the skeleton.

One way to think about this is to consider the potential for “drift.” Suppose we start building our skeleton, and we have two objects c and d in it. We know c ⊗ d exists in C, but its isomorphism class might be represented by an object e that we haven't included in our skeleton yet. If we simply add e to the skeleton, we now have three objects, and we need to consider their tensor products. The process could potentially continue indefinitely, with new objects constantly being generated by tensor products, making it difficult to construct a skeleton that is closed under the tensor operation.

The difficulty is further compounded by the fact that the tensor product doesn't necessarily behave in a straightforward way with respect to isomorphisms. While c ≅ c’ and d ≅ d’ implies c ⊗ d ≅ c’ ⊗ d’, this doesn't tell us whether the representative of the isomorphism class of c ⊗ d is related in any simple way to the representatives of c and d. We need a way to control how the tensor product interacts with our choice of representatives.

This question also highlights the importance of the axiom of choice in category theory. The construction of a skeleton often relies on the axiom of choice to select a representative from each isomorphism class. However, the axiom of choice is a powerful tool that can sometimes lead to non-constructive results. In this case, even if we can prove that a skeleton with the desired property exists, it might be difficult to explicitly construct such a skeleton without additional assumptions or structure on the category C. This brings us to the importance of considering specific examples and counterexamples to gain a deeper understanding of the problem.

My Initial Thoughts: A Guess and the Road Ahead

My initial guess is that the existence of such a skeleton isn't guaranteed in general. I suspect that there might be counterexamples, particularly in categories with complicated tensor structures or where the isomorphism classes are “large” in some sense. However, proving this requires constructing a specific counterexample, which is often the hardest part in these kinds of problems. To build such an example, we might need to consider categories where the tensor product interacts with the isomorphism classes in a non-trivial way, creating a kind of “feedback loop” that prevents the construction of a suitable skeleton.

Alternatively, it might be possible to prove that such a skeleton does exist under certain conditions. For example, if the category C has some additional structure, such as being a monoidal category with a braiding or symmetry, it might be possible to use this structure to guide the choice of representatives and ensure that the tensor product remains within the skeleton. Another approach could be to impose restrictions on the tensor product itself, such as requiring it to be particularly well-behaved with respect to isomorphisms.

To make progress on this question, the next steps would involve exploring specific examples of tensor categories and trying to construct skeletons that preserve the tensor structure. We could start with relatively simple examples, such as the category of finite-dimensional vector spaces over a field, and then move on to more complicated examples, such as categories of modules over an algebra or categories of representations of a group. By examining these examples, we can gain intuition for the kinds of phenomena that can occur and potentially identify the key properties that determine whether a tensor-preserving skeleton exists.

Ultimately, this question touches on some fundamental aspects of category theory and highlights the subtle interplay between different categorical structures. Whether a tensor-preserving skeleton always exists, or only exists under certain conditions, the answer will provide valuable insights into the nature of categories and their applications in various areas of mathematics and physics. So, let's keep exploring and see what we can discover!

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Does a skeleton C' of a category C with a bifunctor ⊗ exist such that c ⊗ d is in C' whenever c and d are in C'?

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Skeletal Subcategories: Tensor Structure Preservation