The terms you’re invoking already assume you’re living in antisymmetric space.
I think this is the other way around. We could decide to only care about the alternating tensors, sure, but your explanation for why we care about this component of the tensor algebra in particular is just “it turns out” that the physical world works like this. I’m trying to explain that we could have known this in advance, it’s natural to expect the alternating component to be particularly nice.
I think my answers to your other two questions are basically the same so I’ll write a longer paragraph here:
We can think of derivatives and integrals as coming from a boundary operator, sure, but this isn’t “the” boundary operator, because it’s only defined up to simplicial structure and we might choose many simplicial structures! In practice we don’t depend on a simplicial structure at all. We care a lot about doing calculus on (smooth) manifolds—we want to integrate over membranes or regions of spacetime, etc. With manifolds we get smooth coordinate charts, local coordinates with some compatibility conditions to make calculus work, and this makes everything nice! To integrate something on a manifold, you integrate it on the coordinate charts via an isomorphism to Rn (we could work over complex manifolds instead, everything is fine), you lift this back to the manifold, everything works out because of the compatibility. Except that we made a lot of choices in this procedure: we chose a particular smooth atlas, we chose particular coordinates on the charts. Our experience from Rn tells us that these choices shouldn’t matter too much, and we can formalize how little they should matter.
There’s a particularly nice smooth atlas, where every point gets local coordinates that are essentially the projection from its tangent space. Now what sorts of changes of coordinates can we do? The integral maps nice functions on our manifold to our base field (really we want to think of this as a map of covectors). That should give us a linear map on tangent spaces, and this map should respect change of basis—that is, change-of-basis should look like some map GLR(n)→R, and functoriality comes from the universal property of the tangent space. We’re not just looking at transformations, locally we’re looking at change of basis, and we expect the effect of changing from basis A to B to C to just be the same as changing from A to C. All we’re asking for is a special amount of linearity, and linearity is exactly what we expect when working with manifolds.
All this to say, we want a homomorphism because we want to do calculus on smooth manifolds, because physics gives us lots of smooth manifolds and asks us to do calculus on them. This is “why” the alternating component is the component we actually care about in the tensor algebra, which is a step you didn’t motivate in your original explanation, and motivating this step makes the rest of the explanation redundant.
(Now you might reasonable ask, where does the fractal story fit into this? What about non-Hausdorff measures? These integrals aren’t functorial, sure. They’re also not absolutely continuous, they don’t respect this linear structure or the manifold structure. One reason measure theory is useful is because it can simultaneously formalize these two different notions of size, but that doesn’t mean these notions are comparable—measure theory should be surprising to you, because we shouldn’t expect a single framework to handle these notions simultaneously. Swapping from one measure to another implies a huge change in priorities, and these aren’t the priorities we have when trying to define the sorts of integrals we encounter in physics.)
(You might also ask, why are we expecting the action to be local? Why don’t we expect GL(n) to do different things in different parts of our manifold? And I think the answer is that we want these transformations to leave the transitions between charts intact and gauge theory formalizes the meaning of “intact” here, but I don’t actually know gauge theory well enough to say more.)
(Third aside, it’s true that derivatives and integrals come from the double differential being 0, but we usually don’t use the simplicial boundary operator you describe, we use the de Rham differential. The fact that these give us the same cohomology theory is really extremely non-obvious and I don’t know a proof that doesn’t go through exterior powers. You can try to define calculus with reference to simplicial structures like this, but a priori it might not be the calculus you expect to get with the normal exterior power structure, it might not be the same calculus you would get with a cubical structure, etc.)
(Fourth aside, I promise this is actually the last one, the determinant isn’t the only homomorphism. It tensor generates the algebraic homomorphisms, meaning det−1,det2,det3,… are also homomorphisms, and there are often non-algebraic field automorphisms, so properly we should be looking at φ∘detk for any integer k and any field automorphism φ. But in practice we only care about algebraic representations and the linear one is special so we care about the determinant in particular.)
I think this is the other way around. We could decide to only care about the alternating tensors, sure, but your explanation for why we care about this component of the tensor algebra in particular is just “it turns out” that the physical world works like this. I’m trying to explain that we could have known this in advance, it’s natural to expect the alternating component to be particularly nice.
I think my answers to your other two questions are basically the same so I’ll write a longer paragraph here:
We can think of derivatives and integrals as coming from a boundary operator, sure, but this isn’t “the” boundary operator, because it’s only defined up to simplicial structure and we might choose many simplicial structures! In practice we don’t depend on a simplicial structure at all. We care a lot about doing calculus on (smooth) manifolds—we want to integrate over membranes or regions of spacetime, etc. With manifolds we get smooth coordinate charts, local coordinates with some compatibility conditions to make calculus work, and this makes everything nice! To integrate something on a manifold, you integrate it on the coordinate charts via an isomorphism to Rn (we could work over complex manifolds instead, everything is fine), you lift this back to the manifold, everything works out because of the compatibility. Except that we made a lot of choices in this procedure: we chose a particular smooth atlas, we chose particular coordinates on the charts. Our experience from Rn tells us that these choices shouldn’t matter too much, and we can formalize how little they should matter.
There’s a particularly nice smooth atlas, where every point gets local coordinates that are essentially the projection from its tangent space. Now what sorts of changes of coordinates can we do? The integral maps nice functions on our manifold to our base field (really we want to think of this as a map of covectors). That should give us a linear map on tangent spaces, and this map should respect change of basis—that is, change-of-basis should look like some map GLR(n)→R, and functoriality comes from the universal property of the tangent space. We’re not just looking at transformations, locally we’re looking at change of basis, and we expect the effect of changing from basis A to B to C to just be the same as changing from A to C. All we’re asking for is a special amount of linearity, and linearity is exactly what we expect when working with manifolds.
All this to say, we want a homomorphism because we want to do calculus on smooth manifolds, because physics gives us lots of smooth manifolds and asks us to do calculus on them. This is “why” the alternating component is the component we actually care about in the tensor algebra, which is a step you didn’t motivate in your original explanation, and motivating this step makes the rest of the explanation redundant.
(Now you might reasonable ask, where does the fractal story fit into this? What about non-Hausdorff measures? These integrals aren’t functorial, sure. They’re also not absolutely continuous, they don’t respect this linear structure or the manifold structure. One reason measure theory is useful is because it can simultaneously formalize these two different notions of size, but that doesn’t mean these notions are comparable—measure theory should be surprising to you, because we shouldn’t expect a single framework to handle these notions simultaneously. Swapping from one measure to another implies a huge change in priorities, and these aren’t the priorities we have when trying to define the sorts of integrals we encounter in physics.)
(You might also ask, why are we expecting the action to be local? Why don’t we expect GL(n) to do different things in different parts of our manifold? And I think the answer is that we want these transformations to leave the transitions between charts intact and gauge theory formalizes the meaning of “intact” here, but I don’t actually know gauge theory well enough to say more.)
(Third aside, it’s true that derivatives and integrals come from the double differential being 0, but we usually don’t use the simplicial boundary operator you describe, we use the de Rham differential. The fact that these give us the same cohomology theory is really extremely non-obvious and I don’t know a proof that doesn’t go through exterior powers. You can try to define calculus with reference to simplicial structures like this, but a priori it might not be the calculus you expect to get with the normal exterior power structure, it might not be the same calculus you would get with a cubical structure, etc.)
(Fourth aside, I promise this is actually the last one, the determinant isn’t the only homomorphism. It tensor generates the algebraic homomorphisms, meaning det−1,det2,det3,… are also homomorphisms, and there are often non-algebraic field automorphisms, so properly we should be looking at φ∘detk for any integer k and any field automorphism φ. But in practice we only care about algebraic representations and the linear one is special so we care about the determinant in particular.)