Add semi-additive categories infrastructure #2305
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Also apologies if there are any nits I missed - I'll give it another lookover soon. |
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Refactored. File is about 20% shorter than before at 800 lines instead of 1000. Going through it again... |
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@jdchristensen In light of yesterday’s disastrous PR attempt, and out of respect for your time, I’d like to hold off on this PR for a few more weeks to ensure it’s truly shipshape. The rest of my contributions to Coq-HoTT will remain focused on the formalization. |
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We're now down to about half the length of the original PR. Continuing. |
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@CharlesCNorton I haven't looked at this yet and will wait until you ok it. But I'll point out a couple of things that may help. First, there's a trick for showing the a binary operation The second thing that sometimes is helpful is to study the triple product The Baer sum formalization should be particularly relevant, since it uses a lot about biproducts of abelian groups which should generalize to biproducts in any category. |
Thank you very much! |
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@jdchristensen Ready for review! Was having a tough time implementing the twist, and so kept a more direct proof for associativity. Please let me know what you think. Happy to make any required changes. 👍 |
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I've only looked over the first 20% or so.
| (biproduct_coprod_mor (semiadditive_biproduct Y Y) Y | ||
| 1%morphism 1%morphism | ||
| o biproduct_prod_mor (semiadditive_biproduct Y Y) X | ||
| f g)%morphism. |
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This is verbose. Can you come up with a better way to express these morphisms, with some arguments implicit and provided via typeclass search? This comes up a lot in the file. (A similar thing appears in the ⊕ Notation.)
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| (** ** Biproduct characterization lemmas *) | ||
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| Section BiproductCharacterization. |
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It seems to me that the lemmas in the section only require a single Biproduct to exist, rather than a SemiAdditiveCategory structure, so they could be proved in Biproducts.v.
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| (** ** Swap morphism for biproducts *) | ||
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| Section BiproductSwap. |
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Again, some or all of this section probably belongs in Biproducts.v, since these results could be useful for a single biproduct object, even if you don't know you are in a semiadditive category.
| Definition biproduct_swap (A : object C) | ||
| : morphism C (A ⊕ A) (A ⊕ A) |
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Why not [A ⊕ B -> B ⊕ A]? Then you are probably close to proving that biproducts are commutative.
| Lemma biproduct_zero_right_is_inl (Y Z : object C) (h : morphism C Z Y) | ||
| : biproduct_prod_mor (semiadditive_biproduct Y Y) Z h (zero_morphism Z Y) | ||
| = (Biproducts.inl (biproduct_data (semiadditive_biproduct Y Y)) o h)%morphism. |
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This and the next lemma are ok, but I think the proof could be built out of a few more conceptual tools. In Biproducts.v, you could prove that Biproducts.inl = biproduct_prod_mor id zero (not typed precisely), which is a useful general fact. Then the third lemma in this section is a naturality result, which tells you that biproduct_prod_mor id zero o h = biproduct_prod_mor (id o h) (zero o h). Then you just use the properties of id and zero. Seems cleaner, and that result about inl is independently useful.
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Also, I think one of the Ys can be changed to a fresh object Y', giving a slightly more general result.
| Lemma biproduct_comp_general (W X Y Z : object C) | ||
| (f : morphism C W X) (g : morphism C W Y) (h : morphism C Z W) | ||
| : (biproduct_prod_mor (semiadditive_biproduct X Y) W f g o h)%morphism | ||
| = biproduct_prod_mor (semiadditive_biproduct X Y) Z (f o h) (g o h). |
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This is a naturality result, so that should be mentioned in the comment and in the name. (Also, this is a result that is just a fact about products, not biproducts. Really products (and coproducts) should have been studied on their own first, and then biproducts could make use of those results, but I think that's a refactoring for another time.)
The examples where I've seen it used are for proving equivalences That said, the proof of associativity still looks really long, and conceptually I don't see why it should be long. I wonder if what you are missing is the associativity of the biproduct itself (for which the twist approach would help). #2016 does this for wild categories using the twist approach (but doesn't get to addition of morphisms). Not sure if it would be worth trying this. It would certainly be a lot of work, but the idea is that instead of pushing through proofs, one tries to develop the basic tools that make later proofs easy. And knowing that biproducts are associative is one of the basic facts one would expect to have. Probably not worth it here, so just take this as an aside. (This also suggests the idea of switching to wild categories. We have more about biproducts there, and they are also defined in terms of coproducts and products, so results are stated at the correct level of generality. But #2016 is still a work in progress, with @ThomatoTomato planning to reorganize things a bit.) |
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@jdchristensen Thanks for the review. Do you think I should switch to WildCat now or stick with the current approach? I suspect I know the answer, but wanted your opinion in case there's some utility in continuing. |
I don't think there's an easy answer. If your current proofs just needed mild touch-ups, I'd lean towards just sticking with Categories, since the code has already been written and it might be quite a burden to rewrite it. But so far it seems that each file needs fairly substantial revisions, which makes me wonder if it wouldn't be that much extra work to switch to WildCat. The advantage of switching is that multiple people will be interested in using some of these results and may be able to contribute to review or even proofs. One disadvantage of switching (besides the fact that you already have code for Categories) is that dealing with the plethora of typeclasses used by WildCat can be a bit confusing. |
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Break semicolon and comma-chained tactics into individual lines per code review feedback from @jdchristensen. Each tactic is now on its own line, making proofs easier to step through interactively.
Make semiadditive_biproduct a typeclass instance (::) so Coq can infer it automatically. Replace explicit (semiadditive_biproduct X Y) arguments with _ throughout, reducing occurrences from 51 to 3. Per code review feedback from @jdchristensen.
Remove incorrect indentation from morphism_addition_commutative (was indented as if inside a Section). Remove trailing whitespace throughout file.
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Extracting SemiAdditive.v from additive categories work per PR #2304 review as part of general formalization for #2288
Provides:
SemiAdditiveCategory: Class with just cat, zero object, and biproductsIsCommutativeMonoidstructure on morphisms using biproductsNote: The file is almost 1000 lines and requires refactoring, especially the associativity proof. @jdchristensen this was the only way I could get it to work - would very much appreciate help simplifying and identifying which helper lemmas can be removed. The associativity proof in particular requires many intermediate lemmas about how biproducts interact. It took me an entire week of 16 hour days to get here, but I assume there is a much more elegant method.