Add semi-additive categories infrastructure by CharlesCNorton · Pull Request #2305 · HoTT/Coq-HoTT

CharlesCNorton · 2025-08-10T16:04:49Z

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 biproducts
Derives IsCommutativeMonoid structure on morphisms using biproducts
Addition defined as: f + g = ∇ ∘ (f ⊕ g) ∘ Δ
Proves identity laws, commutativity, associativity from biproduct axioms
Bilinearity of composition (distributivity)

Note: 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.

CharlesCNorton · 2025-08-10T16:14:50Z

Also apologies if there are any nits I missed - I'll give it another lookover soon.

CharlesCNorton · 2025-08-12T02:09:57Z

Refactored. File is about 20% shorter than before at 800 lines instead of 1000. Going through it again...

CharlesCNorton · 2025-08-12T10:18:52Z

@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.

CharlesCNorton · 2025-08-12T16:11:46Z

We're now down to about half the length of the original PR. Continuing.

jdchristensen · 2025-08-12T19:44:50Z

@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 * is associative and commutative. You prove commutativity as expected, but instead of directly proving associativity, you prove the "twist" law: (x * y) * z <~> (z * y) * x (or some other variant). This is easier than proving associativity, because you can use the same function in both directions, and you only have to prove one composite is the identity (since that covers both composites, just as when you are proving symmetry). Associativity follows from these two. See baer_sum_twist and equiv_trijoin_twist' in the library.

The second thing that sometimes is helpful is to study the triple product x * y * z as its own operation, which is done for both the Baer sum and the join.

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.

CharlesCNorton · 2025-08-12T21:44:04Z

@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 * is associative and commutative. You prove commutativity as expected, but instead of directly proving associativity, you prove the "twist" law: (x * y) * z <~> (z * y) * x (or some other variant). This is easier than proving associativity, because you can use the same function in both directions, and you only have to prove one composite is the identity (since that covers both composites, just as when you are proving symmetry). Associativity follows from these two. See baer_sum_twist and equiv_trijoin_twist' in the library.

The second thing that sometimes is helpful is to study the triple product x * y * z as its own operation, which is done for both the Baer sum and the join.

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!

CharlesCNorton · 2025-08-31T10:00:34Z

@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. 👍

jdchristensen

I've only looked over the first 20% or so.

jdchristensen · 2025-09-02T17:47:45Z

+      (biproduct_coprod_mor (semiadditive_biproduct Y Y) Y 
+         1%morphism 1%morphism
+       o biproduct_prod_mor (semiadditive_biproduct Y Y) X 
+         f g)%morphism.


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.)

jdchristensen · 2025-09-02T17:51:07Z

+
+(** ** Biproduct characterization lemmas *)
+
+Section BiproductCharacterization.


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.

jdchristensen · 2025-09-02T17:57:13Z

+
+(** ** Swap morphism for biproducts *)
+
+Section BiproductSwap.


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.

jdchristensen · 2025-09-02T17:58:57Z

+  Definition biproduct_swap (A : object C) 
+  : morphism C (A ⊕ A) (A ⊕ A)


Why not [A ⊕ B -> B ⊕ A]? Then you are probably close to proving that biproducts are commutative.

jdchristensen · 2025-09-02T18:02:40Z

+  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.


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.

Also, I think one of the Ys can be changed to a fresh object Y', giving a slightly more general result.

jdchristensen · 2025-09-02T18:04:01Z

+  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).


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.)

jdchristensen · 2025-09-02T18:18:28Z

Was having a tough time implementing the twist, and so kept a more direct proof for associativity.

The examples where I've seen it used are for proving equivalences A * (B * C) <~> (A * B) * C, but here you are proving equalities, so maybe it's not as useful. For equalities, you just have to give a path in one direction and you are done. For equivalences, you have to give maps in both directions and then prove that both composites are the identity, and the twist saves you a factor of 2 of work, as the same map works in both directions and the same proof handles both composites.

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.)

CharlesCNorton · 2025-09-02T18:44:26Z

@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.

jdchristensen · 2025-09-02T18:59:24Z

@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.

CharlesCNorton · 2026-03-14T12:16:16Z

This should be ok for review now @jdchristensen - it appears that the alectryon fail is a different issue?

jdchristensen · 2026-03-14T14:51:05Z

@CharlesCNorton Thanks for the update. I'm not sure when I'll get a chance to look at this PR, but it's on my radar.

This should be ok for review now @jdchristensen - it appears that the alectryon fail is a different issue?

Yes, that's a different issue, #2340. (I'm hoping @Alizter can finish that up or revert to the old code.)

jdchristensen

So far I only reviewed the changes to Biproducts.v. After fixing things, can you re-read your other changes to make sure you are not missing generalizations?

jdchristensen · 2026-03-15T00:09:23Z

+  (** Pairing into a self-biproduct. *)
+  Definition biproduct_sum_pair {X Y : object C}
+    `{BYY : @Biproduct C Z Y Y}
+    (f g : morphism C X Y)
+    : morphism C X BYY
+    := biproduct_prod_mor BYY X f g.


Is it useful to have this definition? Except for the different choices of implicit arguments, one can just use biproduct_prod_mor.

(And it would seem reasonable to make the W argument of biproduct_prod_mor implicit, since it can be deduced from the provided maps. Not sure how well it would work to make the biproduct argument implicit, but you could try.)

jdchristensen · 2026-03-15T00:11:38Z

+  (** A morphism induced on self-biproducts by maps on the two summands. *)
+  Definition biproduct_sum_map {Y Y' : object C}
+    `{BY : @Biproduct C Z Y Y} `{BY' : @Biproduct C Z Y' Y'}
+    (a b : morphism C Y Y')
+    : morphism C BY BY'
+    := biproduct_prod_mor BY' BY (a o outl BY) (b o outr BY).


This works equally well for biproducts of non-equal objects, and is just the functoriality of biproducts, so it should be done in that generality.

jdchristensen · 2026-03-15T00:14:50Z

+  Lemma biproduct_inl_sum_pair {Y : object C}
+    `{BYY : @Biproduct C Z Y Y}
+    : inl BYY = biproduct_sum_pair 1%morphism (zero_morphism Y Y).
+  Proof.
+    symmetry.
+    unfold biproduct_sum_pair.
+    rewrite biproduct_prod_zero_r.
+    apply right_identity.
+  Qed.


This and the next result don't need the objects the same. (biproduct_prod_zero_l is the dual result, generalized even further.)

jdchristensen · 2026-03-15T00:16:48Z

+  Qed.
+
+  (** Product pairing is natural in the domain. *)
+  Lemma biproduct_prod_comp (V W : object C)


The name of this and the next result should mention nat for naturality. Not sure how to indicate that it involves the domain. Maybe there are other examples to compare to?

jdchristensen · 2026-03-15T00:19:05Z

+  (** Pairing is natural in the codomain. *)
+  Lemma biproduct_sum_pair_natural {X Y Y' : object C}
+    `{BYY : @Biproduct C Z Y Y} `{BYY' : @Biproduct C Z Y' Y'}
+    (a : morphism C Y Y') (f g : morphism C X Y)
+    : biproduct_sum_pair (a o f) (a o g)
+      = (biproduct_sum_map a a o biproduct_sum_pair f g)%morphism.


This also works when the objects are not the same. Same with most or all later results.

jdchristensen · 2026-03-15T00:25:00Z

+  Lemma biproduct_codiagonal_factor_through_sum_map {Y Y' : object C}
+    `{BYY : @Biproduct C Z Y Y} `{BYY' : @Biproduct C Z Y' Y'}
+    (a b : morphism C Y Y')
+    : (biproduct_codiagonal Y' o biproduct_sum_map a b)%morphism
+      = biproduct_coprod_mor BYY Y' a b.


This is a special case of naturality of biproduct_coprod_mor in the domain, so maybe it should be proved in that generality first?

jdchristensen · 2026-03-15T00:27:26Z

+  Definition biproduct_swap (Y : object C)
+    `{BYY : @Biproduct C Z Y Y}
+    : morphism C BYY BYY
+    := biproduct_sum_pair (outr BYY) (outl BYY).


As I mentioned a few months ago, there's no reason to only define the swap when the two objects are the same.

jdchristensen · 2026-03-15T00:28:22Z

+  Lemma biproduct_prod_swap {X Y : object C}
+    `{BYY : @Biproduct C Z Y Y}
+    (f g : morphism C X Y)
+    : biproduct_sum_pair g f
+      = (biproduct_swap Y o biproduct_sum_pair f g)%morphism.


This also generalizes to different objects.

CharlesCNorton · 2026-06-10T13:39:22Z

Re-read SemiAdditive.v; nothing else generalizes. Also removed biproduct_sum_pair, proved copairing natural in the domain and derived the codiagonal factorization from it, and proved uniqueness of the enrichment.

Copilot

Pull request overview

This PR introduces foundational infrastructure for semi-additive categories in the additive category theory development, extracting a lightweight structure (zero object + biproducts) that induces a commutative monoid structure on hom-sets.

Changes:

Updates additive-category prerequisites (ZeroObjects.v, Biproducts.v) to use the newer Category.Core / Category.Objects imports.
Extends Biproducts.v with additional lemmas/operations for self-biproducts (codiagonal, swap, naturality, interaction lemmas).
Adds SemiAdditive.v, defining SemiAdditiveCategory and deriving commutative monoid structure on morphisms, plus an enrichment uniqueness result.

Reviewed changes

Copilot reviewed 3 out of 3 changed files in this pull request and generated 1 comment.

File	Description
theories/Categories/Additive/ZeroObjects.v	Adjusts imports to `Basics.*` / `Category.Core` / `Category.Objects` for zero objects and zero morphisms.
theories/Categories/Additive/Biproducts.v	Adds lemmas and operations (naturality, codiagonal, swap) supporting semi-additive structure derivations.
theories/Categories/Additive/SemiAdditive.v	Introduces `SemiAdditiveCategory` and morphism-addition + commutative monoid construction from biproducts.

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jdchristensen · 2026-06-10T14:17:22Z

+From HoTT Require Import Basics.Overture.
+From HoTT.Categories Require Import Category.Core.
+From HoTT.Categories.Additive Require Import ZeroObjects Biproducts.
+From HoTT.Classes.interfaces Require Import abstract_algebra.


The reason the build succeeds is because those dependencies must be coming via one of the other imports. But I agree with copilot that we should explicit require the things we use. (I don't remember for sure, but I may have been the one to suggest trimming the imports too aggressively.)

jdchristensen

So far I just skimmed Biproducts.v and there are some obvious mistakes in organization. Please re-read SemiAdditive.v and look for similar issues. Can you also check whether all of my previous comments have been addressed? (Even if github flags them as outdated, they are often still relevant.)

jdchristensen · 2026-06-10T14:05:36Z

+  (** A morphism induced on self-biproducts by maps on the two summands. *)
+  Definition biproduct_sum_map {X Y X' Y' : object C}
+    `{BXY : @Biproduct C Z X Y} `{BXY' : @Biproduct C Z X' Y'}
+    (a : morphism C X X') (b : morphism C Y Y')
+    : morphism C BXY BXY'
+    := biproduct_prod_mor BXY' (a o outl BXY) (b o outr BXY).


This should move elsewhere, and the comment should be updated, as it's not about self-biproducts. (Please check the other results too.)

jdchristensen · 2026-06-10T14:09:00Z

+
+  (** The codiagonal is natural in the codomain. *)
+  Lemma biproduct_codiagonal_natural {Y Y' : object C}
+    `{BYY : @Biproduct C Z Y Y}


It looks like you always have to write @Biproduct C Z Y Y, probably because Z can't be inferred. That suggests that Z should be made an explicit argument, so you could write Biproduct Z Y Y. Can you look for other places where the notation is awkward and can be improved?

jdchristensen · 2026-06-10T14:10:30Z

+  Lemma biproduct_codiagonal_natural {Y Y' : object C}
+    `{BYY : @Biproduct C Z Y Y}
+    (a : morphism C Y Y')
+    : (a o biproduct_codiagonal Y)%morphism


Can these %morphism annotations be avoided? Does the whole Categories library need to always annotate composition? Is that because o is overloaded?

jdchristensen · 2026-06-10T14:11:24Z

+  (** The functorial biproduct map commutes with pairing. *)
+  Lemma biproduct_sum_map_prod {X Y X' Y' W : object C}


Not about self-biproducts.

CharlesCNorton · 2026-06-10T16:51:16Z

Fixed in b32c4cf and 17de95f: the zero object is now explicit (Biproduct Z X Y), the misplaced lemmas moved into functoriality/symmetry/self-biproduct sections with corrected comments, imports are explicit, and swapping twice is proven to be the identity. On %morphism: o is also function composition, and Bind Scope only covers arguments of type morphism, not equations — opening morphism_scope locally per file removes the annotations entirely. I went back through all the earlier comments; the one thing I left alone is making the biproduct argument of the operations implicit, since typeclass search picks an arbitrary instance when two biproducts of the same pair are in scope, as in the swap lemmas.

jdchristensen

I made a quick review through everything. I still want to make another pass through SemiAdditive.v after you respond to my feedback.

jdchristensen · 2026-06-10T17:12:55Z

+Local Notation "X ⊕ Y" :=
+  (semiadditive_biproduct X Y)
+  (at level 40, left associativity).


We usually reserve notations in Basics/Notations.v. (I'm not exactly sure why, but I guess we should do so here too.)

jdchristensen · 2026-06-10T17:14:09Z

+    : biproduct_sum_map a b o inl BXY
+      = inl BX'Y' o a.
+  Proof.
+    etransitivity (biproduct_prod_mor BX'Y' a (zero_morphism X Y')).


I'll push a commit showing a slightly cleaner way to do this using rhs_V. More generally, lhs, lhs_V, rhs, and rhs_V can be used to avoid using rewrite when the thing being rewritten is at the top level of one side of an equality.

jdchristensen · 2026-06-10T17:17:33Z

+  intros f g.
+  unfold sgop_morphism.
+  rewrite (biproduct_prod_swap f g).
+  rewrite <- Category.Core.associativity.


I think Category.Core. can be removed here and elsewhere.

jdchristensen · 2026-06-10T17:19:28Z

+  Lemma addition_of_pairs (X Y : object C) (f1 f2 g1 g2 : morphism C X Y)
+    : sgop_morphism C X (Y ⊕ Y)
+        (biproduct_prod_mor (Y ⊕ Y) f1 g1)
+        (biproduct_prod_mor (Y ⊕ Y) f2 g2)
+      = biproduct_prod_mor (Y ⊕ Y)
+          (sgop_morphism C X Y f1 f2)
+          (sgop_morphism C X Y g1 g2).


I think this generalizes to Y ⊕ Y'.

jdchristensen · 2026-06-10T17:20:14Z

+      rewrite biproduct_prod_beta_l.
+      rewrite biproduct_prod_beta_l.


rewrite 2 here and below?

jdchristensen · 2026-06-10T17:25:12Z

+(** ** Identity laws for morphism addition *)
+
+Section IdentityLaws.
+  Context (C : SemiAdditiveCategory).


I think C should be implicit. Elsewhere too. And X Y below.

jdchristensen · 2026-06-10T17:25:49Z

+    Addition is the codiagonal composed with the pairing morphism. *)
+
+Section MorphismAddition.
+  Context (C : SemiAdditiveCategory) (X Y : object C).


All implicit?

jdchristensen · 2026-06-10T17:26:53Z

+    (X Y : object C) (f g h : morphism C X Y)
+    : (f + g) + h = f + (g + h).
+  Proof.
+    unfold sgop_morphism at 1.


This unfold doesn't seem to do anything.

jdchristensen · 2026-06-10T17:27:52Z

+    etransitivity
+      (biproduct_codiagonal Y
+        o biproduct_prod_mor (Y ⊕ Y) (sgop_morphism C X Y f g)
+            (sgop_morphism C X Y (zero_morphism X Y) h)).


This seems overly complicated. Can this be expressed using + notation? Can we avoid the etransitivity by adjusting the lhs with zero_left_identity directly?

CharlesCNorton · 2026-06-10T18:22:03Z

Arguments are implicit wherever inferable from the morphisms, ⊕ is reserved in Basics/Notations.v, addition_of_pairs is generalized to Y ⊕ Y', the remaining Category.Core. qualifiers are gone, and associativity is restated in the typeclass direction. With the zero-identity adjustments done directly on the lhs, its proof is now four lines and the monoid instance fields are all bare exacts. Thanks for the commits and your patience.

jdchristensen · 2026-06-10T21:06:50Z

LGTM! Can you squash to one commit?

Co-authored-by: Dan Christensen <jdc@uwo.ca>

CharlesCNorton · 2026-06-10T21:59:16Z

done

jdchristensen · 2026-06-11T02:26:14Z

Thanks!

jdchristensen reviewed Aug 13, 2025

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CharlesCNorton force-pushed the patch-8 branch from 9c518b5 to bdcb85a Compare June 10, 2026 21:19

Add semi-additive categories infrastructure

c88dfc2

Co-authored-by: Dan Christensen <jdc@uwo.ca>

CharlesCNorton force-pushed the patch-8 branch from bdcb85a to c88dfc2 Compare June 10, 2026 21:20

jdchristensen merged commit c0ba3f5 into HoTT:master Jun 11, 2026
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CharlesCNorton mentioned this pull request Jun 11, 2026

Add additive categories infrastructure #2304

Merged


		( Biproduct characterization lemmas *)

		Section BiproductCharacterization.


		( Swap morphism for biproducts *)

		Section BiproductSwap.

		Definition biproduct_swap (A : object C)
		: morphism C (A ⊕ A) (A ⊕ A)

		(** The functorial biproduct map commutes with pairing. *)
		Lemma biproduct_sum_map_prod {X Y X' Y' W : object C}

		rewrite biproduct_prod_beta_l.
		rewrite biproduct_prod_beta_l.

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