What is known and/or written about “Frobenius eliminators”?

[Peter LeFanu Lumsdaine]

Briefly: I’m looking for background on the “Frobenius version” of elimination rules for inductive types.  I’m aware of a few pieces of work mentioning this for identity types, and nothing at all for other inductive types.  I’d be grateful to hear if anyone else can point me to anything I’ve missed in the literature — even just to a reference that lays out the Frobenius versions of the rules for anything beyond Id-types.  The proximate motivation is just that I want to use these versions in a paper, and it’d be very nice to have a reference rather than cluttering up the paper by writing them all out in full…

In more detail: Here are two versions of the eliminator for identity types:

Γ, x,y:A, u:Id(x,y)  |– C(x,y,u) type
Γ, x:A |– d(x) : C(x,x,r(x)) type
Γ |— a, b : A
Γ |— p : Id(a,b)
——————————————
Γ |— J(A, (x,y,u)C, (x)d, a, b, p) : C(a,b,p)

Γ, x,y:A, u:Id(x,y), w:Δ(x,y,u) |– C(x,y,u,w) type
Γ, x:A, w:Δ(x,x,r(x)) |– d(x,w) : C(x,x,r(x),w) type
Γ |— a, b : A
Γ |— p : Id(a,b)
Γ |— c : Δ(a,b,p)
——————————————
Γ |— J(A, (x,y,u)Δ, (x,y,u,w)C, (x,w)d, a, b, p, c) : C(a,b,p,c)

(where Δ(x,y,u) represents a “context extension”, i.e. some finite sequence of variables and types w_1 : B_1(x,y,u), w_2 : B_2(x,y,u,w_1), …)

I’ll call these the “simple version” and the “Frobenius version” of the Id-elim rule; I’ll call Δ the “Frobenius context”.  The simple version is a special case of the Frobenius one; conversely, in the presence of Pi-types, the Frobenius version is derivable from the simple one.

Most presentations just give the simple version.  The first mention of the Frobenius version I know of is in [Gambino, Garner 2008]; the connection with categorical Frobenius conditions is made in [van den Berg, Garner 2008], and some further helpful explanatory pointers are given in [Gambino, Sattler 2015].  It’s based on this that I use “Frobenius” to refer to these versions; I’m open to suggestions of better terminology.  (All references are linked below.)

The fact that the Frobenius version is strictly stronger is known in folklore, but not written up anywhere I know of.  One way to show this is to take any non right proper model category (e.g. the model structure for quasi-categories on simplicial sets), and take the model of given by its (TC,F) wfs; this will model the simple version of Id-types but not the Frobenius version.

Overall, I think the consensus among everyone who’s thought about this (starting from [Gambino, Garner 2008], as far as I know) is that if one’s studying Id-types in the absence of Pi-types, then one needs to use the Frobenius version.

One can also of course write Frobenius versions of the eliminators for other inductive types — eg Sigma-types, W-types, …  However, I don’t know anywhere that even mentions these versions!

I remember believing at some point that at least for Sigma-types, the Frobenius version is in fact derivable from the simple version (without assuming Pi-types or any other type formers), which would explain why no-one’s bothered considering it… but if that’s the case, it’s eluding me now.  On the other hand, I also can’t think of a countermodel showing the Frobenius version is strictly stronger — wfs models won’t do for this, since they have strong Sigma-types given by composition of fibrations.

So as far as I can see, if one’s studying Sigma-types in the absence of Pi-types, one again might want the Frobenius version; and it seems likely that the situation for other inductive types would be similar.

But I’m not sure, and I feel I may be overlooking or forgetting something obvious.  What have others on the list thought about this?  Does anyone have a reference discussing the Frobenius versions of inductive types other than identity types, or at least giving the rules for them?

Best,

–Peter.

References:

- Gambino, Garner, 2008, “The Identity Type Weak Factorisation System”, https://arxiv.org/abs/0803.4349
- van den Berg, Garner, 2008, “Types are weak  ω-groupoids”, https://arxiv.org/pdf/0812.0298.pdf
- Gambino, Sattler, 2015, “The Frobenius condition, right properness, and uniform fibrations”, https://arxiv.org/pdf/1510.00669.pdf

[Valery Isaev]

Hi Peter,

I've been thinking about such eliminators lately too. It seems that they are derivable from ordinary eliminator for most type-theoretic constructions as long as we have identity types and sigma types. I did not write down proofs in full detail, but I sketched them for identity types, sigma types, and coproducts and I think this also should be true for pushouts. Now, when I say that we have identity types I mean a strong version of Id:

Γ |– a : A

Γ, y:A, u:Id(a,y)  |– C(y,u) type

Γ |– d : C(a,a,r(a)) type

Γ |— b : A

Γ |— p : Id(a,b)

——————————————

Γ |— J(a, (y,u)C, d, b, p) : C(b,p)

We can prove all the usual constructions for identity types using this version of J including transport and extensionality for sigma types. To define the "Frobenius version" of this J, we construct a term t : (b,p) = (a,r(a)) using extensionality for sigma. Then we transport c along t from Δ(b,p) to Δ(a,r(a)), and substitute this in d. At this point, we have a term d' of type C(a,r(a),t_*(c)). Then we use the fact that the type of triples \Sigma (q : \Sigma (b : A) Id(a,b)) Id((b,p),q) is contractible (this type is of the form \Sigma (x : A) Id(a,x) and we can prove that such types are contractible using the strong version of J). Thus, we have a path from ((a,r(a),t) to ((b,p),r((b,p))) and transporting d' along this path gives us a term of type C(b,p,c).

Similar argument shows that the "Frobenius version" of sigma eliminator is definable. We just prove that p = (\pi_1(p),\pi_2(p)) and then transport terms back and forth along this path. The proof for coproducts is slightly more difficult, but the idea is similar.

I constructed Frobenius versions of various eliminators in my recent preprint https://arxiv.org/abs/1806.08038 (I call Frobenius eliminators "local" and the usual ones "global"). I used there a non-standard version of HoTT, but as I mentioned before I believe that it should be possible to construct them in the ordinary HoTT (with sigma and Id types only).

I also want to mention that Frobenius eliminators are definable only when our constructions are stable under substitution. If we do not assume that the type former, the constructors, and the non-Frobenius eliminator of some construction are stable under substitution, then we can interpret them in any category in which the corresponding categorical constructions are not necessarily stable under pullbacks, but this is not true for Frobenius eliminators.

2018-07-12 18:15 GMT+03:00 Peter LeFanu Lumsdaine <p.l.lu...@gmail.com>:

Briefly: I’m looking for background on the “Frobenius version” of elimination rules for inductive types.  I’m aware of a few pieces of work mentioning this for identity types, and nothing at all for other inductive types.  I’d be grateful to hear if anyone else can point me to anything I’ve missed in the literature — even just to a reference that lays out the Frobenius versions of the rules for anything beyond Id-types.  The proximate motivation is just that I want to use these versions in a paper, and it’d be very nice to have a reference rather than cluttering up the paper by writing them all out in full…

In more detail: Here are two versions of the eliminator for identity types:

Γ, x,y:A, u:Id(x,y)  |– C(x,y,u) type
Γ, x:A |– d(x) : C(x,x,r(x)) type
Γ |— a, b : A
Γ |— p : Id(a,b)
——————————————
Γ |— J(A, (x,y,u)C, (x)d, a, b, p) : C(a,b,p)

Γ, x,y:A, u:Id(x,y), w:Δ(x,y,u) |– C(x,y,u,w) type
Γ, x:A, w:Δ(x,x,r(x)) |– d(x,w) : C(x,x,r(x),w) type
Γ |— a, b : A
Γ |— p : Id(a,b)
Γ |— c : Δ(a,b,p)
——————————————
Γ |— J(A, (x,y,u)Δ, (x,y,u,w)C, (x,w)d, a, b, p, c) : C(a,b,p,c)

(where Δ(x,y,u) represents a “context extension”, i.e. some finite sequence of variables and types w_1 : B_1(x,y,u), w_2 : B_2(x,y,u,w_1), …)

I’ll call these the “simple version” and the “Frobenius version” of the Id-elim rule; I’ll call Δ the “Frobenius context”.  The simple version is a special case of the Frobenius one; conversely, in the presence of Pi-types, the Frobenius version is derivable from the simple one.

Most presentations just give the simple version.  The first mention of the Frobenius version I know of is in [Gambino, Garner 2008]; the connection with categorical Frobenius conditions is made in [van den Berg, Garner 2008], and some further helpful explanatory pointers are given in [Gambino, Sattler 2015].  It’s based on this that I use “Frobenius” to refer to these versions; I’m open to suggestions of better terminology.  (All references are linked below.)

The fact that the Frobenius version is strictly stronger is known in folklore, but not written up anywhere I know of.  One way to show this is to take any non right proper model category (e.g. the model structure for quasi-categories on simplicial sets), and take the model of given by its (TC,F) wfs; this will model the simple version of Id-types but not the Frobenius version.

Are you sure this is true? It seems that we can interpret the strong version of J even in non right proper model categories. Then the argument I gave above shows that the Frobenius version is also definable.

 

Overall, I think the consensus among everyone who’s thought about this (starting from [Gambino, Garner 2008], as far as I know) is that if one’s studying Id-types in the absence of Pi-types, then one needs to use the Frobenius version.

One can also of course write Frobenius versions of the eliminators for other inductive types — eg Sigma-types, W-types, …  However, I don’t know anywhere that even mentions these versions!

I remember believing at some point that at least for Sigma-types, the Frobenius version is in fact derivable from the simple version (without assuming Pi-types or any other type formers), which would explain why no-one’s bothered considering it… but if that’s the case, it’s eluding me now.  On the other hand, I also can’t think of a countermodel showing the Frobenius version is strictly stronger — wfs models won’t do for this, since they have strong Sigma-types given by composition of fibrations.

So as far as I can see, if one’s studying Sigma-types in the absence of Pi-types, one again might want the Frobenius version; and it seems likely that the situation for other inductive types would be similar.

But I’m not sure, and I feel I may be overlooking or forgetting something obvious.  What have others on the list thought about this?  Does anyone have a reference discussing the Frobenius versions of inductive types other than identity types, or at least giving the rules for them?

Best,

–Peter.

References:

- Gambino, Garner, 2008, “The Identity Type Weak Factorisation System”, https://arxiv.org/abs/0803.4349
- van den Berg, Garner, 2008, “Types are weak  ω-groupoids”, https://arxiv.org/pdf/0812.0298.pdf
- Gambino, Sattler, 2015, “The Frobenius condition, right properness, and uniform fibrations”, https://arxiv.org/pdf/1510.00669.pdf

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[Matt Oliveri]

This looks similar to the "left rule" version of inductive elimination, where you eliminate a variable (not an arbitrary term), and the context extension depending on it gets updated. Except the "Frobenious version" allows an arbitrary term, so you say in what way part of the context depends on it. (Your "c : Δ(a,b,p)".) Coq's "destruct" tactic sort of implements this; it picks Δ and c for you by searching subexpressions in the context. In general, "dependent pattern matching" seems to use the left rules, and variants, rather than the "simple versions", where the context never changes. Have you tried looking at the literature on dependent pattern matching? I know this isn't a real answer. I don't know any papers that I know have Frobenious versions. Also, it looks like your Δ is not literally a context extension, but the corresponding dependent record type.

[Thorsten Altenkirch]

Isn't this what is usually called recursion with parameters. E.g. in the simple typed case for natural numbers: the usual recursor can be written using only 1st order functions:

z : X 

s : Nat -> X -> X

-----------------------

R_X(z,s) : N -> X

R(z,s) 0 == z

R(z,s) (suc m) == s m (R(z,s) m)

while recursion with a parameter is:

z : Y -> X

s : Nat -> Y -> X -> X

----------------------------

R'_X,Y(z,s) : Nat -> Y -> X

R'(z,s) 0 y == z y

R'(z,s) (suc m) y == s m y (R'(z,s) m y)

Using functions we can reduce R' to R

R'_X,Y(z,s) = R_(Y -> X)(z,\n f y.s n y (f y))

but without functions R' is stronger and it is what you need to have recursion in every slice.  A simple example is addition over the 1st argument, which with functions we can write as

R_Nat->Nat (\n . n) (\ n fn m . suc (fn m))

but you can use R' without function types

R'_Nat,Nat (\ n . n) (\ n m x . suc m) 

Hence in the absence of Pi-types you need to use the "localized" version of the recursor. I think in the special case of + this gives you distributivity over x even without cartesian closure.

If we don't assume products we need to replace Y by a context. 

I think I have seen the case for Id in Thomas Streicher's habilitation but I am not sure.

Thorsten

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[Valery Isaev]

Such version corresponds simply to the fact that the eliminator is defined in every context (Γ in Peter's original post). Without this context eliminators are undeniably too weak. The point of Frobenius eliminators is that Y in your notation can depend on the eliminated parameter. I don't think that it is possible to define Frobenius for recursive datatypes.

Regards,

Valery Isaev

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[Peter LeFanu Lumsdaine]

On Thu, Jul 12, 2018 at 7:07 PM Valery Isaev <valery...@gmail.com> wrote:

Hi Peter,

I've been thinking about such eliminators lately too. It seems that they are derivable from ordinary eliminator for most type-theoretic constructions as long as we have identity types and sigma types.

Thankyou — very nice observation, and (at least to me) quite surprising!

 

I mean a strong version of Id:

Side note: this is I think more widely known as the Paulin-Mohring or one-sided eliminator for Id-types; the HoTT book calls it based path-induction.

The fact that the Frobenius version is strictly stronger is known in folklore, but not written up anywhere I know of.  One way to show this is to take any non right proper model category (e.g. the model structure for quasi-categories on simplicial sets), and take the model of given by its (TC,F) wfs; this will model the simple version of Id-types but not the Frobenius version.

Are you sure this is true? It seems that we can interpret the strong version of J even in non right proper model categories. Then the argument I gave above shows that the Frobenius version is also definable.

Ah, yes — there was a mistake in the argument I had in mind.  In that case, do we really know for sure that the Frobenius versions are really strictly stronger?

–p.

[Valery Isaev]

2018-07-13 13:38 GMT+03:00 Peter LeFanu Lumsdaine <p.l.lu...@gmail.com>:

On Thu, Jul 12, 2018 at 7:07 PM Valery Isaev <valery...@gmail.com> wrote:

Hi Peter,

I've been thinking about such eliminators lately too. It seems that they are derivable from ordinary eliminator for most type-theoretic constructions as long as we have identity types and sigma types.

Thankyou — very nice observation, and (at least to me) quite surprising!

I was surprised too. The case of coproducts is especially unexpected.

 

 

I mean a strong version of Id:

Side note: this is I think more widely known as the Paulin-Mohring or one-sided eliminator for Id-types; the HoTT book calls it based path-induction.

The fact that the Frobenius version is strictly stronger is known in folklore, but not written up anywhere I know of.  One way to show this is to take any non right proper model category (e.g. the model structure for quasi-categories on simplicial sets), and take the model of given by its (TC,F) wfs; this will model the simple version of Id-types but not the Frobenius version.

Are you sure this is true? It seems that we can interpret the strong version of J even in non right proper model categories. Then the argument I gave above shows that the Frobenius version is also definable.

Ah, yes — there was a mistake in the argument I had in mind.  In that case, do we really know for sure that the Frobenius versions are really strictly stronger?

I don't know how to prove this, but it seems that we can't even define transport without Frobenius J. Also, if we do have transport and we know that types of the form \Sigma (x : A) Id(a,x) are contractible, then the "one-sided J" is definable, so if we want to prove that the Frobenius version does not follows from the non-Frobenius, we need to find a model where either transport is not definable or \Sigma (x : A) Id(a,x) is not contractible.

 

–p.

[Ambrus Kaposi]

Hi,

Sorry for resurrecting an old thread, I just wanted to document that Frobenius J can be derived from Paulin-Mohring J without Pi, Sigma or universes: https://github.com/akaposi/hiit-signatures/blob/master/formalization/FrobeniusJDeriv.agda

This was observed by Rafael Bocquet and András Kovács in October 2019, but maybe other people have known about this.

We mention this briefly in our paper on HIIT signatures where Paulin-Mohring J can be used on previous equality constructors but we don't have Pi, Sigma or (usual) universes: https://lmcs.episciences.org/6100

Cheers,

Ambrus

2018. július 13., péntek 13:05:47 UTC+2 időpontban Valery Isaev a következőt írta:

2018-07-13 13:38 GMT+03:00 Peter LeFanu Lumsdaine <p.l.l...@gmail.com>:

[Rafaël Bocquet]

Hi,

Here's a simple countermodel that shows that the Martin-Löf eliminator (two-sided eliminator) for identity types is not strong enough to derive transport in the absence of Pi-types.

Consider the initial model of a type theory with Id, refl, a closed type A, a type family B over A, and closed terms x,y : A, b : B x and p : Id_A x y.

It can be shown that the only terms of that model are the variables, the weakenings of the generators and their iterated refls.

Because there is no closed term of type B y, transport does not hold in this model.

However, unless I missed some cases, it is still possible to define the Martin-Löf eliminator by case distinction on the elimination target.

Best,
Rafaël

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