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 From: > on behalf of Peter LeFanu Lumsdaine > Date: Thursday, 12 July 2018 at 16:15 To: "homotopytypetheory@googlegroups.com" > Subject: [HoTT] What is known and/or written about “Frobenius eliminators”? 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 -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. 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