Towards a Homotopy Domain Theory (HoDT)

A favourable environment is proposed for the achievement of λ-models with ∞-groupoid structure, which we will call homotopic λ-models, through of an ∞-category of ∞-groupoids with cartesian closure and enough points. Thus establishing the start of a project of generalization of the Domain Theory and λ-calculus; in the sense of the elevation of the concept proof (morphism) of equality of λ-terms to higher proof (homotopy).

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1 Introduction

The purpose of this paper is to give the horizon for to build a -model endowed with a topology, such that any proof of -equality between -terms is not represented by equality between points (extensional equality), but rather by the existence of a continuous path between the terms (intensional equality), where the interpretation of these terms correspondence to two points in the space. By example, given the -equality between different -terms

 λx.(λy.yx)zv=βzv,

these terms are -equal, because there is a proof determined by a finite sequence of -contractions () or inverted -contractions () which to connect the terms and , hence

 (λx.(λy.yx)z)v⊳1β(λy.yv)z⊳1βzv.

So the problem is to build a topological model, such that the interpretations of -terms and are different points, and the proof is a continuous path which to connect both points. This is to establish, when two proofs (two continuous paths) of a -equality between different terms (different points) are “equal” (homotopic). Hence in the example, given an second proof which correspond to finite sequence

 (λx.(λy.yx)z)v⊳1β(λx.zx)v⊳1βzv,

one has that and are two different proofs, so in the model their interpretations should be two different continuous paths. But, would these proof interpretation (paths) are homotopically equal? and how does this equality of paths affect -calculus?. If on -calculus we call higher proof (proof of proofs) to any homotopy of the model, then, when two different higher proof are “equal”?, and so on, answering these questions, we could define in -calculus a theory of higher -equality, with the help of the higher homotopies in the -model. Thus achieving a structure of non-trivial -groupoid essentially different to -groupoid obtained in (Martínez; de Queiroz, 2019).

According to Quillen’s Theorem, each CW complex topological space is homotopically equivalent to a Kan complex (-groupoid), and reciprocally each Kan complex is homotopically equivalent to a CW complex. Then, instead of working directly with topological spaces, we are going to do it with Kan compleces, which are -categories whose 1-simplexes or edges are weakly invertible. Or in other words:

Definition 1.1.

An -category is an -groupoid if its homotopy category is a groupoid. An -category is a simplicial set which has the following property: for any , any map admits an extension .

When it comes to the equivalence of -categories, the equivalence of vertices of an -category and homotopy of functors, we have the following definition:

Definition 1.2.

A functor of -categories is a categorical equivalence if the induced map is a categorical equivalence in the homotopy category of spaces. We say that and are categorically equivalent if there is a categorical equivalence between them, and we write . A morphism in an -category is an equivalence if it determines an isomorphism in the homotopy category . We say that and are equivalent if there is an equivalence between them, and we write . If are two functors of -categories, then and are homotopic if and are naturally isomorphic, and we write .

Finally to find -groupoids that to model -calculus, the strategy would be to generalize the procedure used in (Hyland, 2010) where to show an way to find categories that to model -calculus, though the solution possible of domain equations, which are posed on a bicategory with desirable properties of cartesian closure and enough points.

But, before proposing an -category of -groupoids with properties cartesian closure and enough points, we first introduce in Section 2 the concept -model homotopic on an arbitrary -groupoid with some consequences. In Section 3, we adopt the notion of Kleisli structure to the case of the -categories and we defined the -category Kleisli of an structure. In the Section 4, we fix the -categorical version of the distributivity and extension of monads. And finally in the Section 5, we propose an -category of -groupoids and we prove that it is closed cartesian and has enough points.

1.1 Some ∞-groupoids of presheaves

In the literature, such as can be seen in (Lurie, 2009) and (Cinski, 2019) is defined the -category of the presheaves on a small -category as which set the categorical equivalence

 Fun(A,PB)≃Fun(A×Bop,S),

where is the -category of all small -groupoids. The -category and this equivalence are defined on the -bicategory of all -categories. In this paper, we will consider all the definitions and results on the -category of all the -categories , particularly on (the -category of all the -groupoids). Thus, must be a -groupoid, i.e., or , where is the functor which sends each -category to the largest -groupoid . So is the -groupoid which is obtained by discarding the non-invertible morphisms of . Hence has all the vertices of . For this paper we will consider and the previous equivalence is given by the equivalence of -groupoids

 CAT∞(A,PB)≃CAT∞(A×Bop,S′).

Another fundamental result is the following: given a collection of simplicial sets , and an -category, there exists an -category a map with the following properties:

2. For every -category which admits -indexed colimits, composition with induces an equivalence of -categories

 FunK(PKRA,B)≃FunR(A,B).

If admits all the -indexed colimits, we also have

3. The functor is fully faithful.

where is the full subcategory of spanned by those functors which preserve -indexed colimits, i.e., which preserve -indexed colimits for each ; the same applies to .

For the case , we have that must be an -groupoid and the property (2) would result in the equivalence of -groupoids

 CATK∞(PKRA,B)≃CATR∞(A,B).

where is the full subcategory of spanned by those morphism which preserve -indexed colimits. The situation is similar for the -groupoid .

Example 1.1.

Let and be the class of all small simplicial sets. If is a small -category, then .

Example 1.2.

Let and be the class of all small -filtered simplicial sets for some regular cardinal . If is a small -category, then .

Example 1.3.

Let and be the class of all -small simplicial sets for some regular cardinal . If is a small -category, then . Where is the class of all -compact elements of .

Example 1.4.

Let the class of all -small simplicial sets for some regular cardinal and let the collection of all small simplicial sets. Let be a small -category which admits -small colimits, then . Also we have for some small -category which does not necessarily admits -small colimits.

2 Arbitrary homotopic λ-models

In this section we introduce arbitrary homotopic lambda models as a direct generalization of the traditional structured set models of a closed cartesian category as can be seen in (Barendregt, 1984) and (Hindley; Seldin, 2008), and discuss some consequences of this definition.

Definition 2.1 (Homotopic λ-model).

A homotopic -model is a triple , where is an -groupoid, is a binary operation and is a mapping which assigns, to -term and each assignment , an object of such that

1. for all ;

2. if for ;

3. if ;

4. if , then .

The homotopic model is an extensional homotopic model if it satisfies the additional property: with .

Definition 2.2.

Let be a homotopic -model. The notion of satisfaction in is defined as

 M⊨M=N⟺∀ρ(M,ρ⊨M=N)
Lemma 2.1.

Let be a homotopic -model. Then, for all , , and ,

 ⟦[N/x]M⟧ρ≃⟦M⟧[⟦N⟧ρ/x]ρ.
Proof.

The proof is very similar to the equality as in (Barendregt, 1984) and (Hindley; Seldin, 2008). ∎

Theorem 2.1.

Let be a homotopic -model. Then

 λ⊢M=N⟹M⊨M=N.
Proof.

By induction on the length of proof. For the axiom we precede

 ⟦(λx.M)N⟧ρ =⟦λx.M⟧ρ∙⟦N⟧ρ ≃⟦M⟧[⟦N⟧ρ/x]ρ ≃⟦[N/x]M⟧ρ

The rule follows from Definition 2.1 (6). The other rules are trivial. ∎

Definition 2.3 (∞-category cartesian closed).

Let be an -category of -groupoids. We say that is cartesian closed if:

1. has a terminal object ,

2. For , there exist an object in ,

3. For , there exist an object in such that set the equivalence

 HomC(A×B,C)≃HomC(A,B⇒C).
Definition 2.4 (Enough points).

An cartesian closed -category have enough points if for each pair of morphisms of such for each point , then .

Definition 2.5 (Reflexive ∞-groupoid).

Let be a cartesian closed -category of -groupoids. An object is called reflexive if the -groupoid is an weak retract of i.e., there are functors

 F:C→[C⇒C],G:[C⇒C]→C

such that .

It will be shown that every reflexive -groupoid defines naturally a homotopic -model.

Definition 2.6.

A reflexive -groupoid via the functors , has enough points if for each pair of morphisms such that for each object , then .

Thus, any morphism on a reflexive -groupoid with enough points is determined by all the objects of , which motivates to define the following.

Definition 2.7.

Let be a reflexive -groupoid via the functors , which enough points.

1. For each define

 a∙b=Fab
2. Let a valuation in . Define the interpretation by induction as follows

1. ,

Where

Theorem 2.2.

Let be a reflexive -groupoid via the functors , with enough points and let . Then

1. is a homotopic -model.

2. is extensional iff

Proof.

1. The conditions in Definition 2.1 (1), (2) are trivial. As to (3)

 ⟦λx.M⟧ρ∙a =G(λd.⟦M⟧[d/x]ρ)∙a =F(G(λd.⟦M⟧[d/x]ρ))a ≃(λd.⟦M⟧[d/x]ρ)a ≃⟦M⟧[a/x]ρ

The condition (4) follows an easy induction on . The condition (5), given any object and

 λd.⟦[y/x]M⟧[d/y]ρ ≃λd.⟦(λx.M)y⟧[d/y]ρ ≃λd.⟦λx.M⟧[d/y]ρ∙⟦y⟧[d/y]ρ =λd.⟦λx.M⟧ρ∙d ≃λd.⟦M⟧[d/x]ρ

Applying and by Definition 2.7 (c) follows

 ⟦λy.[y/x]M⟧ρ =G(λd.⟦[y/x]M⟧[d/y]ρ) ≃G(λd.⟦M⟧[d/x]ρ) =⟦λx.M⟧ρ

Condition (6). By the proof of condition (3) and by hypothesis, for all object in

 (λd.⟦M⟧[d/x]ρ)a ≃⟦M⟧[a/x]ρ ≃⟦N⟧[a/x]ρ ≃(λd.⟦N⟧[d/x]ρ)a,

since has enough points, then

 λd.⟦M⟧[d/x]ρ≃λd.⟦N⟧[d/x]ρ

applying and by Definition 2.7 (c)

 ⟦λx.M⟧ρ =G(λd.⟦M⟧[d/x]ρ) ≃G(λd.⟦N⟧[d/x]ρ) =⟦λx.N⟧ρ

2. Suppose that is extensional. Let be an object of . Then for all

 (GFb)∙a=F(GFb)a≃Fba=b∙a,

by extensionality

 GFb≃b≃idCb,

since has enough points, hence

 GF≃IdC.

If . For all by hypothesis and Definition 2.7

 Fab=a∙b≃a′∙b≃Fa′b,

since has enough points, . Applying , it follows that . ∎

3 Kleisli ∞-categories

Next we define the Kleisli structures on the -categories; a general and direct version of those initially introduced by (Hyland, 2014) for the case of bicategories.

Definition 3.1 (Kleisli structure).

Let be an -category and be an -category contained in . A Kleisli structure on is the following.

• For each vertex an arrow in .

• For each a functor

 K(a,Pb)→K(Pa,Pb),f↦f#.

Such that one has the equivalences

where and be edges in .

It is clear that is a functor from to such that for each 1-simplex of , set .

Proposition 3.1.

The functor given by be a Kleisli structure.

Proof.

We have for each there is which preserves small colimits such that , and also it has , see (Lurie, 2009), this is according to equivalence

where is the class of morphisms which preserve small colimits. It only remains to prove that for all and . For the composition , one has . Since , then . So . But the functor is an equivalence, hence . ∎

Definition 3.2 (Kleisli ∞-category).

Given a Kleisli structure on . Define its Kleisli -category as follows. The objects of are the objects of and the -simplex generated for the composable chain of morphisms

 X0g1−→X1g2−→⋯gn−→Xn

in is the -simplex generated for the composable chain of morphism

 X0g1−→PX1g#2−→⋯g#n−→PXn

in the -category .

Proposition 3.2.

in the -category .

Proof.
 PrL(P(A×B),C) =FunL(P(A×B),C) ≃Fun(A×B,C) ≃Fun(A,CB) ≃FunL(PA,FunL(PB,C)) =PrL(PA,PB⊸C) ≃PrL(PA⊗PB,C).

Next we define the distributive laws of (Hyland; Nagayama; Power; Rosolini, 2006) for the case of the -categories. The existence of the Kleisli -category of a monad of -categories is deduced from the existence of -algebras as seen in (Rielh; Verity, 2016), where these monads receive the name of Homotopy coherent monads and the -categories are calls quasi-categories.

Definition 4.1 (Distributivity law).

Let be a Kleisli structure on and a monad such that . Define the distributivity law such that for each object in , set the equivalence , where , and are the Yoneda embeddings.

Let be a Kleisli structure on and a monad such that . Define the extension of along the functor free as the monad such that .

Theorem 4.1.

Let be a Kleisli structure on and a monad such that . Given a distributivity , then there exists an extension of along the functor free .

Proof.

Let be a morphism of . Define as

 LPg=λbLg.

Let’s see what extends to . Let a morphism of , then

 LPFf=LPybf=λbL(ybf)≃λbLybLf,

on the other hand

 FLf=yLbLf,

but is distributive, i.e., , thus

 LPFf≃FLf.

Hence . ∎

5 ∞-groupoidal λ-models

Let be the monad sending each small -groupoid to the smallest -groupoid that contains it and which admits -small limits, i.e., is closure of under -small limits. Let be its corresponding comonad which closes each -groupoid under -small colimits, i.e., , with be Kleisli structure restricted to the -groupoids, i.e., on . Hence, according to the Example 1.3 for be an -groupoid which admitted -small colimits we have the equivalence

 ˆSκ(L∗A,B)≃ˆS(A,B),

where is the class of morphisms which preserve -small colimits.

Proposition 5.1.

There exists an extension of to a monad on , which is denoted by .

Proof.

Let be a small -groupoid. For the composition of the unit with the embedding Yoneda there is the Kan extension . Since , thus . By definition of

 P(ηA)=(yLAηA)#≃(yLAηA)#iPA=(yLAηA)#ηPA.

Hence let be the distributivity law for each small -groupoid . Therefore there is an extension of the monad . ∎

Dually for the comonad , there is an extension on .

Lemma 5.1.

The -category is cartesian closed.

Proof.
 Kl(P)[A×B,C] =ˆS(A×B,PC) ≃ˆS(A,[B,PC]) ≃ˆS(A,[B,S′Cop]) ≃ˆS(A,[B×Cop,S′]) =ˆS(A,P(Bop×C)) =Kl(P)(A,Bop×C) =Kl(P)(A,B⇒C).

Theorem 5.1.

The -category is cartesian closed.

Proof.
 Kl(L∗P)[A×B,C] =Kl(P)[L∗P(A×B),C] =ˆS(L∗(A×B),PC) ≃ˆSκ(L∗(L∗(A×B)),PC) ≃ˆSκ(L∗(A×B),PC) ≃ˆS(A×B,PC) ≃ˆS(A,(PC)B) ≃ˆSκ(L∗A,(PC)B) ≃ˆSκ(L∗(L∗A),(PC)B) ≃ˆS(L∗A,(PC)B) =ˆS(L∗A×B,PC) =Kl(P)[L∗PA×B,C] =Kl(P)[L∗PA,B⇒C] =Kl(L∗P)[A,B⇒C]

Theorem 5.2.

The -category does have enough points.

Proof.

A functor of corresponds to a functor of . Since is a closure of under -small colimits, by Section … such a functor corresponds to a filtered colimit preserving functor . Given any pair of functors such that for all object of . Let be a morphism of . Since is -groupoid, then is a filtered colimit from object . By hypothesis

 Ff≃¯¯¯¯¯¯¯¯¯FX≃¯¯¯¯¯¯¯¯¯GX≃Gf,

Thus (naturally isomorphic). By the Definition 1.2 we have the homotopy of functors . ∎

6 Conclusions

What is exposed in this article is a beginning for the construction of a Homotopy Domain Homotopy (HoDT) which provides techniques to generate homotopic -models that allow to generalize the -equalities to higher -equalities. For future work, we will establish methods to solve domain equations on cartesian closed -categories of -groupoids with enough points and see what repercussions have these homotopic models on the theory of -calculus.

References

1. D-C. Cisinski, Higher Categories and Homotopical Algebra, Cambridge University Press, 2019.

2. H. P. Barendregt, The Lambda Calculus, its Syntax and Semantics, North-Holland Co., Amsterdam, 1984.

3. M. Hyland, Some Reasons for Generalizing Domain Theory, Mathematical Structures in Computer Science, Volume 20, Issue 2, pp.239-265, 2010.

4. J. M. E. Hyland, Elements of a theory of algebraic theories, Theoretical Computer Science 546 (2014) 132-144.

5. M. Hyland, M. Nagayama, J. Power and G. Rosolini, A Category Theoretic Formulation for Engeler-style Models of the Untyped -Calculus, Electronic Notes in Theoretical Computer Science 161 (2006) 43-57.

6. J.R. Hindley and J.P. Seldin, Lambda-Calculus and Combinators, an Introduction, Cambridge University Press, New York, NY, 2008.

7. J. Lurie, Higher Topos Theory, Princeton University Press, Princeton and Oxford, 2009.

8. D. Martínez-Rivillas and R. de Queiroz, A -model with -groupoid structure based on Scott -model , arXiv:1906.05729 [cs.LO], 2020.

9. E. Rielh, D. Verity, Homotopy coherent adjunctions and the formal theory of monads, Advances in Mathematics 286 (2016) 802-888.