Towards a Homotopy Domain Theory (HoDT)

1 Introduction

The purpose of this paper is to give the horizon for to build a $λ$ -model endowed with a topology, such that any proof of $β$ -equality between $λ$ -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 $β$ -equality between different $λ$ -terms

λ x . (λ y . y x) z v =_{β} z v,

these terms are $β$ -equal, because there is a proof $p_{1}$ determined by a finite sequence of $β$ -contractions ( $⊳_{1 β}$ ) or inverted $β$ -contractions ( $⊲_{1 β}$ ) which to connect the terms $λ x . (λ y . y x) z v$ and $z v$ , hence

(λ x . (λ y . y x) z) v ⊳_{1 β} (λ y . y v) z ⊳_{1 β} z v .

So the problem is to build a topological model, such that the interpretations of $λ$ -terms $λ x . (λ y . y x) z v$ and $z v$ are different points, and the proof $p_{1}$ is a continuous path which to connect both points. This is to establish, when two proofs (two continuous paths) of a $β$ -equality between different terms (different points) are “equal” (homotopic). Hence in the example, given an second proof $p_{2}$ which correspond to finite sequence

(λ x . (λ y . y x) z) v ⊳_{1 β} (λ x . z x) v ⊳_{1 β} z v,

one has that $p_{1}$ and $p_{2}$ 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 $λ$ -calculus?. If on $λ$ -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 $λ$ -calculus a theory of higher $β$ -equality, with the help of the higher homotopies in the $λ$ -model. Thus achieving a structure of non-trivial $\infty$ -groupoid essentially different to $\infty$ -groupoid obtained in (Martínez; de Queiroz, 2019).

According to Quillen’s Theorem, each CW complex topological space is homotopically equivalent to a Kan complex ( $\infty$ -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 $\infty$ -categories whose 1-simplexes or edges are weakly invertible. Or in other words:

Definition 1.1.

An $\infty$ -category $C$ is an $\infty$ -groupoid if its homotopy category $h C$ is a groupoid. An $\infty$ -category is a simplicial set $C$ which has the following property: for any $0 < i < n$ , any map $f_{0} : Λ_{i}^{n} \to C$ admits an extension $f : △^{n} \to C$ .

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

Definition 1.2.

A functor $F : S \to T$ of $\infty$ -categories is a categorical equivalence if the induced map $h S \to h T$ is a categorical equivalence in the homotopy category of spaces. We say that $S$ and $T$ are categorically equivalent if there is a categorical equivalence between them, and we write $S ≃ T$ . A morphism $f : X \to Y$ in an $\infty$ -category $C$ is an equivalence if it determines an isomorphism in the homotopy category $h C$ . We say that $X$ and $Y$ are equivalent if there is an equivalence between them, and we write $X ≃ Y$ . If $F, G : S \to T$ are two functors of $\infty$ -categories, then $F$ and $G$ are homotopic if $h F$ and $h G$ are naturally isomorphic, and we write $F ≃ G$ .

Finally to find $\infty$ -groupoids that to model $λ$ -calculus, the strategy would be to generalize the procedure used in (Hyland, 2010) where to show an way to find categories that to model $λ$ -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 $\infty$ -category of $\infty$ -groupoids with properties cartesian closure and enough points, we first introduce in Section 2 the concept $λ$ -model homotopic on an arbitrary $\infty$ -groupoid with some consequences. In Section 3, we adopt the notion of Kleisli structure to the case of the $\infty$ -categories and we defined the $\infty$ -category Kleisli of an structure. In the Section 4, we fix the $\infty$ -categorical version of the distributivity and extension of monads. And finally in the Section 5, we propose an $\infty$ -category of $\infty$ -groupoids and we prove that it is closed cartesian and has enough points.

1.1 Some $\infty$ -groupoids of presheaves

In the literature, such as can be seen in (Lurie, 2009) and (Cinski, 2019) is defined the $\infty$ -category of the presheaves on a small $\infty$ -category $B$ as $P B = [B^{o p}, S]$ which set the categorical equivalence

F u n (A, P B) ≃ F u n (A \times B^{o p}, S),

where $S$ is the $\infty$ -category of all small $\infty$ -groupoids. The $\infty$ -category $P B$ and this equivalence are defined on the $\infty$ -bicategory of all $\infty$ -categories. In this paper, we will consider all the definitions and results on the $\infty$ -category of all the $\infty$ -categories $C A T_{\infty}$ , particularly on $ˆ S$ (the $\infty$ -category of all the $\infty$ -groupoids). Thus, $P B$ must be a $\infty$ -groupoid, i.e., $P B = [B^{o p}, S^{'}]$ or $P B = [B^{o p}, S]^{'}$ , where $C \mapsto C^{'}$ is the functor which sends each $\infty$ -category $C$ to the largest $\infty$ -groupoid $C^{'} \subseteq C$ . So $C^{'}$ is the $\infty$ -groupoid which is obtained by discarding the non-invertible morphisms of $C$ . Hence $C^{'}$ has all the vertices of $C$ . For this paper we will consider $P B = [B^{o p}, S^{'}]$ and the previous equivalence is given by the equivalence of $\infty$ -groupoids

C A T_{\infty} (A, P B) ≃ C A T_{\infty} (A \times B^{o p}, S^{'}) .

Another fundamental result is the following: given a collection of simplicial sets $K$ , $R \subseteq K$ and $A$ an $\infty$ -category, there exists an $\infty$ -category $P_{R}^{K} A$ a map $j : A \to P_{R}^{K} A$ with the following properties:

$P_{R}^{K} A$ admits $K$ -indexed colimits, i.e., admits $K$ -indexed colimits for each $K \in K$ .
For every $\infty$ -category $B$ which admits $K$ -indexed colimits, composition with $j$ induces an equivalence of $\infty$ -categories

$F u n_{K} (P_{R}^{K} A, B) ≃ F u n_{R} (A, B) .$

If $A$ admits all the $R$ -indexed colimits, we also have
The functor $j$ is fully faithful.

where $F u n_{R} (A, B)$ is the full subcategory of $F u n (A, B)$ spanned by those functors which preserve $R$ -indexed colimits, i.e., which preserve $K$ -indexed colimits for each $K \in R$ ; the same applies to $F u n_{K} (P_{R}^{K} A, B)$ .

For the case $C A T_{\infty}$ , we have that $P_{R}^{K} A$ must be an $\infty$ -groupoid and the property (2) would result in the equivalence of $\infty$ -groupoids

C A T_{\infty}^{K} (P_{R}^{K} A, B) ≃ C A T_{\infty}^{R} (A, B) .

where $C A T_{\infty}^{R} (A, B)$ is the full subcategory of $C A T_{\infty} (A, B)$ spanned by those morphism which preserve $R$ -indexed colimits. The situation is similar for the $\infty$ -groupoid $C A T_{\infty}^{K} (P_{R}^{K} A, B)$ .

Example 1.1.

Let $R = \emptyset$ and $K$ be the class of all small simplicial sets. If $A$ is a small $\infty$ -category, then $P_{R}^{K} A ≃ P A$ .

Example 1.2.

Let $R = \emptyset$ and $K$ be the class of all small $κ$ -filtered simplicial sets for some regular cardinal $κ$ . If $A$ is a small $\infty$ -category, then $P_{R}^{K} A ≃ I n d_{κ} A$ .

Example 1.3.

Let $R = \emptyset$ and $K$ be the class of all $κ$ -small simplicial sets for some regular cardinal $κ$ . If $A$ is a small $\infty$ -category, then $P_{R}^{K} A ≃ P^{κ} A$ . Where $P^{κ} A$ is the class of all $κ$ -compact elements of $P A$ .

Example 1.4.

Let $R$ the class of all $κ$ -small simplicial sets for some regular cardinal $κ$ and let $K$ the collection of all small simplicial sets. Let $A$ be a small $\infty$ -category which admits $κ$ -small colimits, then $P_{R}^{K} A ≃ I n d_{κ} A$ . Also we have $A ≃ P^{κ} C$ for some small $\infty$ -category $C$ which does not necessarily admits $κ$ -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 $λ$ -model is a triple $⟨ C, ∙, ⟦ ⟧ ⟩$ , where $C$ is an $\infty$ -groupoid, $∙ : | C | \times | C | \to | C |$ is a binary operation and $⟦ ⟧$ is a mapping which assigns, to $λ$ -term $M$ and each assignment $ρ : V a r \to | C |$ , an object $⟦ M ⟧_{ρ}$ of $G$ such that

$⟦ x ⟧ = ρ (x);$
$⟦ M N ⟧_{ρ} = ⟦ M ⟧_{ρ} ∙ ⟦ N ⟧_{ρ};$
$⟦ λ x . M ⟧_{ρ} ∙ a ≃ ⟦ M ⟧_{[a / x] ρ}$ for all $a \in C$ ;
$⟦ M ⟧_{ρ} = ⟦ M ⟧_{σ}$ if $ρ (x) = σ (x)$ for $x \in F V (M)$ ;
$⟦ λ x . M ⟧_{ρ} ≃ ⟦ λ y . [y / x] M ⟧_{ρ}$ if $y \notin F V (M)$ ;
if $(\forall a \in D) (⟦ M ⟧_{[a / x] ρ} ≃ ⟦ N ⟧_{[a / x] ρ})$ , then $⟦ λ x . M ⟧_{ρ} ≃ ⟦ λ x . N ⟧_{ρ}$ .

The homotopic model $⟨ C, ∙, ⟦ ⟧ ⟩$ is an extensional homotopic model if it satisfies the additional property: $⟦ λ x . M x ⟧_{ρ} ≃ ⟦ M ⟧_{ρ}$ with $x \notin F V (M)$ .

Definition 2.2.

Let $M = ⟨ C, ∙, ⟦ ⟧ ⟩$ be a homotopic $λ$ -model. The notion of satisfaction in $M$ is defined as

M ⊨ M = N ⟺ \forall ρ (M, ρ ⊨ M = N)

Lemma 2.1.

Let $M = ⟨ C, ∙, ⟦ ⟧ ⟩$ be a homotopic $λ$ -model. Then, for all $M$ , $N$ , $x$ and $ρ$ ,

⟦ [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 $M = ⟨ C, ∙, ⟦ ⟧ ⟩$ be a homotopic $λ$ -model. Then

λ ⊢ M = N ⟹ M ⊨ M = N .

Proof.

By induction on the length of proof. For the axiom $(λ x . M) N = [N / x] M$ we precede

	$⟦ (λ x . M) N ⟧_{ρ}$	$= ⟦ λ x . M ⟧_{ρ} ∙ ⟦ N ⟧_{ρ}$
		$≃ ⟦ M ⟧_{[⟦ N ⟧_{ρ} / x] ρ}$
		$≃ ⟦ [N / x] M ⟧_{ρ}$

The rule $M = N ⟹ λ x . M = λ x . N$ follows from Definition 2.1 (6). The other rules are trivial. ∎

Definition 2.3 ( $\infty$ -category cartesian closed).

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

$C$ has a terminal object $T$ ,
For $A, B \in C$ , there exist an object $A \times B$ in $C$ ,
For $A, B \in C$ , there exist an object $[A \Rightarrow B]$ in $C$ such that set the equivalence

$H o m_{C} (A \times B, C) ≃ H o m_{C} (A, B \Rightarrow C) .$

Definition 2.4 (Enough points).

An cartesian closed $\infty$ -category $C$ have enough points if for each pair of morphisms $f, g : X \to Y$ of $C$ such $f x ≃ g x$ for each point $x : T \to X$ , then $f ≃ g$ .

Definition 2.5 (Reflexive $\infty$ -groupoid).

Let $C$ be a cartesian closed $\infty$ -category of $\infty$ -groupoids. An object $C \in C$ is called reflexive if the $\infty$ -groupoid $[C \Rightarrow C]$ is an weak retract of $C$ i.e., there are functors

F : C \to [C \Rightarrow C], G : [C \Rightarrow C] \to C

such that $F G ≃ i d_{[C \Rightarrow C]}$ .

It will be shown that every reflexive $\infty$ -groupoid defines naturally a homotopic $λ$ -model.

Definition 2.6.

A reflexive $\infty$ -groupoid $C$ via the functors $F$ , $G$ has enough points if for each pair of morphisms $f, g : C \Rightarrow C$ such that $f x ≃ g x$ for each object $x : △^{0} \to C$ , then $f ≃ g$ .

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

Definition 2.7.

Let $C$ be a reflexive $\infty$ -groupoid via the functors $F$ , $G$ which enough points.

For each $a, b : △^{0} \to C$ define

$a ∙ b = F a b$
Let $ρ$ a valuation in $C$ . Define the interpretation $⟦ ⟧_{ρ} : Λ \to | C |$ by induction as follows
1. $⟦ x ⟧_{ρ} = ρ (x),$
2. $⟦ M N ⟧_{ρ} = ⟦ M ⟧_{ρ} ∙ ⟦ N ⟧_{ρ}$ ,
3. $⟦ λ x . M ⟧_{ρ} = G (λ d . ⟦ M ⟧_{[d / x] ρ}) .$
Where $λ d . ⟦ M ⟧_{[d / x] ρ} : △^{0} \to (C \Rightarrow C)$

Theorem 2.2.

Let $C$ be a reflexive $\infty$ -groupoid via the functors $F$ , $G$ with enough points and let $M = ⟨ C, ∙, ⟦ ⟧ ⟩$ . Then

$M$ is a homotopic $λ$ -model.
$M$ is extensional iff $G F ≃ i d_{C} .$

Proof.

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

	$⟦ λ x . M ⟧_{ρ} ∙ a$	$= G (λ d . ⟦ M ⟧_{[d / x] ρ}) ∙ a$
		$= F (G (λ d . ⟦ M ⟧_{[d / x] ρ})) a$
		$≃ (λ d . ⟦ M ⟧_{[d / x] ρ}) a$
		$≃ ⟦ M ⟧_{[a / x] ρ}$

The condition (4) follows an easy induction on $M$ . The condition (5), given any object $a \in C$ and $y \notin F V (M)$

	$λ d . ⟦ [y / x] M ⟧_{[d / y] ρ}$	$≃ λ d . ⟦ (λ x . M) y ⟧_{[d / y] ρ}$
		$≃ λ d . ⟦ λ x . M ⟧_{[d / y] ρ} ∙ ⟦ y ⟧_{[d / y] ρ}$
		$= λ d . ⟦ λ x . M ⟧_{ρ} ∙ d$
		$≃ λ d . ⟦ M ⟧_{[d / x] ρ}$

Applying $G$ and by Definition 2.7 (c) follows

	$⟦ λ y . [y / x] M ⟧_{ρ}$	$= G (λ d . ⟦ [y / x] M ⟧_{[d / y] ρ})$
		$≃ G (λ d . ⟦ M ⟧_{[d / x] ρ})$
		$= ⟦ λ x . M ⟧_{ρ}$

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

	$(λ d . ⟦ M ⟧_{[d / x] ρ}) a$	$≃ ⟦ M ⟧_{[a / x] ρ}$
		$≃ ⟦ N ⟧_{[a / x] ρ}$
		$≃ (λ d . ⟦ N ⟧_{[d / x] ρ}) a,$

since $C$ has enough points, then

λ d . ⟦ M ⟧_{[d / x] ρ} ≃ λ d . ⟦ N ⟧_{[d / x] ρ}

applying $G$ and by Definition 2.7 (c)

	$⟦ λ x . M ⟧_{ρ}$	$= G (λ d . ⟦ M ⟧_{[d / x] ρ})$
		$≃ G (λ d . ⟦ N ⟧_{[d / x] ρ})$
		$= ⟦ λ x . N ⟧_{ρ}$

2. Suppose that $M$ is extensional. Let $b$ be an object of $C$ . Then for all $a \in C$

(G F b) ∙ a = F (G F b) a ≃ F b a = b ∙ a,

by extensionality

G F b ≃ b ≃ i d_{C} b,

since $C$ has enough points, hence

G F ≃ I d_{C} .

If $G F ≃ I d_{C}$ . For all $b \in C$ by hypothesis and Definition 2.7

F a b = a ∙ b ≃ a^{'} ∙ b ≃ F a^{'} b,

since $C$ has enough points, $F a ≃ F b$ . Applying $G$ , it follows that $a ≃ b$ . ∎

3 Kleisli $\infty$ -categories

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

Definition 3.1 (Kleisli structure).

Let $K$ be an $\infty$ -category and $A$ be an $\infty$ -category contained in $K$ . A Kleisli structure $P$ on $A ↪ K$ is the following.

For each vertex $a \in A$ an arrow $y_{a} : a \to P a$ in $K$ .
For each $a, b \in A$ a functor

$K (a, P b) \to K (P a, P b), f \mapsto f^{#} .$

Such that one has the equivalences

where $f : a \to P b$ and $g : b \to P c$ be edges in $K$ .

It is clear that $P$ is a functor from $A$ to $K$ such that for each 1-simplex $f : a \to b$ of $A$ , set $P f = (y_{b} f)^{#} : P a \to P b$ .

Proposition 3.1.

The functor $P : C a t_{\infty} \to C A T_{\infty}$ given by $P A = [A^{o p}, S^{'}]$ be a Kleisli structure.

Proof.

We have for each $f : A \to P B$ there is $f^{#} : P A \to P B$ which preserves small colimits such that $f ≃ f^{#} y_{A}$ , and also it has $(y_{A})^{#} ≃ 1_{P A}$ , see (Lurie, 2009), this is according to equivalence

where $C A T_{\infty}^{L} (P A, P B)$ is the class of morphisms which preserve small colimits. It only remains to prove that $(g^{#} f)^{#} ≃ g^{#} f^{#}$ for all $f : A \to P B$ and $g : B \to P C$ . For the composition $g^{#} f : A \to P C$ , one has $g^{#} f ≃ (g^{#} f)^{#} y_{A}$ . Since $f ≃ f^{#} y_{A}$ , then $g^{#} f ≃ g^{#} f^{#} y_{A}$ . So $(g^{#} f)^{#} y_{A} ≃ g^{#} f^{#} y_{A}$ . But the functor $(-) y_{A}$ is an equivalence, hence $(g^{#} f)^{#} ≃ g^{#} f^{#}$ . ∎

Definition 3.2 (Kleisli $\infty$ -category).

Given a Kleisli structure $P$ on $A ↪ K$ . Define its Kleisli $\infty$ -category $K l (P)$ as follows. The objects of $K l (P)$ are the objects of $A$ and the $n$ -simplex generated for the composable chain of morphisms

X_{0} g_{1} - \to X_{1} g_{2} - \to \dots g_{n} - \to X_{n}

in $K l (P)$ is the $n$ -simplex generated for the composable chain of morphism

X_{0} g_{1} - \to P X_{1} g_{2}^{#} - \to \dots g_{n}^{#} - \to P X_{n}

in the $\infty$ -category $K$ .

Proposition 3.2.

$P (A \times B) ≃ P A \otimes P B$ in the $\infty$ -category $P r^{L}$ .

Proof.

	$P r^{L} (P (A \times B), C)$	$= {Fun}^{L} (P (A \times B), C)$
		$≃ Fun (A \times B, C)$
		$≃ Fun (A, C^{B})$
		$≃ {Fun}^{L} (P A, {Fun}^{L} (P B, C))$
		$= P r^{L} (P A, P B ⊸ C)$
		$≃ P r^{L} (P A \otimes P B, C) .$

∎

4 Distributivity and monads extension

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

Definition 4.1 (Distributivity law).

Let $P$ be a Kleisli structure on $A ↪ K$ and $L : K \to K$ a monad such that $L | A : A \to A$ . Define the distributivity law $λ : L P \to P L$ such that for each object $a$ in $A$ , set the equivalence $λ_{a} L y_{a} ≃ y_{L a}$ , where $λ_{a} : L P a \to P L a$ , $y_{a} : a \to P a$ and $y_{L a} : L a \to P L a$ are the Yoneda embeddings.

Definition 4.2 (Monad extension).

Let $P$ be a Kleisli structure on $A ↪ K$ and $L : K \to K$ a monad such that $L | A : A \to A$ . Define the extension of $L$ along the functor free $F : A \to K l (P)$ as the monad $L_{P} : K l (P) \to K l (P)$ such that $L_{P} F ≃ F L$ .

Theorem 4.1.

Let $P$ be a Kleisli structure on $A ↪ K$ and $L : K \to K$ a monad such that $L | A : A \to A$ . Given a distributivity $λ : L P \to P L$ , then there exists an extension of $L$ along the functor free $F : A \to K l (P)$ .

Proof.

Let $g : a \to P b$ be a morphism of $K l (P)$ . Define $L_{P} : K l (P) \to K l (P)$ as

L_{P} g = λ_{b} L g .

Let’s see what $L_{P}$ extends to $L$ . Let $f : a \to b$ a morphism of $K$ , then

L_{P} F f = L_{P} y_{b} f = λ_{b} L (y_{b} f) ≃ λ_{b} L y_{b} L f,

on the other hand

F L f = y_{L b} L f,

but $λ$ is distributive, i.e., $λ_{b} L y_{b} ≃ y_{L b}$ , thus

L_{P} F f ≃ F L f .

Hence $L_{P} F ≃ F L$ . ∎

5 $\infty$ -groupoidal $λ$ -models

Let $L : S \to S$ be the monad sending each small $\infty$ -groupoid $A$ to the smallest $\infty$ -groupoid that contains it and which admits $κ$ -small limits, i.e., $A \subseteq L A$ is closure of $A$ under $κ$ -small limits. Let $L^{*}$ be its corresponding comonad which closes each $\infty$ -groupoid $A$ under $κ$ -small colimits, i.e., $L^{*} A ≃ P^{κ} A$ , with $P$ be Kleisli structure restricted to the $\infty$ -groupoids, i.e., on $S ↪ ˆ S$ . Hence, according to the Example 1.3 for $B$ be an $\infty$ -groupoid which admitted $κ$ -small colimits we have the equivalence

{ˆ S}^{κ} (L^{*} A, B) ≃ ˆ S (A, B),

where ${ˆ S}^{κ} (L^{*} A, B)$ is the class of morphisms which preserve $κ$ -small colimits.

Proposition 5.1.

There exists an extension of $L$ to a monad on $K l (P)$ , which is denoted by $L_{P}$ .

Proof.

Let $A$ be a small $\infty$ -groupoid. For the composition of the unit with the embedding Yoneda $y_{L A} η_{A} : A \to P L A$ there is the Kan extension $(y_{L A} η_{A})^{#} : P A \to P L A$ . Since $P A = L P A$ , thus $η_{P A} = i_{P A}$ . By definition of $P$

P (η_{A}) = (y_{L A} η_{A})^{#} ≃ (y_{L A} η_{A})^{#} i_{P A} = (y_{L A} η_{A})^{#} η_{P A} .

Hence let $λ_{A} = (y_{L A} η_{A})^{#} : L P A \to P L A$ be the distributivity law for each small $\infty$ -groupoid $A$ . Therefore there is an extension $L_{P}$ of the monad $L$ . ∎

Dually for the comonad $L^{*}$ , there is an extension $L_{P}^{*}$ on $K l (P)$ .

Lemma 5.1.

The $\infty$ -category $K l (P)$ is cartesian closed.

Proof.

	$K l (P) [A \times B, C]$	$= ˆ S (A \times B, P C)$
		$≃ ˆ S (A, [B, P C])$
		$≃ ˆ S (A, [B, {S^{'}}^{C^{o p}}])$
		$≃ ˆ S (A, [B \times C^{o p}, S^{'}])$
		$= ˆ S (A, P (B^{o p} \times C))$
		$= K l (P) (A, B^{o p} \times C)$
		$= K l (P) (A, B \Rightarrow C) .$

∎

Theorem 5.1.

The $\infty$ -category $K l (L_{P}^{*})$ is cartesian closed.

Proof.

	$K l (L_{P}^{*}) [A \times B, C]$	$= K l (P) [L_{P}^{*} (A \times B), C]$
		$= ˆ S (L^{*} (A \times B), P C)$
		$≃ {ˆ S}^{κ} (L^{} (L^{} (A \times B)), P C)$
		$≃ {ˆ S}^{κ} (L^{*} (A \times B), P C)$
		$≃ ˆ S (A \times B, P C)$
		$≃ ˆ S (A, (P C)^{B})$
		$≃ {ˆ S}^{κ} (L^{*} A, (P C)^{B})$
		$≃ {ˆ S}^{κ} (L^{} (L^{} A), (P C)^{B})$
		$≃ ˆ S (L^{*} A, (P C)^{B})$
		$= ˆ S (L^{*} A \times B, P C)$
		$= K l (P) [L_{P}^{*} A \times B, C]$
		$= K l (P) [L_{P}^{*} A, B \Rightarrow C]$
		$= K l (L_{P}^{*}) [A, B \Rightarrow C]$

∎

Theorem 5.2.

The $\infty$ -category $K l (L_{P}^{*})$ does have enough points.

Proof.

A functor $A \to B$ of $K l (L_{P}^{*})$ corresponds to a functor $L_{P}^{*} A \to P B$ of $K l (P)$ . Since $L_{P}^{*} A$ is a closure of $A$ under $κ$ -small colimits, by Section … such a functor corresponds to a filtered colimit preserving functor $P A \to P B$ . Given any pair of functors $F, G : P A \to P B$ such that $F X ≃ G X$ for all object $X$ of $P A$ . Let $f : X \to Y$ be a morphism of $P A$ . Since $P A$ is $\infty$ -groupoid, then $f$ is a filtered colimit from object $X$ . By hypothesis

F f ≃ ¯ ¯¯¯¯¯¯¯ ¯ F X ≃ ¯ ¯¯¯¯¯¯¯ ¯ G X ≃ G f,

Thus $h F ≅ h G$ (naturally isomorphic). By the Definition 1.2 we have the homotopy of functors $F ≃ G$ . ∎

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 $λ$ -models that allow to generalize the $β$ -equalities to higher $β$ -equalities. For future work, we will establish methods to solve domain equations on cartesian closed $\infty$ -categories of $\infty$ -groupoids with enough points and see what repercussions have these homotopic models on the theory of $λ$ -calculus.

References

D-C. Cisinski, Higher Categories and Homotopical Algebra, Cambridge University Press, 2019.
H. P. Barendregt, The Lambda Calculus, its Syntax and Semantics, North-Holland Co., Amsterdam, 1984.
M. Hyland, Some Reasons for Generalizing Domain Theory, Mathematical Structures in Computer Science, Volume 20, Issue 2, pp.239-265, 2010.
J. M. E. Hyland, Elements of a theory of algebraic theories, Theoretical Computer Science 546 (2014) 132-144.
M. Hyland, M. Nagayama, J. Power and G. Rosolini, A Category Theoretic Formulation for Engeler-style Models of the Untyped $λ$ -Calculus, Electronic Notes in Theoretical Computer Science 161 (2006) 43-57.
J.R. Hindley and J.P. Seldin, Lambda-Calculus and Combinators, an Introduction, Cambridge University Press, New York, NY, 2008.
J. Lurie, Higher Topos Theory, Princeton University Press, Princeton and Oxford, 2009.
D. Martínez-Rivillas and R. de Queiroz, A $λ$ -model with $\infty$ -groupoid structure based on Scott $λ$ -model $D_{\infty}$ , arXiv:1906.05729 [cs.LO], 2020.
E. Rielh, D. Verity, Homotopy coherent adjunctions and the formal theory of monads, Advances in Mathematics 286 (2016) 802-888.

Towards a Homotopy Domain Theory (HoDT)

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1 Introduction

Definition 1.1.

Definition 1.2.

1.1 Some ∞-groupoids of presheaves

Example 1.1.

Example 1.2.

Example 1.3.

Example 1.4.

2 Arbitrary homotopic λ-models

Definition 2.1 (Homotopic λ-model).

Definition 2.2.

Lemma 2.1.

Proof.

Theorem 2.1.

Proof.

Definition 2.3 (∞-category cartesian closed).

Definition 2.4 (Enough points).

Definition 2.5 (Reflexive ∞-groupoid).

Definition 2.6.

Definition 2.7.

Theorem 2.2.

Proof.

3 Kleisli ∞-categories

Definition 3.1 (Kleisli structure).

Proposition 3.1.

Proof.

Definition 3.2 (Kleisli ∞-category).

Proposition 3.2.

Proof.

4 Distributivity and monads extension

Definition 4.1 (Distributivity law).

Definition 4.2 (Monad extension).

Theorem 4.1.

Proof.

5 ∞-groupoidal λ-models

Proposition 5.1.

Proof.

Lemma 5.1.

Proof.

Theorem 5.1.

Proof.

Theorem 5.2.

Proof.

6 Conclusions

References

1.1 Some $\infty$ -groupoids of presheaves

2 Arbitrary homotopic $λ$ -models

Definition 2.1 (Homotopic $λ$ -model).

Definition 2.3 ( $\infty$ -category cartesian closed).

Definition 2.5 (Reflexive $\infty$ -groupoid).

3 Kleisli $\infty$ -categories

Definition 3.2 (Kleisli $\infty$ -category).

5 $\infty$ -groupoidal $λ$ -models