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
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
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
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:
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:
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
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
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:
admits -indexed colimits, i.e., admits -indexed colimits for each .
For every -category which admits -indexed colimits, composition with induces an equivalence of -categories
If admits all the -indexed colimits, we also have
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
where is the full subcategory of spanned by those morphism which preserve -indexed colimits. The situation is similar for the -groupoid .
Let and be the class of all small simplicial sets. If is a small -category, then .
Let and be the class of all small -filtered simplicial sets for some regular cardinal . If is a small -category, then .
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 .
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 -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 -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
for all ;
if for ;
if , then .
The homotopic model is an extensional homotopic model if it satisfies the additional property: with .
Let be a homotopic -model. The notion of satisfaction in is defined as
Let be a homotopic -model. Then, for all , , and ,
The proof is very similar to the equality as in (Barendregt, 1984) and (Hindley; Seldin, 2008). ∎
Let be a homotopic -model. Then
By induction on the length of proof. For the axiom we precede
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:
has a terminal object ,
For , there exist an object in ,
For , there exist an object in such that set the equivalence
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
such that .
It will be shown that every reflexive -groupoid defines naturally a homotopic -model.
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.
Let be a reflexive -groupoid via the functors , which enough points.
For each define
Let a valuation in . Define the interpretation by induction as follows
Let be a reflexive -groupoid via the functors , with enough points and let . Then
is a homotopic -model.
is extensional iff
1. The conditions in Definition 2.1 (1), (2) are trivial. As to (3)
The condition (4) follows an easy induction on . The condition (5), given any object and
Applying and by Definition 2.7 (c) follows
Condition (6). By the proof of condition (3) and by hypothesis, for all object in
since has enough points, then
applying and by Definition 2.7 (c)
2. Suppose that is extensional. Let be an object of . Then for all
since has enough points, hence
If . For all by hypothesis and Definition 2.7
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
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 .
The functor given by be a Kleisli structure.
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
in is the -simplex generated for the composable chain of morphism
in the -category .
in the -category .
4 Distributivity and monads extension
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.
Definition 4.2 (Monad extension).
Let be a Kleisli structure on and a monad such that . Define the extension of along the functor free as the monad such that .
Let be a Kleisli structure on and a monad such that . Given a distributivity , then there exists an extension of along the functor free .
Let be a morphism of . Define as
Let’s see what extends to . Let a morphism of , then
on the other hand
but is distributive, i.e., , thus
Hence . ∎
5 -groupoidal -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
where is the class of morphisms which preserve -small colimits.
There exists an extension of to a monad on , which is denoted by .
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
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 .
The -category is cartesian closed.
The -category is cartesian closed.
The -category does have enough points.
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
Thus (naturally isomorphic). By the Definition 1.2 we have the homotopy of functors . ∎
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.
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