\aldine  Foundations of Substructural Dependent Type Theory

A comprehension category, viewed as a model of dependent type theory (c.f. Jacobs (jacobs, Jac93)), comprises:

1. (1)

   A category $\mathfrak{C}$ of contexts, with objects $\Gamma,\Delta$ representing contexts and morphisms $f:\Gamma\xrightarrow[]{}\Delta\in\mathfrak{C}$ viewed as substitutions.

2. (2)

   For each context $\Gamma\in\mathfrak{C}$, a category $\mathfrak{T}(\Gamma)$ of types, with objects $S,T$ interpreted as types dependent upon $\Gamma$ and morphisms $t:S\xrightarrow[]{}T\in\mathfrak{T}(\Gamma)$ as terms of type $T$ in context $\Gamma$, additionally parameterized by $S$.

3. (3)

   For each $f:\Gamma\xrightarrow[]{}\Delta\in\mathfrak{C}$, a functor $f^{*}:\mathfrak{T}(\Delta)\to\mathfrak{T}(\Gamma)$ representing the application of the substitution $f$, such that the assignment of functors $f^{*}$ preserves identities and composites of substitutions up to coherent natural isomorphism.

4. (4)

   For each $\Gamma\in\mathfrak{C}$, a context extension functor $\Gamma_{\bullet}:\mathfrak{T}(\Gamma)\to\mathfrak{C}$ mapping each type $S$ to the context $\Gamma_{\bullet}(S)$, to be thought of as $\Gamma$ extended with a fresh variable of type $S$.

5. (5)

   For each substitution $f:\Gamma\xrightarrow[]{}\Delta\in\mathfrak{C}$, a natural transformation $f_{\bullet}$ with components $f_{\bullet}(S):\Gamma_{\bullet}(f^{*}(S))\xrightarrow[]{}\Delta_{\bullet}(S)\in\mathfrak{C}$ for all $S\in\mathfrak{T}(\Delta)$, which – roughly speaking – applies $f$ to turn $\Gamma$ under an extension of $\Gamma$ substituted over $f$. We moreover require that the assignment of natural transformations $f_{\bullet}$ preserves identities and composites modulo the natural isomorphisms described above in item 3.

6. (6)

   For each context $\Gamma\in\mathfrak{C}$, a natural transformation $\pi_{\Gamma}$ with components $\pi_{\Gamma}^{S}:\Gamma_{\bullet}(S)\xrightarrow[]{}\Gamma\in\mathfrak{C}$ for each $S\in\mathfrak{T}(\Gamma)$, to be thought of as projecting out $\Gamma$ from an extension of $\Gamma$, such that for each substitution $f:\Gamma\xrightarrow[]{}\Delta\in\mathfrak{C}$ and type $S\in\mathfrak{T}(\Delta)$, the following square is a pullback:

A comprehension category has Dependent Sums:

1. (1)

   if for each context $\Gamma\in\mathfrak{C}$ and type $S\in\mathfrak{T}(\Gamma)$, the weakening functor $\omega_{\Gamma}^{S}$ has a left adjoint $\Sigma_{S}:\mathfrak{T}(\Gamma_{\bullet}(S))\to\mathfrak{T}(S)$, i.e. there is a natural bijection of homsets:

   $$\text{Hom}_{\mathfrak{T}(\Gamma_{\bullet}(S))}(T,\omega_{\Gamma}^{S}(R))\cong\text{Hom}_{\mathfrak{T}(\Gamma)}(\Sigma_{S}(T),R)$$

2. (2)

   and if the functors $\Sigma_{S}$ satisfy the Beck-Chevalley condition, which essentially states that for any ${f:\Gamma\to\Delta\in\mathfrak{C}}$ along with $S\in\mathfrak{T}(\Delta)$ and $T\in\mathfrak{T}(\Delta_{\bullet}(S))$, there is a canonical isomorphism $f^{*}(\Sigma_{S}(T))\simeq\Sigma_{f^{*}(S)}(f_{\bullet}(S)^{*}(T))$.

A comprehension category has unit types if, for each context $\Gamma\in\mathfrak{C}$, the category $\mathfrak{T}(\Gamma)$ has a terminal object $\top_{\Gamma}$, and the substitution functors preserve terminal objects. These unit types are strong if the projection $\pi_{\Gamma}:\Gamma_{\bullet}(\top_{\Gamma})\xrightarrow[]{}\Gamma\in\mathfrak{C}$ is an isomorphism for all contexts $\Gamma\in\mathfrak{C}$.

An LFDC consists of the same data as items 1-5 of Def. 2.1, along with the following:

1. (1)

   For each context $\Gamma\in\mathfrak{C}$ and type $S\in\mathfrak{T}(\Gamma)$, a dependent pair type functor ${\oplus_{S}:\mathfrak{T}(\Gamma_{\bullet}(S))\to\mathfrak{T}(\Gamma)}$.

2. (2)

   For each substitution $f:\Gamma\to\Delta\in\mathfrak{C}$ and each term $g:f^{*}(S)\xrightarrow[]{}S^{\prime}\in\mathfrak{T}(\Gamma)$, a natural transformation $\langle g,-\rangle$ with components $\oplus_{f^{*}(S)}(\Gamma_{\bullet}(g)^{*}(T))\to\oplus_{S^{\prime}}(T)$.

3. (3)

   For each context $\Gamma\in\mathfrak{C}$, an unit type object $\mathbbm{1}_{\Gamma}\in\mathfrak{T}(\Gamma)$.

4. (4)

   Natural isomorphisms $\eta,\rho,\ell,\alpha,\beta,\gamma,\delta$ with components:

   1. (a)

      $\eta_{\Gamma}:\Gamma_{\bullet}(\mathbbm{1}_{\Gamma})\simeq\Gamma$ for all $\Gamma\in\mathfrak{C}$

   2. (b)

      $\rho_{\Gamma,S}:\oplus_{S}(\mathbbm{1}_{\Gamma_{\bullet}(S)})\simeq S$ for all $\Gamma\in\mathfrak{C}$ and $S\in\mathfrak{T}(\Gamma)$

   3. (c)

      $\ell_{\Gamma,S}:\oplus_{\mathbbm{1}_{\Gamma}}(\eta_{\Gamma}^{*}(S))\simeq S$ for all $\Gamma\in\mathfrak{C}$ and $S\in\mathfrak{T}(\Gamma)$

   4. (d)

      $\alpha_{\Gamma,S,T}:\Gamma_{\bullet}(\oplus_{S}(T))\simeq(\Gamma_{\bullet}(S))_{\bullet}(T)$ for all $\Gamma\in\mathfrak{C}$ together with $S\in\mathfrak{T}(\Gamma)$ and $T\in\mathfrak{T}(\Gamma_{\bullet}(S))$

   5. (e)

      $\beta_{\Gamma,S,T,R}:\oplus_{S}(\oplus_{T}(R))\simeq\oplus_{\oplus_{S}(T)}(\alpha_{\Gamma,S,T}^{*}(R))$ for all
      $\Gamma\in\mathfrak{C},~{}S\in\mathfrak{T}(\Gamma)$ with $T\in\mathfrak{T}(\Gamma_{\bullet}(S))$ and $R\in\mathfrak{T}((\Gamma_{\bullet}(S))_{\bullet}T)$

   6. (f)

      $\gamma_{\Gamma,\Delta,f,S,T}:f^{*}(\oplus_{S}(T))\simeq\oplus_{f^{*}(S)}(f_{\bullet}(S)^{*}(T))$ for all $\Gamma,\Delta\in\mathfrak{C}$ with ${f:\Gamma\xrightarrow[]{}\Delta}\in\mathfrak{C}$ and $S\in\mathfrak{T}(\Delta)$ and $T\in\mathfrak{T}(\Delta_{\bullet}(S))$ (Beck-Chevalley for pair types)

   7. (g)

      $\delta_{\Gamma,\Delta,f}:f^{*}(\mathbbm{1}_{\Delta})\simeq\mathbbm{1}_{\Gamma}$ for all $\Gamma,\Delta\in\mathfrak{C}$ with ${f:\Gamma\xrightarrow[]{}\Delta}\in\mathfrak{C}$ (Beck-Chevalley for unit types)

   subject to certain coherence laws.

One additional piece of type-theoretic structure that has so-far been missing from the above exposition of LFDCs is some notion of empty context from which to build other contexts. To this end, we have the following: a unit context in an LFDC is a context $\epsilon\in\mathfrak{C}$ such that the context extension functor ${\epsilon_{\bullet}}$ is an equivalence of categories between $\mathfrak{T}(\epsilon)$ and $\mathfrak{C}$. We think of types $T\in\mathfrak{T}(\epsilon)$ as closed types, such that contexts and closed types may be treated interchangeably. Hence we shall hereafter restrict our attention mainly to LFDCs equipped with a choice of unit context.

An LFDC has type-level weakening if it is additionally equipped with:

1. (1)

   A functor $\omega_{\Gamma}^{S}:\mathfrak{T}(\Gamma)\to\mathfrak{T}(\Gamma_{\bullet}(S))$ for each context $\Gamma\in\mathfrak{C}$ and type $S\in\mathfrak{T}(\Gamma)$

2. (2)

   A functor $\Omega_{\Gamma}^{S,T}:\mathfrak{T}(\Gamma_{\bullet}T)\to\mathfrak{T}((\Gamma_{\bullet}S)_{\bullet}\omega_{\Gamma}^{S}(T))$ for each context $\Gamma\in\mathfrak{C}$ and types $S,T\in\mathfrak{T}(\Gamma)$

3. (3)

   Subject to certain natural isomorphisms and coherence laws thereupon that ensure the compatibility of the above-given functors with each other and with the structure of the ambient LFDC. In particular, there are natural isomorphisms $\zeta$ and $\theta$ with components:

   • •

     $\zeta_{f,g,T}:\Gamma_{\bullet}(g)^{*}(\omega_{\Gamma}^{S^{\prime}}(T))\simeq\omega^{f^{*}(S)}_{\Gamma}(T)$

   • •

     $\theta_{f,g,T,R}:\Gamma_{\bullet}(g)_{\bullet}(\omega_{\Gamma}^{S^{\prime}}(T))^{*}(\Omega_{\Gamma}^{S^{\prime},T}(R))\simeq\zeta_{f,g,T}^{*}(\Omega_{\Gamma}^{f^{*}(S),T}(R))$

   for all $\Gamma,\Delta\in\mathfrak{C}$ with $f:\Gamma\to\Delta\in\mathfrak{C}$ and $S^{\prime},T,R\in\mathfrak{T}(\Gamma)$ and $S\in\mathfrak{T}(\Delta)$ along with $g:f^{*}(S)\xrightarrow[]{}S^{\prime}\in\mathfrak{T}(\Gamma)$. These isomorphisms effectively witness that a weakened type does not depend upon the type added to its context by weakening. We also require natural isomorphisms $\xi,\chi$ with components

   • •

     $\xi_{\Gamma,S}:\omega_{\Gamma}^{S}(\mathbbm{1}_{\Gamma})\simeq\mathbbm{1}_{\Gamma_{\bullet}(S)}$

   • •

     $\chi_{\Gamma,S,T,R}:\omega_{\Gamma}^{S}(\oplus_{T}(R))\simeq\oplus_{\omega_{\Gamma}^{S}(T)}(\Omega_{\Gamma}^{S,T}(R))$

   for all $\Gamma\in\mathfrak{C}$ with $S,T\in\mathfrak{T}(\Gamma)$ and $R\in\mathfrak{T}(\Gamma_{\bullet}(T))$. These isomorphisms ensure that weakening behaves as we would expect with regard to the type formers available in the ambient LFDC, i.e. the weakening of a unit type is itself (up to isomorphism) a unit type, and likewise for dependent pair types. When positing additional type formers for an LFDC with type-level weakening, we shall therefore generally require these to come equipped with analogous natural isomorphisms to ensure their compatibility with the type-level weakening structure, along with the usual Beck-Chevalley isomorphisms.

In an LFDC with type-level weakening, each category of types $\mathfrak{T}(\Gamma)$ for each context $\Gamma$ is naturally equipped with the structure of a monoidal category whose unit object is $\mathbbm{1}_{\Gamma}$, where the monoidal product $S\otimes_{\Gamma}T$ of $S,T\in\mathfrak{T}(\Gamma)$ is defined as the dependent pair type $\oplus_{S}(\omega_{\Gamma}^{S}(T))$.

An LFDC with type-level weakening has term-level function types if for each context $\Gamma\in\mathfrak{C}$ with $S\in\mathfrak{T}(\Gamma)$, the functors $-\otimes_{\Gamma}S$ and $S\otimes_{\Gamma}-$ have right adjoints $\sslash_{S}$ and $\bbslash_{S}$, respectively, with the associated natural bijections of homsets:

We additionally require the functors $\sslash_{S}$ and $\bbslash_{S}$ to satisfy a Beck-Chevalley condition and compatibility with type-level weakening in the form of canonical isomorphisms

etc.

an LFDC with type-level weakening has type-level function types if for each context $\Gamma\in\mathfrak{C}$ with $S\in\mathfrak{T}(\Gamma)$, the weakening functor $\omega_{\Gamma}^{S}:\mathfrak{T}(\Gamma)\to\mathfrak{T}(\Gamma_{\bullet}(S))$ has a right adjoint $\forall_{S}$, with corresponding natural bijection of homsets

As in the above definition, we also require the functors $\forall_{S}$ to be compatible with substitution and type-level weakening in that there are canonical isomorphisms

an LFDC with type-level weakening has exchange if the monoidal structure defined on each category of types $\mathfrak{T}(\Gamma)$ in Def. 2.7 additionally carries the structure of a symmetric monoidal category, i.e. a natural isomorphism $\sigma$ with components

for all $S,T\in\mathfrak{T}(\Gamma)$, subject to certain coherence conditions.

An LFDC has term-level weakening if for each context $\Gamma$, the type $\mathbbm{1}_{\Gamma}$ is a terminal object in the category $\mathfrak{T}(\Gamma)$. For any context $\Gamma$, write $\top_{S}:S\xrightarrow[]{}\mathbbm{1}_{\Gamma}\in\mathfrak{T}(\Gamma)$ for the unique morphism from any type $S\in\mathfrak{T}(\Gamma)$ to $\mathbbm{1}_{\Gamma}$.

Any LFDC with term-level weakening is naturally equpped with type-level weakening, given by substitution along terminal morphisms. Specifically, given a context $\Gamma\in\mathfrak{C}$ and types $S,T\in\mathfrak{T}(\Gamma)$ and $R\in\mathfrak{T}(\Gamma_{\bullet}(T))$, we may define

and

An LFDC with type-level weakening has contraction if there is a natural transformation of homsets:

for all contexts $\Gamma\in\mathfrak{C}$ and types $S,R\in\mathfrak{T}(\Gamma)$ and $T\in\mathfrak{T}(\Gamma_{\bullet}(S))$, that is suitably compatible with the structure of the LFDC.

By a Yoneda-style argument, the above is equivalent to the existence of a natural family of morphisms

for all contexts $\Gamma\in\mathfrak{C}$ with $S\in\mathfrak{T}(\Gamma)$, suitably compatible with the LFDC structure. In this sense, contraction allows for the term-level use of type-level variables. This makes sense if we think of type-level contexts in LFDCs as consisting of variables that are treated as already having been used elsewhere.

It follows that in an LFDC with contraction, for each context $\Gamma\in\mathfrak{C}$, there is a natural transformation $\Delta$ with components $\Delta_{S}:S\xrightarrow[]{}S\otimes_{\Gamma}S\in\mathfrak{T}(\Gamma)$ for all types $S\in\mathfrak{T}(\Gamma)$, defined as follows:

If we think of $\Delta_{S}$ as duplicating its input, then this suffices to show that the above-defined notion of contraction for LFDCs gives rise to the usual notion of contraction as copying of variables.

An LFDC is Cartesian if 1) it has term-level weakening, and 2) the natural transformations of homsets defined in Def. 2.11 are all invertible, i.e. the type-level weakening functors $\omega_{\Gamma}^{S}$ are right-adjoint to the dependent pair type functors $\varoplus_{S}$.

Moreover, any Cartesian LFDC is naturally equipped with exchange, since for any context $\Gamma\in\mathfrak{C}$ and types $S,T\in\mathfrak{T}(\Gamma)$, we may define a morphism $\sigma:S\otimes_{\Gamma}T\to T\otimes_{\Gamma}S$ as follows:

where $\Delta$ is as defined in Def. 2.12. One then checks that $\sigma$ is an isomorphism satisfying all the properties given in Def. 2.10.

More generally, the monoidal product $\otimes_{\Gamma}$ on each category of types $\mathfrak{T}(\Gamma)$ in a Cartesian LFDC satisfies the universal property of a product. Even if the given LFDC is not Cartesian, it may have such products in its type-theoretic structure, without these necessarily coinciding with the monoidal product, as follows:

An LFDC has product types if:

1. (1)

   For each $\Gamma\in\mathfrak{C}$, there is a functor $\times_{\Gamma}:\mathfrak{T}(\Gamma)\times\mathfrak{T}(\Gamma)\to\mathfrak{T}(\Gamma)$ together with natural transformations $\pi_{1},\pi_{2}$ with components $\pi_{1}^{S,T}:S\times_{\Gamma}T\to S$ and $\pi_{2}^{S,T}:S\times_{\Gamma}T\to T$, such that for all types $S,T,R\in\mathfrak{T}(\Gamma)$ with morphisms $f:R\xrightarrow[]{}S\in\mathfrak{T}(\Gamma)$ and $g:R\xrightarrow[]{}T\in\mathfrak{T}(\Gamma)$, there exists a unique morphism $(f,g):R\xrightarrow[]{}S\times_{\Gamma}T\in\mathfrak{T}(\Gamma)$ which factors $f$ and $g$ through $\pi_{1}^{S,T}$ and $\pi_{2}^{S,T}$, respectively.

2. (2)

   The substitution functors $f^{*}$ are all finite-product-preserving, and if the LFDC has type-level weakening, then similarly the type-level weakening functors preserve finite products.

We think of the product $S\times_{\Gamma}T$ of two types $S,T\in\mathfrak{T}(\Gamma)$ as offering a choice between an element of $S$ and an element of $T$, while the monoidal product $S\otimes_{\Gamma}T$ offers two such elements at once.

An LFDC is strict if:

1. (1)

   The assignment of substitution functors $f^{*}$ strictly preserves identities and composites of substitutions.

2. (2)

   The Beck-Chevalley isomorphisms $\gamma,\delta$ are identities, i.e. substitution strictly preserves unit and dependent pair types.

Assume a countably infinite stock of variables – which I shall write as $a,b,c,d,w,x,y,z$ – and countably infinite supplies of type family symbols and constant symbols – written $P,Q$ and $p,q$, respectively. We then have the following grammar of syntactic expressions:

Fix a strict LFDC with type-level weakening, function types, and products. For each syntactic sort of expressions defined above, we then define a corresponding class of denotations:

• •

  Contexts: the denotation of a context $\Gamma$ is a list $\llparenthesis\Gamma\rrparenthesis$ of type annotations $x:\llbracket S\rrbracket$, which is well-formed if each such $\llbracket S\rrbracket$ is a type in the context corresponding to the list of annotations preceding $x$. We then have the following definition of the semantic context $\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket$ represented by $\llparenthesis\Gamma\rrparenthesis$

  $$\llbracket\epsilon\rrbracket=\epsilon\qquad\llbracket\llparenthesis\Gamma\rrparenthesis,x:\llbracket S\rrbracket\rrbracket=\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket_{\bullet}(\llbracket S\rrbracket)$$

  I write $\llparenthesis\Gamma\rrparenthesis,\llparenthesis\Delta\rrparenthesis$ (or $\llparenthesis\Gamma\rrparenthesis\fatsemi\llparenthesis\Delta\rrparenthesis$ when $\llparenthesis\Gamma\rrparenthesis$ is understood to be the type-level context, and $\llparenthesis\Delta\rrparenthesis$ the term-level context), to mean that 1) $\llparenthesis\Gamma\rrparenthesis$ is well-formed, and 2) the concatenation of $\llparenthesis\Delta\rrparenthesis$ onto the right-hand side of $\llparenthesis\Gamma\rrparenthesis$ is also well-formed, and say that $\llparenthesis\Delta\rrparenthesis$ is a telescope for $\llparenthesis\Gamma\rrparenthesis$. We then have the following recursive definition of the semantic type $\llbracket\llparenthesis\Delta\rrparenthesis\rrbracket_{\llparenthesis\Gamma\rrparenthesis}$ represented by a telescope $\llparenthesis\Delta\rrparenthesis$ for a semantic context $\llparenthesis\Gamma\rrparenthesis$:

  $$\qquad\llbracket\epsilon\rrbracket_{\llparenthesis\Gamma\rrparenthesis}=\mathbbm{1}_{\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket}\quad\llbracket x:\llbracket S\rrbracket,\llparenthesis\Delta\rrparenthesis\rrbracket_{\llparenthesis\Gamma\rrparenthesis}=\varoplus_{\llbracket S\rrbracket}(\llbracket\llparenthesis\Delta\rrparenthesis\rrbracket_{\llparenthesis\Gamma\rrparenthesis,x:\llbracket S\rrbracket})$$

• •

  Types: given a semantic context $\llparenthesis\Gamma\rrparenthesis$, the denotation of a type $S$ wrt $\llparenthesis\Gamma\rrparenthesis$ is defined as an object $\llbracket S\rrbracket_{\llparenthesis\Gamma\rrparenthesis}\in\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket)$.

• •

  Introduction & Elimination Forms: given a semantic context $\llparenthesis\Gamma\rrparenthesis$ and a telescope $\llparenthesis\Gamma\rrparenthesis\fatsemi\llparenthesis\Delta\rrparenthesis$ together with a semantic type $\llbracket S\rrbracket_{\llparenthesis\Gamma\rrparenthesis}$, the denotation of an introduction/elimination form $\digamma$ wrt $\llparenthesis\Gamma\rrparenthesis\fatsemi\llparenthesis\Delta\rrparenthesis$ and $\llbracket S\rrbracket_{\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket}$ is defined as a morphism $\llbracket\digamma\rrbracket^{\llparenthesis\Gamma\rrparenthesis\fatsemi\llparenthesis\Delta\rrparenthesis}_{\llbracket S\rrbracket}:\llbracket\llparenthesis\Delta\rrparenthesis\rrbracket_{\llparenthesis\Gamma\rrparenthesis}\xrightarrow[]{}\llbracket S\rrbracket_{\llparenthesis\Gamma\rrparenthesis}\in\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket)$.

In order to capture structure of the LFDC in which the type theory is interpreted, we allow for the type theory to additionally be parameterized by a signature of constant terms and atomic type families. Such a signature $\Sigma$ is defined to be a set of bindings, where each binding has one of the following forms:

• •

  $P(\llbracket S\rrbracket)\mapsto\llbracket P\rrbracket$, where $\llbracket S\rrbracket\in\mathfrak{T}(\epsilon)$ is a closed semantic type, and $\llbracket P\rrbracket\in\mathfrak{T}(\epsilon_{\bullet}(\llbracket S\rrbracket))$ is a type family dependent upon $\llbracket S\rrbracket$.

• •

  $p:\llbracket S\rrbracket\mapsto\llbracket p\rrbracket$, where $\llbracket S\rrbracket\in\mathfrak{T}(\epsilon)$ is a closed semantic type, and $\llbracket p\rrbracket:\mathbbm{1}_{\epsilon}\xrightarrow[]{}\llbracket S\rrbracket\in\mathfrak{T}(\epsilon)$ is a closed term of type $\llbracket S\rrbracket$.

We conceive of judgments as procedures taking inputs (here written in blue) and yielding outputs (here written in red), with one input of each judgment designated the subject (written in violet). I follow McBride (mcbride, McB16) in writing inputs to a judgment to the left of the subject, and outputs to the right, using $\ni$ and $\in$ for the judgments in which types are checked and inferred, respectively. The four main judgments of the type theory are then as follows:

Observe that within these judgments all inputs/outputs other than the subject are semantic objects. Hence it is only ever the subject of a judgment which requires checking, and all dependence of type checking upon semantic information about types/contexts/etc. is suitably encoded in the structure of these judgments.

Before proceeding with the rules of the type theory, I first note some useful constructions on semantic contexts / types / terms (see Appendix A for full definitions):

• •

  Functoriality of telescopes: given a telescope $\llparenthesis\Gamma\rrparenthesis,\llparenthesis\Delta\rrparenthesis$, there is an associated functor $\varoplus_{\llparenthesis\Delta\rrparenthesis}:\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis,\llparenthesis\Delta\rrparenthesis\rrbracket)\to\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket)$ given by iterative formation of dependent pair types.

• •

  Context substitution/weakening: given a telescope $\llparenthesis\Delta\rrparenthesis,\llparenthesis\Theta\rrparenthesis$ and substitution $f:\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket\to\llbracket\llparenthesis\Delta\rrparenthesis\rrbracket$, there is a telescope $\llparenthesis\Gamma\rrparenthesis,f^{*}(\llparenthesis\Theta\rrparenthesis)$ given by substituting into all types in $\llparenthesis\Theta\rrparenthesis$. Similarly, for a telescope $\llparenthesis\Gamma\rrparenthesis,\llparenthesis\Delta\rrparenthesis$ and type $\llbracket S\rrbracket\in\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket)$, there is a telescope $\llparenthesis\Gamma\rrparenthesis,x:\llbracket S\rrbracket,\omega_{\llparenthesis\Gamma\rrparenthesis}^{\llbracket S\rrbracket}(\llparenthesis\Delta\rrparenthesis)$ given by weakening all types in $\llparenthesis\Delta\rrparenthesis$.

• •

  Reassociation: for each telescope $\llparenthesis\Gamma\rrparenthesis,\llparenthesis\Delta\rrparenthesis$ there is an isomorphism ${\bullet}\mathsf{asc}_{\llparenthesis\Gamma\rrparenthesis}^{\llparenthesis\Delta\rrparenthesis}:\llbracket\llparenthesis\Gamma\rrparenthesis,\llparenthesis\Delta\rrparenthesis\rrbracket\simeq\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket_{\bullet}(\llbracket\llparenthesis\Delta\rrparenthesis\rrbracket_{\llparenthesis\Gamma\rrparenthesis})$. Similarly, for each telescope $\llparenthesis\Gamma\rrparenthesis,\llparenthesis\Delta\rrparenthesis$ and type $\llbracket S\rrbracket\in\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis,\llparenthesis\Delta\rrparenthesis\rrbracket)$ there is an isomorphism

  $$\qquad\quad{\oplus}\mathsf{asc}_{\llparenthesis\Gamma\rrparenthesis}^{\llparenthesis\Delta\rrparenthesis\mid\llbracket S\rrbracket}:\varoplus_{\llparenthesis\Delta\rrparenthesis}(\llbracket S\rrbracket)\simeq\varoplus_{\llbracket\llparenthesis\Delta\rrparenthesis\rrbracket}((({\bullet}\mathsf{asc}_{\llparenthesis\Gamma\rrparenthesis}^{\llparenthesis\Delta\rrparenthesis})^{-1})^{*}(\llbracket S\rrbracket))$$

• •

  Term substitution: given a term $\llbracket s\rrbracket:\llbracket\llparenthesis\Delta\rrparenthesis\rrbracket\xrightarrow[]{}\llbracket S\rrbracket\in\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket)$, there is a substitution ${\downarrow}(\llbracket s\rrbracket):\llbracket\llparenthesis\Gamma\rrparenthesis,\llparenthesis\Delta\rrparenthesis\rrbracket\to\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket_{\bullet}(\llbracket S\rrbracket)$, and given a term $\llbracket e\rrbracket:\llbracket\llparenthesis\Delta\rrparenthesis\rrbracket\xrightarrow[]{}\llbracket S\rrbracket\in\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket)$ and a telescope $\llparenthesis\Gamma\rrparenthesis,x:\llbracket S\rrbracket,\llparenthesis\Theta\rrparenthesis$, there is a substitution

  $$\qquad{\Downarrow}(\llbracket e\rrbracket):\llbracket\llparenthesis\Gamma\rrparenthesis,\llparenthesis\Delta\rrparenthesis,{\downarrow}(\llbracket e\rrbracket)^{*}(\llparenthesis\Theta\rrparenthesis)\rrbracket\xrightarrow[]{}\llbracket\llparenthesis\Gamma\rrparenthesis,x:\llbracket S\rrbracket,\llparenthesis\Theta\rrparenthesis\rrbracket$$

• •

  Lifting: for each telescope $\llparenthesis\Gamma\rrparenthesis,\llparenthesis\Delta\rrparenthesis$ and type $\llbracket S\rrbracket\in\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis)$, there are lifting functors $\uparrow_{\llparenthesis\Gamma\rrparenthesis}^{\llparenthesis\Delta\rrparenthesis}:\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket)\to\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis,\llparenthesis\Delta\rrparenthesis\rrbracket)$ and $\Uparrow_{\llparenthesis\Gamma\rrparenthesis}^{\llparenthesis\Delta\rrparenthesis\mid\llbracket S\rrbracket}:\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket_{\bullet}(\llbracket S\rrbracket))\to\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis,\llparenthesis\Delta\rrparenthesis\rrbracket_{\bullet}(\uparrow_{\llparenthesis\Gamma\rrparenthesis}^{\llparenthesis\Delta\rrparenthesis}(\llbracket S\rrbracket)))$ that lift types into extended contexts by iterative weakening, and a telescopic weakening functor

  $$\qquad\mathsf{W}_{\llparenthesis\Gamma\rrparenthesis}^{\llbracket S\rrbracket\mid\llparenthesis\Delta\rrparenthesis}:\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis,\llparenthesis\Delta\rrparenthesis\rrbracket)\to\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis,x:\llbracket S\rrbracket,\omega_{\llparenthesis\Gamma\rrparenthesis}^{\llbracket S\rrbracket}(\llparenthesis\Delta\rrparenthesis)\rrbracket)$$

• •

  Projection: in an LFDC with term-level weakening, for a telescope $\llparenthesis\Gamma\rrparenthesis,\llparenthesis\Delta\rrparenthesis$ and a type $\llbracket S\rrbracket\in\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket)$, there is a projection

  $$\mathsf{proj}_{\llparenthesis\Gamma\rrparenthesis}^{\llbracket\Delta\rrbracket\mid\llbracket S\rrbracket}:\llbracket\llparenthesis\Delta\rrparenthesis,x:{\uparrow}_{\llparenthesis\Gamma\rrparenthesis}^{\llparenthesis\Delta\rrparenthesis}(\llbracket S\rrbracket)\rrbracket\xrightarrow[]{}\llbracket S\rrbracket\in\mathfrak{T}(\llbracket\llparenthesis\Gamma\rrparenthesis\rrbracket)$$