# Questions tagged [type-theory]

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21
questions with no upvoted or accepted answers

**14**

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**0**answers

999 views

### What's the point of cubical type theory?

I have been following through the development of homotopy type theory since 2013 because I was really interested in the foundation of mathematics. The novel idea of combining programming with homotopy ...

**8**

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260 views

### What metatheory proves cut elimination for Simple Type Theory?

Gaisi Takeuti's book Proof Theory proves cut elimination for Simple Type Theory (a combined result of several researchers). Thus it proves consistency of Simple Type Theory with full comprehension in ...

**8**

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175 views

### Ends and parametricity

It is well known that a set of natural transformations can be expressed as an end:
$$\int_{A \in \mathcal{A}} \mathcal{B}(FA, GA) =_{\operatorname{Set}} \operatorname{Nat}(F, G)$$
This holds for ...

**7**

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143 views

### Generalized (co)-presheaves for Generalized Multicategories?

A $T$-multicategory in the sense of Crutwell-Shulman, where $T$ is a monad on a virtual double category $C$ is a monoid in the "horizontal kleisli category", i.e., an object of objects $O$, a ...

**7**

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248 views

### Feasible Type Theories

I am looking for references about efficient type theories,
efficiency in the sense of computational complexity,
and type theory in the sense of Martin-Lof's type theories.
Has there been any studies ...

**6**

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**0**answers

277 views

### What does first-order co-intuitionistic logic look like (and does it have an equivalent type theory)?

So, this is where I'm at so far:
Heyting algebras model propositional intuitionistic logic (IL)
so do Cartesian closed categories which also model the simply typed lamda calculus
co-Heyting algebras ...

**6**

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152 views

### Preservation of universes in presheaves

In Lifting Grothendieck universes Hofmann and Streicher construct a universe in the category of presheaves over a small category given a Grothendieck universe in $\mathbf{Set}$.
Suppose now I have ...

**5**

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**0**answers

174 views

### Are any formal systems based upon the idea of “iterated characterization pushing” currently in existence? If not, is anyone working on them?

I had an idea in regards to the design of formal systems with foundational aspirations.
To convey the idea, let's talk a bit about the second-order Peano axioms. The way these axioms work, we have a ...

**5**

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195 views

### Proper full submodels of full models of type theory

Let $N$ be the standard full model of the simply typed lambda calculus with infinite base type $o$ and let $X$ be an infinite and coinfinite subset of $N(o)$. I want to know if there's a full ...

**4**

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421 views

### New foundation in homotopy type theory

Is there any model of NF (New Foundations) on HoTT (homotopy type theory)?
Because there is a model of ZF(C) on HoTT (The HoTT book, Section 10.5) and NF on ZFC (by this Wikipedia articile), I think ...

**4**

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116 views

### Problem with a proof in Wellfounded trees in categories

I'm reading the paper Wellfounded trees in categories by Moerdijk and Palmgren. I'm having trouble understanding the proof of Theorem 7.2 (page 216), i.e. that in $\mathbf{ML}_{< \omega} \mathbf{W}$...

**3**

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98 views

### Was Martin-Löf inspired by Peirce when he introduced the dependent sum and dependent product types?

In the following article:
https://plato.stanford.edu/entries/peirce-logic/
it is mentioned that Peirce's introduced the use of the symbols $\Sigma$ and $\Pi$ to express logical sums and products, ...

**3**

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**0**answers

144 views

### Internal language type of power objects

It is a basic fact that in a category with finite limits the following are equivalent
Each object $X$ has a (membership relation to a) power object $PX\times X\hookleftarrow\ni_X$ subject to the ...

**3**

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149 views

### Logical framework for type theories like ML and CIC

I'm looking for a logical framework in which it is possible to easily present both intensional and extensional theories of dependent types with a partially ordered set of universes à la Russell ...

**3**

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202 views

### An elegant formulation for typed sets

Fix a poset $T$, which we'll think of as a set of "types," interpreting $a \leq b$ as "$a$ is more general than $b$." Construct a category of TSet as follows.
Objects: Pairs ($X$, $\tau : X \...

**2**

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133 views

### Categorical semantics of the identity type

In Appendix B of the article Simplicial Model of Univ. Foundations, Definition B.1.3 specifies the structure for identity types in a contextual category $\mathcal{C}$ (which in particular is equipped ...

**2**

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151 views

### What is the consistency limit of accumulative typing below $\omega_1^{CK}$?

Let $Th_\zeta$ be a Mono-sorted first order theory with $\zeta$ representing some recursive ordinal notation system.
Primitives: =, $\in$, $T_0, T_1, ..,T_i,..$ where i is an $\zeta$ ordinal, and ...

**2**

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66 views

### What is the relation of total functions in second order arithmetic and fast growing hierarchies?

Answer to this questions shows that fast growing hierarchies can grow arbitrarily fast for some definition of 'arbitrary'.
Can second order arithmetic define all these functions (for any ordinal) ...

**2**

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**0**answers

180 views

### Type theory: can multiple elimination rules be defined, in principle?

I'd like to ask a question on type theory:
Consider the usual type theoretical definition of the natural numbers. We could give an elimination rule in the form:
or in the form:
I called the ...

**1**

vote

**0**answers

83 views

### Natural relations between substitutions

Consider two contexts $\Gamma,\Delta$ (from some background type theory), and substitutions $s_1,s_2:\Gamma\rightarrow \Delta$. In the case of $1$-element contexts, we get that a substitution is ...

**0**

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128 views

### An equation involving multisets

For finite multisets $A, B, C, A', B', C'$, if $A \uplus B \uplus \{B \uplus C\} \uplus \{A \uplus \{C\}\}$ = $A' \uplus B' \uplus \{B' \uplus C'\} \uplus \{A' \uplus \{C'\}\}$, must $A=A',B=B',C=C'$, ...