Questions tagged [category-theory]
Categories are structures containing objects and arrows between them. Many mathematical structures can be viewed as objects of a category, with structure preserving morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seemingly different areas of mathematics.
22
questions
1
vote
2
answers
232
views
Formalization of a model of ∞-category in a proof assistant
I am aware of similar question in MO whose comment nicely lists zoo of possible models and only such models can be formalized. But I have not found any implementation so far which I could use or ...
- 133
2
votes
0
answers
74
views
Formally verified email or communication?
This question is contextualized by having an account hacked which is prompting me to move towards something I’ve long wanted to anyway.
I would like to consider the simplest possible formally verified ...
- 175
4
votes
2
answers
295
views
Reasoning about CwFs in a proof assistant
I've been chatting with folks on Mastodon about this but the perspective there is markedly Agda-focused, so I thought I'd ask here for some broader opinions.
What tools/libraries are there for ...
- 773
1
vote
1
answer
104
views
Formal proof of “functional” / purposive law
I am really interested in exploring if (legal) laws can be proven to be self-consistent using a formal programming language / proof assistant, and modeling the laws using category theory, perhaps ...
- 175
1
vote
1
answer
77
views
Is every type-theoretic function ℕ → A extensionally equal to one written in terms of the ℕ-eliminator
In Category Theory the Natural Numbers object ℕ has the universal mapping-out property that tells us how to build arrows out of ℕ into an arbitrary object A. But it doesn't say that every arrow ...
- 345
4
votes
1
answer
417
views
Semantics of type theory
There are many tools regarding the semantics of type theory. On one hand, we may organize the structure of substitutions explicitly, resulting in notions such as CwFs, CwAs, natural models, display ...
- 3,688
3
votes
1
answer
139
views
Stacks versus universes
This is a vague question, and I apologize in advance for it. I do not need a definite answer, I am happy to just get some general ideas.
An elementary topos can model higher order logic over a ...
- 722
9
votes
1
answer
154
views
Interpretation of dependent types: Coherence
I have trouble understanding precisely how dependent type theories are intepreted in categories (in the most simple case, for example in a locally cartesian closed category). I know that a type in ...
- 722
11
votes
1
answer
243
views
Seven Trees in One, or How to formalize the Semiring of Types?
This is somewhat conceptual beginner's question about proof assistants.
I've been re-reading the famous Seven Trees in One / Objects of categories as complex numbers. The gist: The type $T$ of binary ...
- 213
1
vote
1
answer
79
views
Proving that applicative functors compose
For simplicity, here an applicative functor means (in a proof assistant based on dependent type theory) the Haskellian applicative functor, bundled with its equational laws.
This I can of course brute ...
- 3,688
2
votes
0
answers
69
views
Construction of "free Lawvere theory" on a collection of function symbols
I think I've been able to puzzle out the free Lawvere theory on a collection of function symbols ought to have maps being just tuples of the free monad.
In Coq
...
- 2,768
20
votes
1
answer
283
views
Examples of formalisation of abelian categories
The question
I would be interested to hear about examples of formalisation of the theory of abelian categories in theorem provers, and in particular formalisations of things like the zig-zag lemma and ...
- 2,209
10
votes
0
answers
183
views
Is there a proof assistant (or an embedding) for the (co)end calculus?
A Higher-Order Calculus for Categories describes a system where you can conveniently perform manipulations with categories, functors, Yoneda embeddings etc. An example of the rules is: $$\frac{\Gamma ,...
- 3,688
14
votes
6
answers
1k
views
Is there any example of a dependent product that makes sense in a non-type-theory context?
Dependent products are said to be the right adjoints of reindexing functors according to nlab. However, I can only make sense of this explanation in the context of type theory, where dependent ...
- 5,857
18
votes
2
answers
488
views
What are the motivations for different variants of categorical models of dependent type?
I am new to the categorical semantics for dependent type theories, so it is surprising for me to see nLab introduces so many variants of categorical models, including comprehension categories, display ...
- 429
17
votes
2
answers
906
views
What are "fibration/cofibration" in type theory and what are their intuitions?
I keep seeing these phrasing in some proof assistants/elaborators and their issues/internal discussions (e.g. Github search results in cooltt), that seems not that related to the actual proofs/...
- 181
6
votes
0
answers
111
views
Is every logical theory embeddable in a dependently typed extensional type theory?
In category theory by the Yoneda embedding every category $\mathcal{C}$ is a subcategory of a category of presheafs $[\mathcal{C}^{\text{op}}, \text{Set}]$. Every category of presheafs is a topos and ...
- 2,768
11
votes
1
answer
195
views
Ergonomic use of multicategories in proof assistants
I've been looking at syntactic multicategories for mechanizing some type theory stuff.
But multicategories are pretty messy to work with in a two sorted definition like the usual dependently typed ...
- 2,768
10
votes
1
answer
266
views
What is realignment and is it useful in non-univalent theories?
From my rough understanding, an (external/internal) realignment property says that given a type $A$, a proposition $p : \Omega$, and a partial isomorph $B : p \to \sum_{A'} A' \cong A$, we can extend ...
- 3,688
4
votes
0
answers
88
views
What is the relation of $\lambda^\to$ and $\lambda^{\to\times}$ to cartesian closed categories?
I am learning about the categorical semantics of type theory. I've written some preliminary results in Agda. In particular, I partially proved that the contexts and substitutions of simply-typed ...
- 3,688
19
votes
0
answers
247
views
Can we automatically get around set-theoretic difficulties?
One of the main technical annoyances of working with (large) categories is the variety of set-theoretic difficulties that come about with it: if we use ZFC as background logic, then those large ...
- 1,038
23
votes
2
answers
647
views
What is Artin gluing, and how is it useful in proving meta-theoretic properties?
I came across this notion in several places, especially in recent papers that establish important meta-theoretic properties of type theories like CuTT. The entry in nLab focuses on the geometric ...
- 3,688