Questions tagged [cartesian-closed-categories]
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20 questions with no upvoted or accepted answers
8
votes
0
answers
370
views
An obscure case of Curry-Howard
It is a theorem of the Intuitionistic Propositional Calculus that
$$
(p\to q)\to p = (q\to p) \land ((p\to q)\to q).
$$
The Curry-Howard correspondence realizes this as a pair of operators (for any ...
- 18.4k
7
votes
0
answers
151
views
Strictifying closed monoidal categories?
Let $C$ be a cartesian closed category. It's well known that $C$ is equivalent to a category where the product is strict monoidal; i.e. where there are equalities of the functors given by the ...
5
votes
0
answers
142
views
Does Cantor Bernstein hold in a Closed Symmetric Monoidal Category?
In a closed symmetric monoidal category with $[I,X] \cong X$ for all $X$ is it true that having monomorphisms $m :A \rightarrow B$ and $m: B \rightarrow A$ is enough to imply $A \cong B$ ?
I tried to ...
- 51
5
votes
0
answers
676
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Cartesian-closed categories of algebras
If the Kleisli-category of a monad is Cartesian-closed, can we say when the category of Eilenberg-Moore algebras is?
- 15.5k
4
votes
0
answers
121
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Is there a faithful functor from the freely generated bicartesian closed category to $\mathbf{Set}$?
Does there exist a faithful (bicartesian closed) functor $\operatorname F$ from the freely generated bicartesian closed category $\mathbf B$ to $\mathbf{Set}$? Preferably, $\mathbf B$ should contain ...
4
votes
0
answers
74
views
Need to know if a certain full subcategory of Top is cartesian closed
Consider the full subcategory of Top consisting of all spaces $X$ such that a subset $A$ of $X$ is closed if and only if $A \cap K$ is closed in $K$ for all subspaces $K$ of $X$ which are countably ...
- 2,125
3
votes
0
answers
54
views
A new(?) kind of 2-adjunction for relating cartesian closed functors using dinatural hexagons
$\newcommand{\A}{\operatorname{A}}
\newcommand{\B}{\operatorname{B}}
\newcommand{\Cat}{\mathcal{Cat}}
\newcommand{\Cart}{\mathcal{Cart}}
\newcommand{\C}{\mathbf{C}}
\newcommand{\F}{\operatorname{F}}
\...
3
votes
0
answers
162
views
When a monoidal closed category is cartesian closed
Let $C$ be a monoidal closed category with tensor $\otimes$ and internal hom $[-, -]$.
Suppose that
$C$ acts by adjoint monads, i.e. $- \otimes X$ is a comonad and $[X, -]$ is a monad, and each $F : ...
user30211
3
votes
0
answers
115
views
Categories in which finite powers commute with filtered colimits
If $\mathcal{C}$ is a category with finite products and filtered colimits, then we say that finite powers commute with filtered colimits in $\mathcal{C}$ if for each natural number $n$, the $n$th ...
- 203
3
votes
0
answers
134
views
Is there a construction capturing indexed families of adjunctions?
I'm sorry in advance if this question does not belong on this site. I am curious as to what is "really" going on when you have a family of functors indexed by elements in a base category, all of which ...
- 713
2
votes
0
answers
74
views
Double categories and fibrations
Is there a way in which Conduche fibrations can lead to completeness in double categories? I know that Conduche conditions on functors play a role in completeness or cocompleteness in pseudo-double ...
- 615
2
votes
0
answers
240
views
Existence and explicit descriptions for left and right Kan extensions and lifts in bicategories of spans
Given a category $\mathcal{C}$, it follows from Proposition 4.1 of Day's
Brian Day, Limit spaces and closed span categories, Lecture Notes in Mathematics, 420, 1974 (doi:10.1007/BFb0063100).
That ...
- 11.8k
2
votes
0
answers
19
views
Reference request for dinatural transformations arising from free Cartesian closed categories
Let $g_n$ be a discrete graph with $n$ nodes and $\operatorname{F}$ the free functor of the adjunction between the category of graphs and the category of Cartesian closed categories and functors, as ...
2
votes
0
answers
92
views
Group ring objects in a Cartesian closed category
Let $\mathcal{C}$ be a Cartesian closed category, with $R$ a ring object in $\mathcal{C}$ and $G$ a group object in $\mathcal{C}$.
Is there literature on the notion of the 'group ring object' $R^G$?
...
- 10.1k
1
vote
0
answers
22
views
Are the categories of definable dinatural transformations freely generated from discrete graphs?
It is well known that the dinatural transformations between multivariant functors defined in Functorial polymorphism don't form a category, because they do not compose in general, but some do. For any ...
1
vote
0
answers
32
views
Any papers on the Lambek graph-$\lambda$ calculus-adjunction and the semantics of the Hindley Milner type system?
Joachim Lambek has described an adjunction between the category of graphs and the category of positive intuitionistic calculi with iteration, see e. g. Introduction to Higher Order Categorical Logic ...
1
vote
0
answers
67
views
Does lambda polymorphism have some universal property?
To evaluate some typed lambda calculus applications, the type of the function might have to be "lifted" in order to match the type of the value it is applied to. For example, in the ...
1
vote
0
answers
66
views
Second order lambda calculus as dinatural transformations in some category of CCCs
Let $\textbf{CART}$ be a category where the objects are all Cartesian closed categories (henceforth shortened as CCC). Is there any way to define the arrows so that $\textbf{CART}$ itself becomes ...
1
vote
0
answers
97
views
Is Set complete for the free CCC/lambda calculus over a monoidal signature?
To be precise, given a monoidal signature $S$ (i.e, a set of generating objects $O$ and morphisms with source and target taken in the free monoid over $O$) , we can generate the free Cartesian closed ...
- 73
0
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0
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Are there cartesian closed monads that also preserve the closed structure of the CCC
When I look for cartesian closed monads, I only find monads where the endofunctor preserves the cartesian structure of a cartesian closed category
$$
\operatorname T\ (a \times b) = (\operatorname T\ ...