Questions tagged [cartesian-closed-categories]
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62 questions
6
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1
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Condition for a functor to induce a cartesian closed functor between categories of presheaves
We denote the category of presheaves on a small category ${\cal C}$ (set-valued functor-category) by $$\widehat{\cal C}:=[{\cal C}^{op},{\bf Set}].$$
Such a category is cartesian closed, i.e. it ...
- 567
2
votes
2
answers
172
views
Can the Category of that kind of small sets in $\sf NFU$ be Cartesian closed?
Working in Quine's $\sf NFU$, with urelements being at least as many as sets. Formally the latter is: $|Ur| \geq |Set|$.
Where $Ur$ is the set of all urelements and $Set$ is the set of all sets. We ...
- 11.3k
2
votes
2
answers
203
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Boolean algebra object structure on coproduct of terminal object
I am asking for some clarification on this old question.
The context for my question is a cartesian closed category $ C $ with a binary coproduct and a terminal object $ I $. One of the answers claims ...
- 143
8
votes
0
answers
370
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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
3
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2
answers
401
views
$R$-Module objects in cartesian closed categories
I am looking for a reference for the following statement.
Theorem. Let
$C$ be a regular, well-powered, countably complete cartesian closed category,
$R$ be a (commutative) ring object in $C$,
$R\...
user103549
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
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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
8
votes
2
answers
551
views
Mention of Bernoulli principle by Bill Lawvere
In the Author Commentary to the reprint of the paper paper Diagonal Arguments and Cartesian Closed Categories in Theory and Applications of Categories Bill Lawvere wrote:
Although the cartesian-...
- 774
13
votes
1
answer
1k
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Are condensed sets (locally) cartesian closed?
The category of condensed sets is the colimit of the toposes of $\kappa$-condensed sets over all cardinals $\kappa$, or equivalently the category of "small sheaves" on the large site of all ...
- 66.8k
3
votes
0
answers
54
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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}}
\...
4
votes
0
answers
121
views
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 ...
0
votes
0
answers
35
views
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\ ...
3
votes
1
answer
140
views
Where can I learn about Cartesian closed functors between categories of simply typed lambda calculus?
I'll try to describe the subject I am looking for literature on, or concept names that I can Google.
For each $n \geq 1$, let $\mathbf{STLC}_n$ be the category where the objects are all simply typed ...
7
votes
2
answers
644
views
Existence of nontrivial categories in which every object is atomic
An object $X$ of a cartesian closed category $\mathbf C$ is atomic if $({-})^X \colon \mathbf C \to \mathbf C$ has a right adjoint (hence is also internally tiny). Intuitively, atomic objects are &...
- 10.7k
1
vote
0
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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 ...
2
votes
0
answers
19
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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 ...
8
votes
2
answers
314
views
When is a locally presentable category (locally) cartesian-closed?
Let $\kappa$ be a regular cardinal. A category $\mathscr C$ is locally $\kappa$-presentable iff it is the free completion of a small $\kappa$-cocomplete category under $\kappa$-filtered colimits. Is ...
- 10.7k
15
votes
3
answers
1k
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Why it is convenient to be cartesian closed for a category of spaces?
In 1967 Steenrod wrote what later became a quite celebrated paper, A convenient category of topological spaces (Michigan Math. J. 14 (1967) 133–152). The paper conveys the work of many (among the most ...
- 9,111
0
votes
1
answer
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Examples of cartesian-closed model categories
One of the main settings of my research are Cartesian-closed model categories. I would like to know as many interesting and/or important examples of such categories as possible. "Interesting"...
- 6,962
8
votes
1
answer
284
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Cartesian monoidal star-autonomous categories
Disclaimer: This is a crosspost (see MathStackexchange). Apologies if cross-posting is frowned upon. However, it seems that on Stackexchange there are not many people familiar with star-autonomous ...
- 764
1
vote
0
answers
66
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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
22
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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 ...
3
votes
0
answers
162
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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
10
votes
3
answers
1k
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Is $\mathrm{Graph}$ cartesian-closed?
Let $\mathrm{Graph}$ be the category of simple, undirected graphs with graph homomorphisms. For any graphs $G, H$ we denote by $\text{Hom}(G, H)$ the set of graph homomorphisms $f:G\to H$. (Note that $...
- 48.9k
8
votes
1
answer
487
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Does the morphism of composition have some universal property?
Let $A$, $B$ and $C$ be three objects in the category Set. For simplicity, assume that their underlying sets contain a finite number of elements, a, b and c respectively. Using the usual Haskell ...
17
votes
4
answers
1k
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What is the monoidal equivalent of a locally cartesian closed category?
If a closed monoidal category is the monoidal equivalent of a Cartesian closed category, is there an analogous equivalent for locally cartesian closed categories? Is there a standard terminology or ...
- 171
15
votes
3
answers
1k
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Enriched cartesian closed categories
Let $V$ be a complete and cocomplete cartesian closed category. Feel free to assume more about $V$ if necessary; in my application $V$ is simplicial sets, so it is a presheaf topos and hence has all ...
- 66.8k
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
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
29
votes
1
answer
2k
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Is there a topological space X homeomorphic to the space of continuous functions from X to [0, 1]?
In general, we might ask when we can find interesting spaces $X, Y$ such that $X$ is homeomorphic to $[X, Y]$. By the Lawvere fixed point theorem $Y$ must have the fixed point property. Happily, $Y = [...
- 118k
1
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0
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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 ...
3
votes
1
answer
599
views
Alternative definition of power object in a category
The standard definition of a power object seems to be: objects $\mathcal{P}X, K \in \mathbf{C}$ and a monic $\in: K \hookrightarrow X \times \mathcal{P}X$ such that for every monic $r: A \...
- 798
1
vote
0
answers
97
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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
8
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2
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275
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The union of all coreflective Cartesian closed subcategories of $\mathbf{Top}$
Very often, in topology, one restricts to a coreflective Cartesian closed subcategory of $\mathbf{Top}$ in order to freely use exponential laws for mapping spaces, which imply things like "the ...
- 7,203
15
votes
2
answers
470
views
A locally presentable locally cartesian closed category that is not a quasitopos
This question asks for a locally presentable locally cartesian closed category that is not a topos. All the answers given (at least in the 1-categorical case) are quasitoposes. What is an example of ...
- 66.8k
7
votes
1
answer
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Example of a locally presentable locally cartesian closed category which is not a topos?
The only way I know to get a locally cartesian closed category which is not a topos is to start with a topos and then throw out some objects so that the category is not sufficiently cocomplete to be a ...
- 64k
9
votes
1
answer
327
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Simplicially enriched cartesian closed categories
In this question I asked whether for a complete and cocomplete cartesian closed category $V$, there can be a complete and cocomplete $V$-category $C$ (with powers and copowers) whose underlying ...
- 66.8k
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
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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
7
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1
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About cartesian closed categories of models of a cartesian theory
Let $T$ be a small category, and $\mathrm{Mod}(T)\subset\mathrm{Fun}(T,\textbf{Set})$ the category of cartesian (finite-limit preserving) copresheaves on $T$. If $T$ is a commutative algebraic theory, ...
- 4,165
5
votes
1
answer
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Is the category of hypergraphs cartesian-closed?
If $H_i = (V_i, E_i)$ for $i=1,2$ are hypergraphs then a map $f:V_1\to V_2$ is said to be a hypergraph homomorphism if $f(e_1)\in E_2$ for all $e_1\in E_1$. Hypergraphs together with hypergraph ...
- 48.9k
4
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0
answers
74
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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
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3
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When is the projective model structure cartesian? When is the internal hom invariant?
If M is a sufficiently nice model category and D is a small category then there are two natural model structures we can impose on the functor category $Fun(D,M)$ where the weak equivalences are the ...
- 27.5k
10
votes
1
answer
215
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Weak colimits in locally cartesian closed categories
The general adjoint functor theorem implies that a complete locally small category has a weak colimit of a diagram if and only if it has a colimit of this diagram. It seems that this is also true for ...
- 4,459
2
votes
1
answer
118
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When is the derived category $D(A)$ locally cartesian closed?
Let $D(A)$ be the derived $(\infty,1)$-category of some abelian category $A$. For which $A$ is $D(A)$ locally cartesian closed?
Replace $D$ with $D^b$ or similar if appropriate.
I essentially want ...
2
votes
1
answer
428
views
Seems like Reader monad composed with a strong monad produces a monad, am I right?
Take a Cartesian (or monoidal) closed category; define Reader monad for a given object $E$ as
$X \mapsto X^E$; and take a strong monad $M$ (strong means preserves product or tensor product).
Now the ...
- 562
3
votes
1
answer
238
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Is the category of convergence spaces cartesian-closed?
Convergence spaces are a generalization of topological spaces; we denote the category of convergence spaces with continuous maps with ${\bf Conv}$. Is ${\bf Conv}$ cartesian-closed?
- 48.9k
2
votes
1
answer
327
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Substructural types, the lambda calculus, and CCCs
It's well known that the simply-typed lambda calculus corresponds to a cartesian closed category. How would substructural type systems be characterized in category theory?
For example, linear type ...
- 31
3
votes
1
answer
150
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Internal characterizations of lifting properties?
This is basically a restatement of this question.
Two arrows $f,g$ are orthogonal, i.e satisfy $f\perp g$, iff the square below is a pullback
$$\require{AMScd} \begin{CD}
\mathsf C(B,X) @>{f^\...
- 10.5k
3
votes
1
answer
2k
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Cartesian closed category
Let $\bf{C}$ be a category with finite products.
(1) An object $X$ of $\bf{C}$ is called cartesian if the functor $(-)\times X$ has a right
adjoint.
(2) A morphism $s:X\rightarrow B$...
- 41