# Questions tagged [cartesian-closed-categories]

The cartesian-closed-categories tag has no usage guidance.

58
questions

4
votes

1
answer

224
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\...

3
votes

0
answers

46
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}}
\...

0
votes

0
answers

28
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\ ...

2
votes

0
answers

206
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 ...

2
votes

0
answers

17
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 ...

1
vote

0
answers

17
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 ...

4
votes

0
answers

117
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 ...

1
vote

0
answers

28
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 ...

0
votes

0
answers

22
views

### For which categories can the Cartesian closedness of a category be described as a family of covariant functors?

The paper Functorial Polymorphism describes how parametric polymorphism can be described as dinatural transformations. It involves second order lambda calculus but my question is restricted to simply ...

3
votes

1
answer

114
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 ...

1
vote

0
answers

65
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

64
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 ...

8
votes

2
answers

533
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-...

0
votes

1
answer

129
views

### 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"...

12
votes

1
answer

1k
views

### 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 ...

3
votes

0
answers

147
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 : ...

8
votes

1
answer

252
views

### 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 ...

7
votes

2
answers

626
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 &...

2
votes

0
answers

89
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' $R^G$?
...

3
votes

0
answers

100
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 ...

1
vote

0
answers

90
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 ...

15
votes

3
answers

1k
views

### 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 ...

8
votes

1
answer

443
views

### 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 ...

6
votes

2
answers

228
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 ...

3
votes

1
answer

491
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 \...

7
votes

0
answers

141
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 ...

4
votes

0
answers

68
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 ...

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 ...

7
votes

2
answers

260
views

### 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 ...

9
votes

1
answer

293
views

### 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 ...

15
votes

3
answers

1k
views

### 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 ...

10
votes

3
answers

1k
views

### 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 $...

5
votes

1
answer

200
views

### 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 ...

10
votes

1
answer

210
views

### 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 ...

2
votes

1
answer

114
views

### 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 ...

14
votes

2
answers

439
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 ...

6
votes

1
answer

389
views

### 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 ...

7
votes

1
answer

252
views

### 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, ...

3
votes

1
answer

229
views

### 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?

2
votes

1
answer

303
views

### 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 ...

29
votes

1
answer

2k
views

### 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 = [...

3
votes

0
answers

127
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 ...

3
votes

1
answer

149
views

### 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^\...

17
votes

4
answers

1k
views

### 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 ...

2
votes

1
answer

328
views

### Are lax functor categories into a cartesian closed 2-category cartesian closed?

Suppose that $C$ is a complete closed monoidal category and $I$ is any small category. Then the functor category $Fun(I,C)$ is again a closed monoidal category with the pointwise tensor product $F \...

3
votes

1
answer

1k
views

### 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 $s:X\rightarrow B$...

2
votes

1
answer

337
views

### Product in the free CCC over a CCC

When you start with a CCC $C$, take the underlying graph of $C$ via the forgetful $U : Cat \to Graph$, and then construct the free CCC over $U(C)$ via $Free : Graph \to Cat$: what's the relationship ...

9
votes

3
answers

1k
views

### 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 ...

5
votes

2
answers

483
views

### A (too?) simple notion of "closed multicategory"

Suppose I define a multicategory $M=(Ob(M),Hom_M)$ to be simply closed if
for every sequence $S=(b_1,\ldots,b_n;x)$ of $n+1$ objects in $M$, we provide an object $Exp(S)\in Ob(M)$, and
for every ...

4
votes

1
answer

442
views

### Example of a non-closed cocomplete symmetric monoidal category

Background
By a cocomplete symmetric monoidal category $C$ I mean a symmetric monoidal category whose underlying category is cocomplete and such that $- \otimes X : C \to C$ is cocontinuous for all $X ...