Questions tagged [cartesian-closed-categories]

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$R$-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\...
B.P.'s user avatar
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3 votes
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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}} \...
Johan Thiborg-Ericson's user avatar
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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\ ...
Johan Thiborg-Ericson's user avatar
2 votes
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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 ...
crystalline cohomology's user avatar
2 votes
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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 ...
Johan Thiborg-Ericson's user avatar
1 vote
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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 ...
Johan Thiborg-Ericson's user avatar
4 votes
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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 ...
Johan Thiborg-Ericson's user avatar
1 vote
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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 ...
Johan Thiborg-Ericson's user avatar
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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 ...
Johan Thiborg-Ericson's user avatar
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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 ...
Johan Thiborg-Ericson's user avatar
1 vote
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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 ...
Johan Thiborg-Ericson's user avatar
1 vote
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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 ...
Johan Thiborg-Ericson's user avatar
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-...
Evgeny Kuznetsov's user avatar
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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"...
Arshak Aivazian's user avatar
12 votes
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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 ...
Mike Shulman's user avatar
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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 : ...
Ronald J. Zallman's user avatar
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 ...
Max Demirdilek's user avatar
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 &...
varkor's user avatar
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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' $R^G$? ...
Alec Rhea's user avatar
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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 ...
User7819's user avatar
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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 ...
FeralX's user avatar
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15 votes
3 answers
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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 ...
Ivan Di Liberti's user avatar
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 ...
Johan Thiborg-Ericson's user avatar
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 ...
varkor's user avatar
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3 votes
1 answer
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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 \...
Jordan Mitchell Barrett's user avatar
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 ...
ClosedCoherence's user avatar
4 votes
0 answers
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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 ...
Rupert's user avatar
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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 ...
Cat_W's user avatar
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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 ...
Jeremy Brazas's user avatar
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 ...
Mike Shulman's user avatar
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15 votes
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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 ...
Mike Shulman's user avatar
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10 votes
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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 $...
Dominic van der Zypen's user avatar
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 ...
Dominic van der Zypen's user avatar
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 ...
Valery Isaev's user avatar
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2 votes
1 answer
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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 ...
FlatulentCategoryTheorist's user avatar
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 ...
Mike Shulman's user avatar
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6 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 ...
Tim Campion's user avatar
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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, ...
Buschi Sergio's user avatar
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?
Dominic van der Zypen's user avatar
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 ...
C. Bednarz's user avatar
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 = [...
Qiaochu Yuan's user avatar
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 ...
Mathemologist's user avatar
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^\...
Arrow's user avatar
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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 ...
pnips's user avatar
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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 \...
Chris Schommer-Pries's user avatar
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 $s:X\rightarrow B$...
Hina's user avatar
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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 ...
user36899's user avatar
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 ...
Chris Schommer-Pries's user avatar
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 ...
David Spivak's user avatar
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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 ...
Martin Brandenburg's user avatar