Questions tagged [cartesian-closed-categories]

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1 answer
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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 ...
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8 votes
2 answers
595 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 &...
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2 votes
0 answers
85 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' $R^G$? ...
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3 votes
0 answers
80 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 ...
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7 votes
0 answers
112 views

Cartesian closed Isbell envelope

This is maybe my third question here (MathStack included). So, I hope I am doing everything right. I have failed to find the answer to the following question in the literature. When is the Isbell ...
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1 vote
0 answers
84 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 ...
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15 votes
3 answers
998 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
367 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 ...
5 votes
2 answers
176 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 ...
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3 votes
1 answer
281 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
124 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
65 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 ...
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5 votes
0 answers
141 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 ...
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7 votes
2 answers
234 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 ...
8 votes
1 answer
267 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 ...
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14 votes
3 answers
907 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 ...
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10 votes
3 answers
942 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
183 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
194 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 ...
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2 votes
1 answer
108 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
407 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 ...
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6 votes
1 answer
342 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 ...
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7 votes
1 answer
215 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
201 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?
3 votes
1 answer
272 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 ...
28 votes
1 answer
1k 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
108 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
146 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^\...
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16 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 ...
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2 votes
1 answer
313 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 $s:X\rightarrow B$...
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2 votes
1 answer
332 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
449 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 ...
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4 votes
1 answer
429 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 ...
1 vote
1 answer
383 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 ...
9 votes
1 answer
589 views

Exponentiable objects in a category, valued in a larger, containing category

Recall that when dealing with topological spaces one usually likes dealing with a subcategory of $Top$ which is convenient, one facet of which is that it is cartesian closed. However to get to a ...
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1 vote
1 answer
265 views

Do Categorical Quotients Preserve Covering Maps?

Before asking a question, please let me write down settings. SETTINGS: Let $C$ be a category with fiber products and $B$ be a closed subcategory of $C$ (i.e. $B$ contains any isomorphism of $C$, and ...
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8 votes
2 answers
396 views

Is the category of quotient of countably based topological spaces cartesian closed ?

In "Handbook of categorical algebra Vol 2" from Francis Borceux, the author gives a proof that $Top$ is not cartesian closed. It seems to me that this proof can be adapted to show that the category $\...
14 votes
2 answers
2k views

Propositional logic with categories

I have some vague sense that certain types of categories are related to certain types of logic. I've been meaning to learn more about this, so I thought I'd ask about the simplest case, propositional ...
7 votes
2 answers
974 views

Is the category of affine schemes (over a fixed field) Cartesian closed?

This is probably a trivial question, but I don't see the answer, and I haven't found it on Wikipedia, nLab, nor MathOverflow. Let $\text{ComAlg}$ denote the category whose objects are commutative ...
0 votes
1 answer
580 views

Bicartesian closed categories and Heyting algebras

In Lambek and Scott's "Introduction to higher order categorical logic" (1988), they state that every Heyting Algebra can be understood as a bicartesian closed category. On the other hand, fixing a ...
5 votes
0 answers
605 views

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?