# Questions tagged [cartesian-closed-categories]

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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 ...
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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 ...
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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 ...
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### Cartesian monoidal category where only initial objects don't receive maps from terminal objects

Let $C$ be a cartesian monoidal category. Assume any object in $C$ that doesn't receive a map from a terminal object is an initial object. Must $C$ have exponential objects then?
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### 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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### 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 ...
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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 ...
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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 ...
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### 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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### 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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### 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, ...
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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?
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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 ...
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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 $s:X\rightarrow B$...
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### 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 ...
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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 ...
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### 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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### 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 ...