# Tag Info

Accepted

### Is there a topological space X homeomorphic to the space of continuous functions from X to [0, 1]?

There cannot be such a homeomorphism, because Lawvere's fixed point theorem would give us something too constructive: a continuous map $[0,1]^{[0,1]} \to [0,1]$ that assigns a fixed point to each ...
Accepted

### Is $\mathrm{Graph}$ cartesian-closed?

There are many categories of graphs, so perhaps it's best to take a synoptic view (though far from exhaustive). The table below surveys several categories of directed multigraphs (DM), directed graphs ...
• 52.8k

### What is the monoidal equivalent of a locally cartesian closed category?

In a certain sense a monoidal version of a slice category is a category of comodules over a cocommutative comonoid object. If $C$ is a cocommutative comonoid object in a monoidal category, then the ...
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• 60.9k

### Is $\mathrm{Graph}$ cartesian-closed?

For the sake of completeness, the interested reader of this question can find more details in the following paper, which addresses in Section 4 the topic in question: "Categories of graphs are ...

### Does the morphism of composition have some universal property?

First of all, as David Roberts pointed out in a comment above, the composition map c :: hom a b -> hom b c -> hom a c c f g = g . f and the evaluation map <...
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### The union of all coreflective Cartesian closed subcategories of $\mathbf{Top}$

I contacted Juraj Činčura and he kindly wrote back and directed me to the following observation that is a consequence of results in the paper that David White noted in his answer. Cartesian closed ...
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Accepted

### When is the derived category $D(A)$ locally cartesian closed?

This will almost never happen. Since $D(A)$ has a terminal object 0, if it's locally cartesian closed, then it's also cartesian clsoed. To be cartesian closed means that $x \oplus (-) : D(A) \to D(A)$ ...
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Accepted

### Simplicially enriched cartesian closed categories

$\newcommand{\y}{\mathbf{y}}$Take $C = \mathcal{P}(a \stackrel{t}{\to} b) = \mathrm{Set}^{\cdot \leftarrow \cdot}$, so $C$ is freely generated under colimits by a morphism $\y t : \y a \to \y b$. ...
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### Enriched cartesian closed categories

$\require{AMScd}$As Theo noted, the question amounts to whether the action $A \odot Y$ is given by the formula $(A \odot 1) \times Y$. The algebraic analogue would be to ask whether a module $C$ over ...
• 24.1k
Accepted

### Is the category of hypergraphs cartesian-closed?

I think the answer is yes, although the internal-hom may be a little surprising. First let's describe the cartesian product. I believe the category $\rm HyGph$ is a topological concrete category ...
• 60.9k

### What is the monoidal equivalent of a locally cartesian closed category?

I think there is another, more logically related way around the problem of defining a "locally monoidal closed category", if your intuition is coming from Type Theory/Logic. Have a look at &...
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Accepted

### Is the category of convergence spaces cartesian-closed?

There seemed to be several slightly different notions of convergence spaces that were considered extensively in the 70ies. So I apologize if the following references do not actually answer your ...
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### Is $\mathrm{Graph}$ cartesian-closed?

You may be interested in the following thesis "Functorial Approach to Graph and Hypergraph Theory" by M. Schmidt looks at when categories of uniform hypergraphs have exponentials. In ...
Accepted

### Cartesian monoidal star-autonomous categories

[I'm going to assume $S' \cong S$, which holds in every symmetric monoidal $*$-autonomous category. (See e.g. Lemma 5.6 of this paper.) This applies here since cartesianness implies symmetry. Part of ...
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### About cartesian closed categories of models of a cartesian theory

I'll answer the question for algebraic theories, or Lawvere theories, which is the context in which "commutative theories" are typically discussed. This question is then the topic of ...
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I might try to improve my answer later this day. For the moment, a sufficient condition was given by Pedicchio and Borcerux in A characterization of quasi-toposes, JoA 139 (1991). Prop 4.1. If $C$ is ...