30 votes
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 ...
23 votes
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
19 votes

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 ...
11 votes

Why it is convenient to be cartesian closed for a category of spaces?

A classical procedure for replacing an arbitrary map with a fibration (preceding Quillen's small object argument) relies on the space of paths in $X$ being a topological space, and the evaluation map $...
11 votes

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

I think there is something intriguing and slightly mysterious going on here. First, my proposed definition would be slightly different from Dimitri Chikladze's. I agree that the natural ...
  • 52.8k
11 votes

Existence of nontrivial categories in which every object is atomic

Building on Maxime's answer -- if $C$ is cartesian closed and has an initial object $0$, and if $0$ is atomic (or even just tiny), then $C$ is the terminal category. For $1 = 0^0 = 0$ (the former ...
  • 52.8k
9 votes
Accepted

Example of a locally presentable locally cartesian closed category which is not a topos?

Every Grothendieck quasitopos is presentable and locally cartesian closed. These are categories of separated presheaves on a site. The simplest example of a site whose separated presheaves do not form ...
  • 7,984
8 votes

A locally presentable locally cartesian closed category that is not a quasitopos

Thomas Holder points out to me that this question is answered in Borceux and Pedicchio, A characterization of quasi-toposes, with an example that is reproduced in C4.2.4 of Sketches of an Elephant: ...
8 votes

A locally presentable locally cartesian closed category that is not a quasitopos

The 1-categorical version of motivic spaces is locally cartesian closed but not a quasitopos. The idea is that there exists an $\mathbb A^1$-contractible scheme that is covered by $\mathbb A^1$-rigid ...
  • 7,984
8 votes
Accepted

Enriched cartesian closed categories

I believe I managed to cook up an actual counterexample where both $C$ and $V$ are presheaf toposes. I'm going to leave my original attempt below since I still think it is instructive. Let $V$ be ...
7 votes
Accepted

The union of all coreflective Cartesian closed subcategories of $\mathbf{Top}$

The relevant paper is Cartesian closed coreflective subcategories of the category of topological spaces by Juraj Cincura. The first line of the abstract says "Answering the first part of Problem 7 in [...
  • 24.1k
6 votes
Accepted

Alternative definition of power object in a category

Let's first consider the analogous case of the exponential object $Y^X$. You might want to give an "alternative" definition that an exponential object should be an object $Y^X$ together with ...
6 votes

Why it is convenient to be cartesian closed for a category of spaces?

One thing to keep in mind is that cartesian closedness implies that the functors $X \times (-)$ preserve colimits. Of course, modulo the applicability of the adjoint functor theorem, these two ...
  • 52.8k
6 votes
Accepted

Internal characterizations of lifting properties?

The second square being a pullback is a strictly stronger condition than $f\perp g$. Mapping out of the unit object shows that it implies $f\perp g$; mapping out of other objects says that the strong ...
  • 60.9k
6 votes

Existence of nontrivial categories in which every object is atomic

This is a partial answer : if $C$ has finite coproducts, then $C$ must be posetal. In fact, I only need biproducts of the form $X\coprod X$. Indeed, because $C$ is cartesian closed, $X\times -$ ...
  • 10.2k
5 votes

Enriched cartesian closed categories

Here is an interesting construction that may be relevant, though it doesn't quite answer the question. Firstly, note that as Theo says, it suffices to assume $X=1$, i.e. to ask whether $C(1,Z^Y) \...
  • 60.9k
5 votes

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 ...
5 votes

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 <...
  • 11.9k
5 votes

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 ...
5 votes
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 $x \oplus (-) : D(A) \to D(A)$ ...
  • 52.8k
5 votes
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 $\y t : \y a \to \y b$. ...
  • 24.1k
5 votes

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
4 votes
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
3 votes

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 &...
  • 169
3 votes
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 ...
3 votes

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 ...
3 votes
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 ...
  • 4,553
2 votes

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 ...
  • 5,789
2 votes

When is a locally presentable category (locally) cartesian-closed?

I think this is the Day reflection theorem when viewing the LFP as a reflective subcategory of a presheaf topos. For the locally Cartesian closed case, I guess you would just apply some fibred ...
  • 1,233
1 vote

When is a locally presentable category (locally) cartesian-closed?

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 ...

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