33
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 ...
- 546
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 ...
- 63.9k
13
votes
Accepted
Are condensed sets (locally) cartesian closed?
Condensed sets are indeed locally cartesian closed. On the other hand, for no cardinal $\kappa$ (no matter how inaccessible) the functor from $\kappa$-condensed sets to condensed sets preserves all ...
- 21.3k
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 ...
- 63.9k
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 $...
- 10.9k
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 ...
- 63.9k
11
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 ...
- 8,972
11
votes
Accepted
Can the Category of that kind of small sets in $\sf NFU$ be Cartesian closed?
No. The same obstruction to cartesian closedness exists as in the case of the category of all sets and functions in NFU. In brief, the problem is that the set of functions from a one element set to ...
- 496
9
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 ...
- 8,972
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
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 ...
- 7,656
8
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 [...
- 30.3k
6
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 ...
- 6,406
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, $X\times -$ ...
- 15.8k
6
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
<...
- 13.6k
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 ...
- 63.9k
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 ...
- 118k
6
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 ...
- 7,193
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 $\y t : \y a \to \y b$. ...
- 25.2k
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 ...
- 25.2k
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) \...
- 66.7k
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
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 $x \oplus (-) : D(A) \to D(A)$ ...
- 63.9k
5
votes
Accepted
Mention of Bernoulli principle by Bill Lawvere
Since the topic of Lawvere paper is differential geometry, the Bernoulli is likely Jacob Bernoulli, or his brother Johann, and refers to their calculus of variations and the principle of virtual work. ...
- 188k
5
votes
Mention of Bernoulli principle by Bill Lawvere
Extended remarks in support of Carlos' answer:
From "The Legendre-Fenchel transform from a category theoretic perspective" by Simon Willerton (pg. 2),
... Lawvere [10] showed using enriched ...
- 10.5k
5
votes
Can the Category of that kind of small sets in $\sf NFU$ be Cartesian closed?
Not an answer, but it is legitimately everything that the literature knows on the topic. Check out:
Category Theory with Stratified Set Theory,
by Forster, Lewicki, and Vidrine.
- 9,111
4
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 &...
- 179
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 ...
- 66.7k
4
votes
$R$-Module objects in cartesian closed categories
References for all the properties of $RMod(C)$ that you ask for can be found in Borceux's excellent book Handbook of Categorical Algebra 2: Categories and Structures, which is very worth having a copy ...
- 30.3k
4
votes
Accepted
Boolean algebra object structure on coproduct of terminal object
I am not entirely sure what a boolean algebra object
There are many ways to make this precise, but an intuitive one is: you can define an internal lattice in $\cal C$ as an object $L$ equipped with $\...
- 13.6k
Only top scored, non community-wiki answers of a minimum length are eligible