All Questions
Tagged with axiom-of-choice category-theory
49
questions
4
votes
2
answers
84
views
Internally Inhabited Object Which Is Not Externally Inhabited
What is the simplest example of a topos with an object $X$ which is internally ($X\to \mathbf{1}$ is epi) but not externally (no $\mathbf{1}\to X$) inhabited?
I think that for all categories $C$, in ...
- 3,584
6
votes
1
answer
166
views
Cohomology and axiom of choice
I was reading this paper talking about how cohomology theories are related to the axiom of choice. I would like to have a better understanding of one of the first topics discussed.
At the bottom of ...
- 765
4
votes
1
answer
68
views
Uniform Choice Functions and Naturality
Some choice functions can be specified explicitly, while in other cases no definite choice function is known. An example of the former is a choice function for non-empty subsets of natural numbers, ...
- 91
6
votes
2
answers
181
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Defining subcategories and axiom of choice
Questions: 1. When do objects of a category form a set?
Is there a choice function when I have a set of categories (as opposed to a set of sets)? Is there an axioma schema of separation for defining (...
- 1,274
2
votes
2
answers
234
views
Where is choice used in "Every category monadic over Set is regular"
From Johnstone's notes: If $\mathbb{T}$ is a monad on $\mathcal{C}$ whose functor part $T$ preserves covers then the functoriality of image factorisation induces a unique algebra structure on the ...
- 682
3
votes
0
answers
62
views
Could a small category have non-isomorphic skeletons without the axiom of choice?
Without the axiom of choice, could there be a small category with two non-isomorphic skeletons?
Let $C$ be a small category and $S$ and $S'$ be two skeletons of $C$ (i.e., full subcategories ...
- 10.1k
7
votes
0
answers
305
views
(Proper) classes and the General Adjoint Functor Theorem
Original problem
A few days ago I asked the following question on Stackexchange. Essentially, I was asking how to verify via one of the adjoint functor theorems that the forgetful functor from $\...
- 1,769
6
votes
2
answers
289
views
Characterization of basis in terms of universal property: axiom of choice
I wonder if the proof of the following statement requires the axiom of choice:
(Characterization of basis in terms of universal property) Let $V$ be a vector space, and let $S$ be a non-empty subset ...
- 361
8
votes
1
answer
259
views
Which presheaf toposes satisfy the axiom of choice?
Which toposes of presheaves $\mathbf{Set}^{C^\mathrm{op}}$ satisfy the axiom of choice (every epimorphism splits)?
Can one formulate a condition on $C$ that yields a necessary or sufficient condition ...
- 401
10
votes
1
answer
525
views
Epic morphisms in the category of vector spaces. Is AC needed?
In $\mathsf{FinVect}_k$, the category of finite-dimensional $k$-vector spaces, all epis are surjective, by the argument given in this answer. I know how to generalize this argument to $\mathsf{Vect}_k$...
5
votes
1
answer
96
views
Isomorphism of products in a category: Does it involve the axiom of choice?
Let $(X_i), (Y_i)$ be two families of objects indexed by the same index set $I$ in a category with products. It seems obvious that if $X_i, Y_i$ are isomorphic for all $i$, then also the products $\...
- 4,624
1
vote
1
answer
138
views
A proof about skeletons without axiom of choice
I'm trying to prove this statement and I found two different proofs: the first seems to work without explicit use of AC while the second uses it precisely where I expected it.
(T) Let $i$ be a ...
- 2,512
2
votes
1
answer
125
views
Axiom of choice in sheafification?
Let $(\mathbb{C}, J)$ be a small site, and let $F : \mathbb{C}^{\mathsf{op}} \to \mathsf{Set}$ be a separated presheaf with respect to $(\mathbb{C}, J)$. In the usual proof that $F^+ : \mathbb{C}^\...
- 1,621
1
vote
2
answers
175
views
The axiom of choice for a category
I am currently studying the counterpart of axiom of choice in ETCS which is the axiom that every surjective function has a right inverse.
In category of sets the surjective functions are epimorphsims ...
- 183
0
votes
0
answers
47
views
Every surjective function has a right inverse [duplicate]
I am studying the counterpart of the axiom of choice in ETCS which is that "Every surjective fucntion has a right inverse". I am trying to see why it implies the axiom of choice and this is ...
- 183