All Questions
Tagged with topos-theory limits-and-colimits
9 questions
6
votes
0
answers
153
views
Covering categories with posets
Let $C$ be a small (1-)category.
There is always a poset $D$ and a functor $p : D \to C$ such that:
$p$ is surjective on objects, i.e. for every $c$ in $C$ there is a $d$ in $D$ such that $p (d) = c$,...
- 15.9k
2
votes
1
answer
74
views
Conditions for partially applied induced product functor to preserve colimits
Let $\mathcal{C}$ have products, $A, B \in \mathcal{C}_0$, ${\times}\colon \mathcal{C}^2\to \mathcal{C}$ be the functor that sends objects to their product.
Then the induced product ${\boxtimes}\colon ...
- 25
2
votes
1
answer
168
views
Inclusion of $1$-presheaves into $\infty$-presheaves preserves pushouts?
Let $\mathcal{R}$ be a $1$-category. Assume that one has a pushout of representable $1$-presheaves $\mathrm{y} A \cup_{\mathrm{y} B} \mathrm{y} C$ in $\mathsf{PSh}(\mathcal{R})$. Under which ...
- 355
9
votes
1
answer
389
views
Is the coproduct $N=1+N$ universal?
Let $\mathcal{C}$ be a category with finite limits and a (parameterized) natural numbers object $(N,0,s)$. Let $1$ denote the terminal object of the category. It's easy to show that the following is a ...
- 285
6
votes
1
answer
243
views
Stability properties of essential geometric morphisms
Notation.
$\mathsf{Topoi}$ is the bicategory of topoi, geometric morphisms and natural transformations between left adjoints.
$\mathsf{Topoi}_{\text{ess}}$ is the bicategory of topoi, essential ...
- 9,111
11
votes
1
answer
352
views
Weak descent and effective equivalence relations
I want to prove that weak descent of a $1$-category implies the classical Giraud axioms.
More precisely, let $\mathsf{C}$ be a cocomplete, finitely complete $1$-category. We say that $\mathsf{C}$ ...
8
votes
0
answers
191
views
Yoneda embedding and Horn sentences
The following is taken from Borceux and Bourn's Mal'cev, Protomodular, Homological, and Semi-Abelian Categories.
Metatheorem 0.1.3. Let $\mathcal P$ be a statement of the form $\varphi\implies \psi$, ...
- 10.5k
6
votes
2
answers
926
views
Is the category of toposes cocomplete ?
Hello.
[Edits between brackets.]
Does the [1-]category of [elementary] toposes [with logical morphisms] admit any [1-]colimits ?
[By colimit I mean initial object in the category of outgoing ...
- 61
7
votes
2
answers
2k
views
Is there a category in which finite limits and directed colimits *don't* commute
Andrew Critch asks at the 20-questions seminar:
In Set and AbGrp (the categories of sets and abelian groups, respectively), finite limits commute with directed colimits. As an example, if you're ...
- 1,059