All Questions
Tagged with limits-colimits topos-theory
19
questions
6
votes
1
answer
112
views
Are all unions in a topos with complete subobject lattices secretly colimits? On a logical analogue of the AB5 axiom
To clarify, here “topos” always means an elementary topos; I do not assume sheaves on a site, where I already knew my question to have a positive answer.
It is known but not so immediate from the ...
- 42.7k
1
vote
1
answer
110
views
$k:A\to 0$ implies $A\cong 0$
$
\def\aut{\text{Aut}}
\def\hom{\text{Hom}}
\def\id{\text{id}}
\def\true{\mathsf{true}}
\def\xra{\xrightarrow}
$In the SiGaL book, they prove that in an elementary topos, every $k:A\to 0$ is an iso, ...
- 3,584
1
vote
1
answer
91
views
Interpreting Diagrams About Power Object Functor
I'm reading Sheaves In Geometry And Logic and this diagram confuses me:
First, do those diagrams really live in $\mathcal{E}$ resp. $\mathcal{E}^\text{op}$ or rather in $[J^\text{op},\mathcal{E}]$ ...
- 3,584
1
vote
1
answer
490
views
Limits and colimits in the category of presheaves
We know that the category of presheaves (i.e. $Fct(C^{Op}, Set)$ ) is an elementary topos and in particular it is finitely complete and finitely cocomplete and so it has all finite limits and colimits....
- 11
2
votes
1
answer
88
views
Given a sub-presheaf $(F,\alpha)\subseteq X$, how does one create subobjects $y_{A,x}\subseteq\scr{A}$ $(\cdot,A)$ for all $(A,x)\in\int_\scr{A}$ $X$?
$\newcommand{\E}{\mathbb{E}}\newcommand{\A}{\mathscr{A}}\newcommand{\psh}{\operatorname{Psh}}\newcommand{\set}{\mathsf{Set}}\newcommand{\op}{^{\mathsf{op}}}\require{AMScd}$The TLDR is: Given that we ...
- 42.7k
2
votes
1
answer
118
views
Does the (pseudo)functor that assigns a commutative monoid $M$ to the topos of $M\text{-Sets}$ preserve limits? [closed]
Let $\mathrm{CMon}$ be the category of commutative monoids, and $\mathrm{Topos}$ be the bicategory of (Grothendieck) toposes with geometric morphisms. Consider the (pseudo)functor $\mathrm{CMon}\to\...
- 1,278
5
votes
1
answer
222
views
How to show a diagram is a pushout?
Let the $U$, $V$ be two subobjects of an object $A$ of a topos. I want to show the diagram
$$\require{AMScd}
\begin{CD}
U\cap V @>>>U \\
@VVV @VVV \\
V@>>>U\cup V
\end{CD}$$
is a ...
- 121
6
votes
1
answer
80
views
Cocontinuity of the fiber functor on topological spaces
Let $X$ be a topological space and $x \in X$. Is the fiber functor
$$\mathbf{Top}/X \to \mathbf{Top},\quad (f : Y \to X) \mapsto f^{-1}(x)$$
cocontinuous?
I already checked that coproducts are ...
- 165k
3
votes
1
answer
48
views
Why is $[\top,\bot]$ the character of the coproduct inclusion $i_1$?
$\require{AMScd}$In an elementary topos $\mathbf{C}$, why should $m=[\top,\bot]$ be the character morphism of the first coproduct inclusion $i_0:\mathbf{1}\to\mathbf{1}+\mathbf{1}$?
I have tried to ...
- 3,226
1
vote
1
answer
340
views
Show that $\Gamma$, $\Lambda$, and the associated sheaf functor are all left exact.
This is Exercise II.6 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to the first few pages of this Approach0 search, it is new to MSE.
The Details:
The ...
- 45.7k
1
vote
1
answer
247
views
A presheaf $P$ on $X$ is a sheaf iff for every covering sieve $S$ on an open set $U$ of $X$ one has $PU=\varprojlim_{V\in S}PV.$
This is Exercise II.2 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]".
The Details:
Adapted from Adámek et al.'s, "Abstract and Concrete Categories: The Joy of Cats&...
- 45.7k
1
vote
1
answer
211
views
The forgetful functor $U:\mathbf{B}G\to\mathbf{Sets}$ need not preserve infinite limits.
This is Exercise I.7 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]".
Here $\mathbf{B}G$ is the category of all continuous $G$-sets, where $G$ is a topological group.
The ...
- 45.7k
2
votes
1
answer
342
views
Prove that a "tensor product" principal $G$-bundle coincides with a "pullback" via topos morphism
From Moerdijk, Classifying spaces and classifying topoi, page 22. Consider a right $G$-set $S$ with the discrete topology. Let $E$ be a principal $G$-bundle over the topological space $X$. One can ...
- 3,120
1
vote
1
answer
367
views
Show that a certain functor preserves colimits and finite limits by verifying it on the stalks of sheaves
In Moerdijk, Classifying spaces and classifying topoi, on page 22, we have a functor from the topos $\mathcal BG$ of right $G$-sets ($G$ is a group) to the topos $Sh(X)$, the sheaves (étale spaces) ...
- 3,120
2
votes
1
answer
130
views
Are binary unions of regular monomorphisms in regular categories effective?
In the presence of pullbacks, the intersection of subobjects is given by their pullback. Whenever image factorization exist, the union of subobjects is given by the image of their coproduct. Sometimes,...
- 13.9k