All Questions
Tagged with limits-colimits representable-functor
20
questions
3
votes
1
answer
69
views
Orbit functor is not co-representable
Let ${1}\neq H\le G$ be groups. Denote by $G\textit{-}\mathsf{Set}$ the category of sets with a $G$ action, with $G$-equivariant maps as morphisms. Let $(-)/H: G\textit{-}\mathsf{Set}\to \mathsf{Set}$ ...
- 596
1
vote
1
answer
43
views
Condition that limit of a diagram $F$ exists if $\alpha: Mor(-,L) \to \lim Mor(X,F(i))$
Suppose $F : I \to \mathcal C$ is a diagram for a small category $I$ and a category $\mathcal C$. Suppose also that there is an object $L$ of $\mathcal C$ such that for all $X \in Obj(\mathcal C)$ ...
- 313
1
vote
1
answer
77
views
Example of non representable limits-preserving functor
Let $\mathcal{C}$ be a locally small category, and $F : \mathcal{C} \rightarrow \mathcal{Set}$ functor. Then :
$$F \ \text{has left adjoint} \implies F \ \text{is representable} \implies F \ \text{ ...
- 811
5
votes
3
answers
727
views
Definition of Limits in Category Theory
I was reading Kashiwara, Schapira's book Categories and Sheaves, in that limit of a projective system,
$$P:\mathcal{I}^{\text{op}}\to \textbf{Sets}$$
Is defined as follows,
$$\lim P = \text{Hom}_{\...
- 325
3
votes
1
answer
106
views
When representables are adjoints
Let $\mathbf C$ be a complete category and let $U:\mathbf C\to\mathbf{Set}$ be a representable functor. Show that $U$ preserves limits.
In general representables preserve limits, but the hypothesis ...
- 1,649
2
votes
1
answer
205
views
Limits/Colimits as representing objects and size of functor category
Let $J,C$ be categories and $T\in C$, we define the functor $\Delta_J(T):J \to C, j \to T $ to be the constant $T$-valued $J$-diagram.
Let $F:J \to C$ be a functor, my professor defined the limit of $...
- 1,220
1
vote
0
answers
55
views
Limit cones and representations (Leinster's Proposition 6.1.1)
I can see two ways to read this proposition (it's probably the language barrier problem -- I'm not an English native speaker), and I suppose the first way below is what Leinster intended to convey. Is ...
- 12k
0
votes
0
answers
112
views
Observation about powers and copowers (in category theory)
Suppose that the category $\mathbf C$ has powers and copowers of every object. Fixed a set $x$ one can define the functor $F:\mathbf C\to \mathbf C$, whose object function is $c\mapsto \coprod_x c$ (...
- 2,523
1
vote
2
answers
163
views
Dual version of "representables preserve limits"
The fact that $\mathscr A(A,-):\mathscr A\to \textbf{Set}$ preserves $D$-indexed limits translates to $$\lim\mathscr A(A,D(-))\simeq \mathscr A(A,\lim D)$$
I'm trying to prove that the dualized ...
- 12k
7
votes
1
answer
443
views
Relationship between two definitions of pro-representable functors
Edit:
I'm pretty sure that my conjecture
$$
\operatorname{Hom}(\varprojlim_i R/\mathfrak{m}^i, A) = \operatorname{colim}_i \operatorname{Hom}(R/\mathfrak{m}^i, A),
$$
is true. To prove it, just use ...
- 3,931
1
vote
0
answers
182
views
Presheaves are the Free Cocompletion - Proving that the functor preserves colimits
I am trying to understand a proof that, for any small category $\mathcal{C}$, the category $\widehat{C} = [\mathcal{C}^\mathrm{op}, \textbf{Set}]$ is the free cocompletion of $\mathcal{C}$. In ...
- 3,931
1
vote
1
answer
160
views
Extension of a functor by colimits: Cisinski - Higher Categories and Homotopical Algebra - Remark 1.1.11
First, I state premilinary results.
For a presheaf $X\colon A^{op}\to\mathsf{Set}$, it's category of elements, denoted by $\int X$, has pairs $(a,s)$ where $a \in A$ and $s \in X(a)$ as objects and $...
- 4,528
3
votes
1
answer
239
views
Every presheaf is a colimit of representables using point-wise computation of colimits
Let $C$ be a small category and let $F \in \text{Fun}(C^{op}, \text{Set})$ be a presheaf.
I'm trying to show it is a colimit of representables using the fact
that colimits in functor categories ...
- 3,770
1
vote
1
answer
448
views
Riehl's "Category Theory in Context" - Exercise 3.4.i
Let $\mathsf{I}$ be a small category, let $\mathsf{C}$ be a locally small category and let $F\colon\mathsf{I\to C}$ be a functor. Emily Riehl in her book "Category Theory in Context" defines a limit ...
- 4,528
1
vote
0
answers
38
views
How to arrive at unique factorization through the limit given naturality compatibility conditions?
If $\alpha : I \to C$ from a small category to any category $C$. Define a functor $\lim\limits_{\rightarrow} \alpha : X \mapsto \lim\limits_{\leftarrow} \text{Hom}_C(\alpha, X)$ from $C^{op}$ to $\...
