Questions tagged [representable-functor]
For questions about and related to Representable Functors, that are set-valued functors which can be "represented" by the hom-set of a single object from the domain category. Should be used together with the (category-theory) tag.
151
questions
0
votes
0
answers
31
views
Yoneda lemma in a functor semicategory
I've read that the Yoneda lemma does not hold in general for semicategories (i.e., 'categories' possibly lacking identity morphisms)[1]. However, I'm wondering about a related situation, where there ...
- 1
0
votes
0
answers
36
views
Proving that $f: T\to S$ factors through $V$ if and only if $f^{*}(v) = 0$ ( for some representable functor )
First note next Proposition ( Gortz's Algebraic Geometry book ):
Proposition 8.4. Let $S$ be a scheme and let $v: \mathcal{E} \to \mathcal{F}$ be a homomorphism of quasi-coherent $\mathcal{O}_S$-...
- 2,656
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
51
views
Tiny objects in a power set
I'm constructing a toy example in order to get into Cauchy completeness for categories. Suppose to have a set $X$ (=a discrete preorder) and compute its powerset $\mathcal{P}(X)$, which is its $\{0<...
1
vote
1
answer
107
views
Prove that any functor $F : \mathcal C \to \text{Sets}$ where $\mathcal C$ is small is a colimit of representable functors
Prove that for any small category $\mathcal C$ and any functor $F:\mathcal C^\text{op}\to\textbf{Set}$, $F$ can be written as a colimit of representable functors $h_x=\text{Hom}_{\mathcal C}(-,x)$.
I ...
- 313
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
0
votes
3
answers
150
views
Prove that the functor $P : \text{Set}^\text{op} → \text{Set}$ , $F(f)=f^{-1}:P(Y)\rightarrow P(X)$ for $f: X \rightarrow Y$ is representable
Let $P(A)$ denote the power set of a set $A$
For a map $f : X \mapsto Y$ of sets, I can define a map $P( f ): P(Y) → P(X)$ s.t that I obtain a functor $P : \text{Set}^\text{op} \to \text{Set}$ in ...
- 589
0
votes
1
answer
118
views
How do I show that the forgetful functor $F : \text {Grp} \rightarrow \text {Set} $ is co-representable?
Let $F : \text {Grp} \rightarrow \text {Set} $ be the forgetful functor. I am trying to show that $F$ is co-representable
My definition of co-representability reads:
A functor $F: C \rightarrow \text{...
- 589
2
votes
0
answers
31
views
Representable morphisms of algebraic spaces
Let $X,Y$ be algebraic spaces, and let $X\longrightarrow Y$ be a representable morphism.
Let $S$ be a scheme and let there be a morphism $S\longrightarrow Y$. The fibre product, $X\times_Y S$, is said ...
- 1,260
1
vote
0
answers
67
views
Representable presheaves on the slice category
$\def\sfC{\mathsf{C}}
\def\op{\mathrm{op}}
\def\set{\mathsf{Set}}
\def\psh{\operatorname{PSh}}
\def\ob{\operatorname{Ob}}
\def\hom{\operatorname{Hom}}$Let $\sfC$ be a category. A presheaf over $\sfC$ ...
0
votes
0
answers
73
views
Existence of a right adjoint functor of the inverse image via a morphism of schemes between the categories of quasi-coherent modules
Let $f:X\to Y$ a morphism of schemes. It induces a covariant functor:
$$f^*:Qcoh(Y)\to Qcoh(X)$$
Which happens to be the inverse image. Now, fixing any quasi-coherent $O_X$-module N we can define ...
2
votes
1
answer
111
views
Corepresentable functor definition.
I have been searching for the definition of a Corepresentable functor here https://ncatlab.org/nlab/show/small+object but still I did not get what exactly is its definition. I also love the definition ...
- 2,087
2
votes
0
answers
54
views
Equivalence relation functor representable?
Let $E : \mathbf{Set} \to \mathbf{Set}$ be the contravariant functor taking a set $X$ to the set $E(X)$ of distinct equivalence relations on $X$. It takes (I assume) a function $f:X \to Y$ to a ...
- 6,938
2
votes
1
answer
75
views
Understanding a proof of representability and exactness of tensor product functor
$\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\Mod}{Mod}$ $\DeclareMathOperator{\End}{End}$ $\DeclareMathOperator{\coker}{coker}$I'm going through the proof of the following theorem :
Let $...
- 1,048
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