Questions tagged [functors]
The functors tag has no usage guidance.
0
votes
0answers
20 views
A concrete example of Schur functor.
Could any one recommend a book for me that contain a concrete example of Shur Functor or give me an example of a Schur functor.
Any help will be appreciated.
Thanks!
0
votes
0answers
9 views
Functor to represent directed acyclic graphs for (co)induction
The general theory of induction and coinduction is usually presented in terms of initial algebras and finial coalgebras for certain endofunctors (monads) on the category of sets. (See for example ...
0
votes
1answer
69 views
Proving naturality after determining a collection of isomorphisms.
Let $F : C \to D, \ \ G : D \to C$ be two functors we're trying to prove are quasi-inverse to each other.
Suppose I've proved that $F \circ G(x) \simeq \text{id}_D(x) = x$ via the collection of ...
3
votes
2answers
68 views
There's an equivalence of simplicial categories $\Delta \to \tilde{\Delta}^{\text{op}}$.
Let $\Delta$ be the simplicial category, that is, the category of finite totally ordered sets and order-preserving maps. Let $\tilde{\Delta}$ be the subcategory where objects are those of $\Delta$ ...
0
votes
1answer
22 views
How to see that $S(\sigma) = \text{Hom}_{\Delta}(\sigma, [0,1])$ maps to the category $\tilde{\Delta}^{\text{op}}$.
Let $\Delta$ be the simplicial category. Let $\tilde{\Delta}$ be the subcategory of non-empty totally ordered sets as objects and order-preserving maps that also preserve the smallest and largest ...
0
votes
4answers
38 views
Counterexample: functor does not preserve monomorphisms
I need to show that functors need not preserves mono's and epi's. For epi's, I have as counterexample the forgetful functor $F : \mathbf{Ring} \to \mathbf{Set}$. We have that $f: \mathbb{Z} \...
2
votes
1answer
58 views
Not able to make sense of gluing via the functor of points in EGA
I am trying to use a well known result of Grothendieck to show that if $S$ is a scheme, and $\mathcal{B}$ is a quasi coherent sheaf of $\mathcal{O}_{S}$-algebras, then there is a relative affine ...
0
votes
0answers
163 views
EGA I (Springer), Proposition 0.4.5.4.
I do not understand one argument in the proof of Proposition 0.4.5.4. in the new version by Springer of EGA I.
When proving that the functor $F$ is representable by $(X, \xi)$, where we obtained $X$ ...
0
votes
0answers
30 views
Does every covariant functor on module category preserve inclusion?
Does every covariant functor $F : ~_R\mathcal{M} \rightarrow \mathcal{C}$ on module category $_R\mathcal{M}$ preserve inclusion? I have proceeded in the following way.
Suppose $A \subseteq B$ in $_R\...
0
votes
0answers
23 views
Order preserving mapping between lattices
I would appreciate some help in solving this problem.
Problem
Let $F:R-\mathbf{Mod}\longrightarrow S-\mathbf{Mod}$ be an additive covariant functor. Let $M$ be an $R$-Module and for each $K\leq M$ ...
3
votes
0answers
26 views
What is the Krull dimension of the Burnside ring of $\mathbb N$?
A contravariant functor $F$ from monoids to commutative rings was defined there.
Question. What is the Krull dimension of $F(\mathbb N)$?
(Here $\mathbb N$ denotes the additive monoid $(\mathbb N,+...
4
votes
1answer
103 views
From monoids to commutative rings
We shall first define a functor
$$
F:\mathsf{Mon}^{\text{op}}\to\mathsf{CRing},
$$
where $\mathsf{Mon}^{\text{op}}$ is the category opposite to the category of monoids and $\mathsf{CRing}$ is the ...
2
votes
1answer
40 views
Currying in a locally small category with coproducts
While studying for category theory course I stumbled upon the following question taken from a previous exam:
Let $\mathcal{D}$ be a locally small category with all coproducts. Show that for every ...
3
votes
2answers
41 views
Are diagrams of a quiver the same as small diagrams
I've just started to wrap my head around category theory, and came across two (from my perspective not obviously equivalent) definitions of a (small) diagram in a category $\mathcal{C}$:
Definition 1 ...
0
votes
1answer
55 views
Do subobjects in concrete categories correspond to subsets?
A concrete category is a category $C$ endowed with a faithful functor $U:C\rightarrow Set$. And if $a$ is an object in $C$, then a subobject of $a$ is an isomorphism class of monomorphisms with ...
6
votes
1answer
117 views
Is there a notion of a transversal of subobjects?
If $a$ is an object in a category $C$, then a subobject of $a$ is an isomorphism class of monomorphisms with codomain $a$. The subobjects of all the objects in $C$ partitions the class of ...
1
vote
1answer
36 views
Symmetric monoidal category which is not closed?
A monoidal category is symmetric if its tensor product is commutative up to natural isomorphism. And a symmetric monoidal category is closed if the tensor product functor has a right adjoint. We ...
3
votes
1answer
66 views
Does an adjoint of the Hom functor make a category monoidal?
In the category of modules, the tensor product functor is the left adjoint of the covariant Hom functor. Similarly in the category of sets, the Cartesian product functor is the left adjoint of the ...
3
votes
0answers
22 views
Terminology for “quasi-constant” functors
Is there a commonly agreed upon terminology for a functor $F:A\to B$ such that
all $F(X)$ are isomorphic in $B$, where $X$ is an object of $A$ ? I would guess something like "quasi-constant" or "...
1
vote
0answers
42 views
Degrees of freedom for injective endofunctor keeping objects constant?
Suppose we have a category $C$ with at least 2 objects. In fact, I have in mind the category Set, but my question applies to all categories.
Suppose we define a functor $F:C\to C$ that
Maps objects ...
1
vote
1answer
30 views
Show that a functor with a domain A × B can be regarded as a pair of families of functors
The following is an exercise 1.2.25 from T. Leistner's "Basic Category Theory" that I am trying to solve.
Let $F : \mathscr{A \times B} \to \mathscr{C}$ be a functor. Prove that for each $A \in \...
2
votes
2answers
420 views
Does every functor from Set to Set preserve products?
In general, not all functors preserve products. But my question is, is it at least true that all functors from Set to Set preserve products?
If not, does anyone know of a counterexample?
2
votes
2answers
71 views
Is a natural transformation uniquely determined by a single morphism?
Let $C$ and $D$ be categories, let $F$ and $G$ be functors from $C$ to $D$, and let $\gamma$ and $\delta$ be natural transformations from $F$ to $G$. Then my question is, if $\gamma_a=\delta_a$ for ...
4
votes
2answers
51 views
Is the identity functor naturally isomorphic to a covariant dual functor?
It is often said that vector spaces are not naturally isomorphic to dual spaces, because the dual functor is not naturally isomorphic to the identity functor. But the latter is a rather trivial ...
0
votes
1answer
39 views
Can natural transformations be viewed as functors between images of functors?
Let $C$ and $D$ be categories, and let $F,G:C\rightarrow D$ be functors. Then a natural transformation $\tau$ from $F$ to $G$ is a family of morphisms $\{\tau_x\}_{x\in C}$ where for each $x\in X$, $\...
0
votes
1answer
36 views
Equivalent definitions of a contravariant functor
Let $\mathcal C,\mathcal D$ be categories. A contravariant functor from $\mathcal C$ to $\mathcal D$ is a functor $F$ from $\mathcal C^{op}$ to $\mathcal D$ . Another definition is given on Wikipedia: ...
3
votes
0answers
89 views
Homology of the complex of exact sequences of homotopy groups
By naturality of the connecting homomorphism, relative homotopy $\pi_{\bullet}$ may be regarded as a functor from the category of pointed pairs of topological spaces to the category of exact sequences ...
4
votes
0answers
36 views
Proof verification: An arrow which is monic under a faithful functor is itself monic
Context:
To introduce some symbols and such, what I'm seeking to prove is this:
Let $F$ be a faithful functor. Suppose $F(f)$ is a monic arrow. Show $f$ is monic.
This came up as part of a class ...
2
votes
2answers
63 views
Can I define a functor F and a “ΔF” of sorts, which will uniquely determine a new functor?
Let $F: \mathcal{C} \rightarrow \mathcal{D}$ be a functor.
Natural transformation between $F$ and some other functor is defined as an assignment of a morphism in $\mathcal{D}$ to each object in $\...
2
votes
0answers
52 views
Well-powered/subobject functor?
Is there a construction which maps each object in a category to the set of its subobjects?
Concretely, I'm interested in mapping an object $M$ in the category of manifolds $\mathbf{Man^1}$ to the set ...
1
vote
0answers
27 views
Adjoint functors for Lie Algebras
Let's restrict to finite dimensional case.
Functor $(-)^{\mathrm{ab}}: \mathrm{LieAlg} \to \mathrm{AbLieAlg}$ is left adjoint of the inclusion functor $i: \mathrm{AbLieAlg} \to \mathrm{LieAlg}$. ...
6
votes
1answer
107 views
Showing that $F$ is not representable [closed]
As I'm trying to find (counter)examples of representable functors, I tried looking up some instructive examples.
One of the counterexamples I'm having trouble with, is the following: Show that the ...
7
votes
5answers
143 views
What constructions of “elementary” mathematics are actually functors?
I'm not looking for the usual simple examples of functors like the fundamental group or forgetful functors, what I'm looking for is some interesting examples of constructions from "elementary" ...
2
votes
1answer
33 views
Is additivity necessary for a left exact functor to preserve pullbacks?
I'm having a bit of difficulty with exercise 5.16 from Rotman's An Introduction to Homological Algebra (second edition).
The exercise (at least the relevant part) reads
Prove that every left exact ...
2
votes
1answer
50 views
How is it obvious that $\times : C \times C \to C$ is right adjoint to the diagonal functor?
This is from "Sheaves in Geometry & Logic".
$\times : C \times C \to C$ is the cartesian product of two objects. So assume that finite products exist in $C$ the above is a functor.
To say ...
2
votes
1answer
45 views
checking the functor $\texttt{Nil}_n$ is represented by $(\mathbb{Z}[x]/(x^n), \tau_R)$
This is the continuation of another question I did some days ago. Here.
I have been working on it and I would like to know if my try to prove it is right or not. I would appreciate a lot any feedback ...
3
votes
1answer
29 views
What does the endofunctor/monad that sends a set to the set of finite words on the set do to morphisms?
Suppose we have a monad $T:Set \rightarrow Set$ that sends a set X to the set of finite words on the set X, with the unit and multiplication being inclusion and concatenation respectively. What does ...
1
vote
0answers
32 views
Join of simplicial sets induces a functor.
Given a simplicial set $X$, denote by $(\mathrm{Set}_{\Delta})/_X$ the over category, whose objects are morphisms of simplicial sets with source $X$. I want to show that the joint $X \star Y$ of ...
1
vote
1answer
70 views
checking that there is a functor $\texttt{Nil}_n: Ring \longrightarrow Set$
I want to show that $\forall n \geq 1$, there is a covariant functor $\texttt{Nil}_n: Ring \longrightarrow Set$ that sends a ring $R$ to the set $\{x\in R | x^n = 0\}$.
I have being thinking that the ...
1
vote
2answers
102 views
$\mathcal{Nil}_n$ isomorphic to $h^A$ with $A=\mathbb{Z}[x]/(x^n)$
I would like to see that $\texttt{Nil}_n$ is isomorphic to $h^{\mathbb{Z}[x]/(x^n)}$ as categories.
$\texttt{Nil}_n: \texttt{Rings} \longrightarrow \texttt{Sets}$ is the functor that sends a ring $R$ ...
1
vote
1answer
88 views
Why is $\mathbb{G}_m$ is a representable functor?
What does it mean that multiplication $\mathbb{G}_m$ is a representable functor, with $\mathbb{G}_m = \text{Spec}(\mathbb{Z}[x,x^{-1}])$ ?
When I looked at the stacks project page on $\mathbb{G}_m$ ...
0
votes
0answers
38 views
Relation between functors under an equivalence
Let $A$, $B$, and $C$ be three categories and $G\colon A\to C$ and $H\colon B\to C$ two functors. Assume that $F\colon A\to B$ and $E\colon B\to A$ form an equivalence between $A$ and $B$.
Suppose ...
2
votes
2answers
77 views
Product in the category of functors.
Let $A$ be a category and $C= Fun(A, Set)$ (i.e. the objects are functors and morphisms are natural transformations between them). I want to know if this category has a product. For given $X \in A$ ...
1
vote
0answers
41 views
$[\mathsf{I},\mathsf{C}]$ and $[\mathsf{J},\mathsf{C}]$ are equivalent if $\mathsf{I}$ and $\mathsf{J}$ are. [duplicate]
Let $\mathsf{I}$ and $\mathsf{J}$ be equivalent categories. Let $\mathsf{C}$ be another category. I need to prove that the categories of functors $[\mathsf{I},\mathsf{C}]$ and $[\mathsf{J},\mathsf{C}]$...
4
votes
0answers
78 views
Model category of all model categories
Is there some natural model category structure on the category of all small model categories and some specified functors among them (i.e. not necessarily all functors), say Quillen adjunctions? What ...
0
votes
0answers
92 views
What is an algebra over a monad?
The category of algebras over a monad (also: “modules over a monad”) is traditionally called its Eilenberg–Moore category (EM)
In that context
What exactly does the word "Algebra" mean?
What ...
0
votes
0answers
41 views
Sandwich natural transformation between two functors
In the Kleisli adjunction we have:
$G\varepsilon F = \mu$ where $\varepsilon$ is a natural transformation called the counit.
How exactly is $G\varepsilon F$ defined? I understand $G\varepsilon$ and $...
2
votes
1answer
61 views
If the Yoneda lemma essentially says that $\text{Hom}(\text{Hom}(\cdot, x), A) \simeq A(x)$, then what about higher iterates of $\text{Hom}$?
Assume that the $C$ in $\text{Hom}_C(x,y)$ can always be inferred from $x,y$ so that we can change our notation to $\text{H}(x,y) := \text{Hom}_C(x,y)$
Then the Yoneda lemma "looks at a single step ...
1
vote
0answers
30 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 $\...
1
vote
1answer
50 views
Definition of symmetric power of a linear represenation
I'm reading Kowalski's Representation theory, and there's a part about the symmetric and antisymmetric powers of a representation, and I'd like to ask a question about those.
So there's a proposition ...
