Questions tagged [functors]

This tag is for questions relating to functors, which is a mapping from one category into another that is compatible with the category structure. Functors exist in both covariant and contravariant types.

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34 views

Functorial isomorphism with a direct sum

I need help on a exercise: Let $M$ be a $A$-module with finite presentation, and $(N_{\lambda})_{\lambda \in \Lambda}$ a family of $A$-modules. Show that exist a functorial isomorphism of $\mathbb{K}$-...
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1answer
50 views

Induced cochain equivalence by $Hom(C_{n},R)$

Given a chain complex $(C_n,\partial_n)$ of free modules over $R$ PID, we can apply the $Hom(,R)$ functor to induce a cochain complex $(C^n,\delta^n$) where $C^n := Hom(C_n,R)$ and $\delta^{n-1} : Hom(...
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1answer
62 views

$Hom(K,R) \otimes Hom(C,R) \simeq Hom(K \otimes C,R)$

Is the following statement true when $K$ is a chain complex of finitely generated free module over $R$ PID and $C$ is a chain complex over $R$ ? $$Hom(K,R) \otimes Hom(C,R) \simeq Hom(K \otimes C,R)$$ ...
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1answer
32 views

$Tor(A,B) = 0$ if $A,B$ is torsion free

I don't understand the proof given in Hatcher p.265 of $Tor(A,B) = 0$ if $A,B$ is torsion free. The proof is the following : The line I don't get is "This means [...] can be reduced to $0$ by a ...
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0answers
37 views

Why is the functor $S\mapsto$ isomorphism classes of curves of genus $g$ over $S$ not representable?

Whenever $R_{1}\hookrightarrow R_{2}$ is an injection of rings, \begin{equation*} \operatorname{Hom}(R,R_{1})\hookrightarrow\operatorname{Hom}(R,R_{2}) \end{equation*} is an injection for any ring ...
4
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0answers
43 views

Schanuel topos an equivalent condition

Let $I$ be the category of finite sets and monomorphisms. The functor $P:I\rightarrow Sets$ is a sheaf for the atomic topology on $I^{op}$ iff $P$sends every morphism of $I$ to a monomorphism and $P$ ...
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3answers
39 views

Is there a simple interpretation of a ring action like a group or monoid one?

A group action corresponds to a group of automorphisms. A monoid action corresponds to a monoid of endomorphisms. Is there a similar way to think about ring actions?
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0answers
43 views

What are the morphisms in $\mathbf{Fun}(\mathbb N,\text{Ab})$? And am I correct about this?

It is the category of functors from $\mathbb N_{\geq }$ to abelian groups. Are the morphisms natural transformations? I am learning basic homology theory but am not quite sure if I have understood it ...
2
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1answer
33 views

Subobject classifier in $Sets^Q$

Let$Q$ be the linearly ordered set of all rational numbers con-sidered as a category, while $R^+$ is the set of reals with a symbol $\infty$ adjoined. In $Sets^Q$, prove that the subobject classifier ...
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1answer
57 views

What is the right derived functor of $\lim_\leftarrow$?

Suppose we are talking about $\lim_\leftarrow$ on category of abelian groups. By definition, the right derived functor of $\lim_\leftarrow$ would be $R_1\lim_\leftarrow \{G_n\}=\text{Ker } \lim_\...
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0answers
43 views

Restriction functor from sheaves on $X$ to sheaves on basis is an equivalence of categories

This is exercise 4 of chapter 2, Mac lane Moerdijk which I stuck in.... For a basis $B$ of the topology on a space $X$, the restriction functor $r: Sh(X)\rightarrow Sh(B)$ is an equivalence of ...
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0answers
28 views

Having trouble with the definition of functors in terms of nerves

In a brief note nlab mentions functors can be thought of maps between nerves. https://ncatlab.org/nlab/show/functor#definition I don't really get this definition. I'm interested because I'm playing ...
2
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1answer
48 views

Why doesn't right exactness of a functor (say tensor product) imply exact?

Let's work in $R$-mod for the remainder of this question. I know I'm probably missing something super basic, but here goes. Let $F$ be the functor $- \otimes_R M$ for some $R$-module $M$. Let's use ...
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1answer
54 views

What does it mean that a functor preserves infinite limits?

What does it mean that a functor preservers infinite limits? Can you please give an example of a functor which preserves finite limits but not infinite ones?
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1answer
26 views

Definition of functor through diagram

I am studying commutative algebra and right now I'm learning a bit of category theory. I have encountered many times diagrams (commutative ones as well: e.g. generally to express universal mapping ...
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1answer
39 views

Kernel of $\phi \colon $Hom$_R(\prod_{i \in I}A_i, B) \to \prod_{i \in I}\text{Hom}_R(A_i, B)$ is isomorphic to Hom$_R(\prod A_i/\sum^\oplus A_i, B)$.

Let $R$ be a ring, $\{A_i\}, B$ be $R$-modules. Let $\iota_i \colon A_i \to \prod_{i \in I} A_i$ be the canonical injection. Let $\phi \colon \text{Hom}_R(\prod_{i \in I}A_i, B) \to \prod_{i \in I}\...
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1answer
30 views

The functor $U\circ C:\mathbf{Top}^{op}\to \mathbf{Set}$ is representable

Consider the functor $C:\mathbf{Top}^{op}\to \mathbf{Ring}$ that sends an object $X$ to a continuous function $X\to \mathbb R$. Consider also the composition $U\circ C$ where $U:\mathbf{Ring}\to\...
5
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1answer
64 views

pair homotopic maps induce the same homology

I'd like to prove that given $f,g : (X,A) \longrightarrow (Y,B)$ homotopic as map of pair, i.e $H(A\ \times I) \subset B$ then they induce the same homology. I already know the theorem which states ...
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1answer
53 views

Connection between universal properties and the existence of a left adjoint

I've noticed that in many cases, whenever a functor $G:\mathscr B\to\mathscr A$ has a left adjoint, there's some kind of universal property around (I don't know how to state this more precisely other ...
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1answer
64 views

Universal Property of the functor $q:\mathbf{TOP}\rightarrow \mathbf{HO(TOP)}$

Let $\mathbf{TOP}$ be the category of topological spaces and $\mathbf{HO(TOP)}$ be the category whose objects are topological spaces and morphisms are equivalence classes of continuous maps. We have a ...
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0answers
58 views

Polynomial functors, Type Theory and Homotopy

I am finding that there is a bit of a battle going on to provide a "foundation" of Type Theory, and perhaps for Mathematics, either with polynomial functors or Homotopy Type Theory. There ...
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1answer
37 views

Limits in functor categories

Let $C,C’,D$ be categories and $u:C\to C’$ be a functor. The functor $u^*:\mathbf{Hom}(C’^\circ,D)\to\mathbf{Hom}(C^\circ,D)$ that sends a functor $G$ to $G\circ u$ commutes with limits and colimits, ...
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0answers
35 views

Induced representations built on sections of an associated vector bundle. Questions on notations

Consider a group $\,G\,$, a vector space $\,{\mathbb{V}}\,$, and a space $\,{\cal{L}}^G\,$ of functions $\varphi$ on this group: $$ {\cal{L}}^G\;=\;\left\{~\varphi~\Big{|}~~~\varphi:\,~G\...
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1answer
60 views

Examples on why isomorphism of categories are “too restrictive”

While I'm reading the Algebra: Chapter 0 by Paolo Aluffi, I've encountered the following paragraph where he discuss that the isomorphism on categories is too restrictive: When are two categories ‘...
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1answer
38 views

Question about the definition of faithful functors

In TOM LEINSTER's Basic Category Theory, Page 25, it says I got very confused here. Doesn't the definition of injective says for distinct $f_1, f_2$ we have $F(f_1)\neq F(f_2)$? Why this is false? I ...
2
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1answer
53 views

Equivalence of $R$-linear additive categories are $R$-linear?

Let $R$ be a commutative ring. Let $C,D$ be $R$-linear categories (https://stacks.math.columbia.edu/tag/09MJ ) such that they are also additive categories. Let $F: C \to D$ be an equivalence of ...
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0answers
53 views

Does the assignment sending every topological space to its sheaf hom functor a functor?

Let $X$ be a topological space. Given sheaves $F,G\in\text{Sh}(X)$, the sheaf hom of $F$ and $G$ is the sheaf $\text{Hom}_X(F,G)$ of $X$ defined on objects by $$\text{Hom}_X(F,G)(U)=\text{Nat}(F|_U,G|...
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2answers
87 views

What conditions ensure that a functor has a left / right inverse.

A functor $F : C \to D$ has a left inverse $G : D \to C$ if $G \circ F : C \to C$ is naturally isomorphic to $\mathrm{id}_C$ and ditto for a right inverse. Are there nice criteria for when a functor ...
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1answer
87 views

Does this functor commute with inverse limits?

Let $F$ a contravariant functor from the category $A$ of pointed connected CW-complexes up to homotopy to the category $B$ of pointed sets, with $F$ sending coproducts of $A$ to products of $B$. Let $...
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2answers
38 views

Intuitive understanding of a non-full functor

A functor $T : C \to B$ is full if for each $c,c' \in C$ and for each $g : T c \to T c'$, there is at least a morphism $f : c \to c'$ such that $g = T f$. What does this tell me about non-full ...
2
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1answer
121 views

Functoriality of the $hom$-functor?

(Originally posted on stackoverflow, but I got no satisfying (for me) answer there.) I'm a bit lost in Section 8.8 from Category Theory for Programmers. The book reads […] the mapping that takes a ...
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1answer
116 views

Let $F$ be a functor. Is the image of an object a category? [closed]

Let $F$ be a functor between two categories $A$ and $B$. Let the morphisms of those categories be structure ismorphisms. Is the image $F(a)$ of $a\in ob(A)$ a category on its own?
3
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2answers
102 views

The forgetful functor $U: \text{Con}(\mathcal C) \to \mathcal C$ has a left adjoint.

Let $\mathcal C$ a category. We define the category of cones $\text{Con}(\mathcal C)$ in the following way: Objects: quadruples $(\mathcal Z, \underline{M}, M, m)$ consisting of a small category $\...
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0answers
35 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 ...
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0answers
46 views

Uniqueness of Categorical Equivalences

For two categories $\mathcal{C}$ and $\mathcal{D}$, and a functor $F: \mathcal{C} \to \mathcal{D}$, can there exist two distinct (there does not exist a natural isomorphism between them) fucntors $G,G'...
2
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2answers
129 views

Equivalence between two definitions of affine algebraic set.

In my algebraic geometry course, we have seen two definitions of an affine algebraic set. The first says that $X$ is an affine algebraic set if $$X = \{x \in \mathbb{A}^n(k)~|~f_i(x) = 0, \forall i \...
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1answer
46 views

Equivalence of categories preserves colimits

Is it true that if $F: \cal{C} \rightarrow \cal{D}$ is an equivalence of categories (I mean, is full, faithfull and essentially surjective over objects), then $F$ preserve colimits?
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0answers
37 views

Exact functor and chain complexes

Let $F \colon \mathcal{A} \to \mathcal{B}$ an exact functor between abelian categories, $X^\bullet$ and $Y^\bullet$ chain complexes and $f \colon X^\bullet \to Y^\bullet$ a morphism. I have to prove ...
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1answer
307 views

'Contravariance has more mathematical structure in it whereas covariance is more geometrical and easy to understand.' Is it a serious claim?

In Shastri's Basic Algebraic Topology Remark 1.8.8, the author wrotes that The difference between covariance and contravariance is simply in the fact that covariance preserves the direction of the ...
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0answers
67 views

Morphisms of locally ringed spaces and functors of points

I'm learning algebraic geometry and I'm trying to reconcile the locally ringed space and functor of points perspectives. Often when one defines a morphism $X \to Y$ of schemes thought of as locally ...
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1answer
58 views

What is a non-artificial example of a bifunctor that is contravariant in both arguments?

The Hom functor is a bifunctor that’s covariant in one argument and contravariant in the other. One can view the construction of group rings as a bifunctor that is covariant in both arguments. But ...
2
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0answers
64 views

Composition structure on ${\bf Fun}$

Let Fun denote the category of functors and natural transformations. Does composition of functors together with the Godement product of natural transformations amount to some sort of canonical ...
2
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2answers
179 views

Eilenberg-Watt's theorem reference.

I'm looking for a reference (with proof!) for the following result: Let $A$ and $B$ be unital, associative rings. (Eilenberg-Watt) Let $F: {}_{A}\mathrm{Mod} \to {}_{B}\mathrm{Mod}$ be a functor that ...
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0answers
11 views

concrete functor in accessible categories

What is a concrete functor $U$ in the context of accessible categories. Everyvere used, never defined. From where to where it leads ?
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1answer
72 views

Why aren't the homotopy group functors a generalized homology theory?

I am currently enrolled in an introductory course on algebraic topology. In a recent research seminar at my home institution, it was stated that the homotopy group functors $\pi_1, \pi_2, \pi_3, ...$ ...
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0answers
99 views

Local systems and Monodromy representation functor

Let sheaf $\mathcal{F}$ on $X$ be a local system, ie. for every $x$ in $X$, there is an an open set $U$ containing $x$ such that $\left.\mathcal{F}\right|_U$ is isomorphic to a constant sheaf $\...
5
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1answer
81 views

Exponentials in the category of graphs?

Let $\Gamma$ be the category $e\overset{s}{\underset{t}{\rightrightarrows}}v$ and $\mathbf{Graphs}=\mathbf{Sets}^{\Gamma}$ the category of (directed multi) graphs. For graphs $G$ and $H$, it's my ...
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0answers
67 views

Sheaf morphism whose cokernel is not a sheaf

Let $\mathcal{F}$ and $\mathcal{G}$ be sheafs of vector spaces. I am looking for an example of a sheaf morphism $\theta:\mathcal{F}\longrightarrow\mathcal{G}$ whose cokernel $\text{Coker}(\theta)$ is ...
2
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0answers
63 views

Natural transformation $\mathbb{A}^n \setminus \{ 0 \} \to X$.

Let $\text{Ring}$ the category of commutative ring and $\text{Set}$ the category of set. Denote by $\Omega : \text{Ring} \to \text{Set}$, the functor $R \mapsto \{ \text{Ideal of $R$} \}$, for a ring ...
2
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1answer
62 views

Problem book on functors and category theory?

I am trying to find a book I saw many years ago at a university library. Unfortunately, I remember so little of the book that I haven't been able to effectively search. What I do recall is that: the ...

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