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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Is there a functor between $\mathbf{Set}$ and $\mathbf{Set}/C$?

In my PhD thesis, I'm interpreting some of my code in Category Theory. I ended up modeling some stuff as a slice category $\mathbf{Set}/C$, and ended up with the question whether I could define a ...
Davi Barreira's user avatar
2 votes
1 answer
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Is "up to natural isomorphism" crucial?

Let $\mathcal{C}$ be a category. By diagram I mean a covariant functor $F\colon\mathcal{J}\to\mathcal{C}$ for some category $\mathcal{J}$. In this source it is said that a commutative diagram is a ...
Andrew Paul's user avatar
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1 answer
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Are (commutative) squares in some sense universal among edge-symmetric double categories?

Definitions: Given a category $\mathcal{C}$, and a double category $\mathbb{D}$, i.e. a category internal to $Cat$ with an “objects category” $\mathcal{D}_0$ and a “morphisms category” $\mathcal{D}_1$,...
hasManyStupidQuestions's user avatar
2 votes
1 answer
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Is there an idempotent e in a ring $R$ such that the functor $T_e$, induced by $e$, is not full?

Let $R$ be a ring and let $e$ be a non-zero idempotent in $R$. For each $R$-module $M$, define $T_e: M \rightarrow eM$, where $eM$ is the left $eRe$-module. For each pair of $R$-modules $M, N$ and ...
Liang Chen's user avatar
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1 answer
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Among morphisms of morphisms, what makes commutative squares special?

Given two (1-)categories $\mathcal{C}, \mathcal{D}$, and given the 0-category (class) of funtors $\mathcal{C} \to \mathcal{D}$, denoted $Func(\mathcal{C} \to \mathcal{D})$, let's say we want to make ...
hasManyStupidQuestions's user avatar
3 votes
1 answer
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Subcategory of left-right projective bimodules

Let $\mathbb{K}$ be a fixed field. Take a finite-dimensional $\mathbb{K}$-algebra $A$ and look at the category $\text{mod}(A^e)$ of finite-dimensional $A$-bimodules. It has a full subcategory $\text{...
Margaret's user avatar
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1 answer
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Diagrams as visualizations of functors

Can this quiver $A\to B\to C$ be a diagram in some category? (Let's hope that now my question is "specific" so the community bot won't close it) This is the background for me asking this: my ...
rutruttt's user avatar
3 votes
2 answers
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Quivers as the underlying structure of categories

The structure of a small category is a quiver - every vertex of the quiver represents an object with its identity morphism, and the edges of the quiver represent the non trivial morphisms (I know you ...
rutruttt's user avatar
2 votes
2 answers
76 views

(Co)Products are bifunctors, but are general (co)limits also functors?

In a category with all products or coproducts, the (co)product operation can be understood as a bifunctor. More generally let $\mathcal{C}$ be a category with all limits of shape $D$, where for ...
Zoltan Fleishman's user avatar
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When We say a diagram in a category is a functor, then how do we treat the category? Do we take it as 2-catgegory?

In a Category $\textbf{C}$, we say that an $\cal{I}$-diagram in a category is a functor $F: \cal{I} \to \textbf{C}$. In fact, Any object A can be viewed as a diagram by defining a functor $F: \cal{I} \...
Rabia Sagheer's user avatar
1 vote
1 answer
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Prove that $F$ is left adjoint to $G$ iff $G$ is right adjoint to $F$.

I started the definition of adjoint functor using universal morphisms. A functor $F:\mathcal C\to \mathcal D$ is a left adjoint functor if given any object $Y\in D$, there is an object $G(Y)$ in $\...
William's user avatar
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Does an exact sequence of holonomic D-modules imply an exact sequence in the solution space?

Here is a probably quite basic question on holonomic D-modules, but I am only a physicist so please bear with me. If I have the weyl algebra $D$ in $n$ variables, and an exact sequence of holonomic $D$...
A.H's user avatar
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GL$_n$ schemes representation

Let $n \geq 0$ be a natural number. Consider the schemes $$ \mathrm{GL}_n:=\operatorname{Spec}\left(\mathbb{Z}\left[X_{i, j} \mid 1 \leq i, j \leq n\right]_f\right),\quad \mu_m :=\operatorname{Spec}\...
Mario's user avatar
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ETCS direct image functor

In Lawvere's paper on Elementary Theory of Category of Sets (https://artscimedia.case.edu/wp-content/uploads/2013/07/14182623/Lawvere-ETCS.pdf), at page 27, he proves Lemma 3. by constructing a ...
hahaha123's user avatar
2 votes
1 answer
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Module structure on $R$-algebra $S$ by restriction of scalars and splitting of the algebra map

Let $R, S$ be commutative Noetherian rings. Let $f: R \to S$ be a ring homomorphism. Consider the $R$-module $f_* S$ whose underlying abelian group is $S$ itself but the $R$-module structure is given ...
uno's user avatar
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Why are the isomorphisms induced by the category with finite products natural?

A follow-up question to this. Source: Categories for the Working Mathematician, second edition by Saunders Mac Lane. Proposition: If a category $C$ has a terminal object $t$ and a product diagram $a\...
Dick Grayson's user avatar
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What are the natural transformations induced by a category with finite products?

Source: Categories for the Working Mathematician, second edition by Saunders Mac Lane. Proposition: If a category $C$ has a terminal object $t$ and a product diagram $a\leftarrow a\times b\rightarrow ...
Dick Grayson's user avatar
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Composition of two right exact functors is also right exact [duplicate]

I have a question regarding composition of exact functors in $Abelian$ categories. lets say I have the two right exact functors: $F: \mathscr C_1 \to \mathscr C_2$, $G: \mathscr C_2 \to \mathscr C_3$....
yoyok3's user avatar
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Question about automorphisms of simplicial category $\mathbf{\Delta}$

Let $\rho: \mathbf{\Delta} \to \mathbf{\Delta}$ be the functor defined as the identity on objects and by the formula $$ \rho(f)(i)=n-f(m-i) $$ for any map $f:[m] \to [n]$, with $0 \leq i \leq m$. Then ...
Siyuan Yin's user avatar
-1 votes
1 answer
72 views

Can the functor to empty sets be defined?

Can these functors from C to Sets be defined? Functor $F$ from one object $A$ and its identity arrow to a empty set $F(A)$ = (empty set) $F(id_A)$ = (function from empty set to empty set) Functor G ...
qS2's user avatar
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2 answers
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Does the category of Boolean algebras imbed in $\operatorname{Set}$

While trying to come up with some examples of functors, I realised that any function $f:X\to Y$ induces a function $P_f: \mathbb{P}(X) \to \mathbb{P}(Y)$ in a natural way, simply define $f_p(A) = f(A) ...
Carlyle's user avatar
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Induced inverse limit sequence

This proposition states that given the functor $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ and the short exact sequence $0 \to A_\bullet \to B_\bullet \to C_\...
June in Juneau's user avatar
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1 answer
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$\mathcal{F}$ left-exact functor of abelian categories, then $\mathcal{F}(\ker (f))\cong\ker (\mathcal{F}(f))$

Let $\mathcal{C},\mathcal{D}$ be two abelian categories. Fix a morphism $f: M\rightarrow N$ in $\mathcal{C}$ and write $(\ker(f),\iota)$ for its kernel. Let $\mathcal{F}:\mathcal{C}\longrightarrow \...
kubo's user avatar
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2 votes
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If $\mathcal C$ is a small category and $\mathcal D$ is a category then the class of functors from $\mathcal C$ to $\mathcal D$ is a set

It's stated in a lecture note that if $\mathcal C$ is a small category and $\mathcal D$ is a category, then the class of functors from $\mathcal C$ to $\mathcal D$ is a set. I asked a chatbot and it ...
Squirrel-Power's user avatar
1 vote
2 answers
76 views

Endomorphisms of an equivalence of categories

For $F:C^{op}\rightarrow C$ an isomorphism of categories, it is easy to see that $End(F)\cong End(\operatorname{id}_C)$ and $Aut(F)\cong Aut(\operatorname{id}_C)$ as sets, where we consider the hom-...
Margaret's user avatar
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$Ext^n(A,B)\cong H^n(Hom(A,I^\bullet)) $, with $I^\bullet$ an injective resolution of B

I was reading this post Weibel 2.5.1 Equivalent statements of injective $R$-module. but he answer uses a fact I am not familiar with: Fact 1 If $B\to I^0\to I^1\to \cdots \to I^n\to\cdots$ is any ...
darkside's user avatar
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1 vote
1 answer
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The free 2-category on a 1-category with respect to pseudo-functors into 2-categories

Let $\mathbb{C}$ be an ordinary 1-category. I'm interested in the following potential construction. Is there a 2-category $\widetilde{\mathbb{C}}$ (equipped with a pseudo-functor $\eta : \mathbb{C} \...
User7819's user avatar
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1 vote
1 answer
35 views

The R-module $P$ is projective (the functor Hom$_R(P, −)$ : R-Mod $\to$ Ab is exact) iff the lifting property holds

A module $P$ is projective if and only if for every diagram there exists a homomorphism $\tilde h : P \to N$ that lifts $h$, i.e., such that $g \circ \tilde h = h$ I know many sources give this ...
darkside's user avatar
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1 answer
31 views

Functor $−\otimes_R L$ is right exact

Consider (*) A sequence of $R$-modules $M′ \to M \to M′′ \to 0$ is exact if and only if for all $R-$ modules $N$ the induced sequence $0 \to Hom_R(M′′, N) \to Hom_R(M, N) \to Hom_R(M′, N)$ is exact ...
some_math_guy's user avatar
1 vote
0 answers
47 views

$F : C \to D$ is an equivalence of categories with quasi-inverse $G: D \to C$. Does the pair $(F,G)$ necessarily form an adjunction?

Suppose $F : C \to D$ is an equivalence of categories with quasi-inverse $G: D \to C$. Does the pair $(F,G)$ necessarily form an adjunction? How would I prove this? I think I can use the unit-counit ...
some_math_guy's user avatar
1 vote
1 answer
53 views

Proving the existance and uniqueness of the unit of an adjunction

Let $L: C \to D$ be left adjoint to the functor $R: D \to C.$ For an object $X$ of $C$, we have an isomorphism $Hom_D(L(X),L(X)) \xrightarrow{\sim} Hom_C(X,(R\circ L)X)$. Show that there exists a ...
some_math_guy's user avatar
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0 answers
55 views

Intersection of vector spaces and inclusion-exclusion

Let $V$ be a finite dimensional vector space. Let $U,W$ be subspaces of $V$. Now $$ \text{dim}(U \cap W) = \text{dim}(U) + \text{dim}(W) - \text{dim}(U + W). $$ What is the the category theory ...
étale-cohomology's user avatar
3 votes
0 answers
109 views

Covariant power object functor?

How would one go about constructing a covariant power object functor (like powerset functor in category Set). Obviously, on objects, the mapping would equal $$ A \mapsto \Omega^A. $$ But what about ...
tses's user avatar
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1 vote
1 answer
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$F : I → C$ is a functor from a filtered category with any colimit. $ι : J → I$ is the embedding with $J$ cofinal. Then colim $F \cong$ colim($F ◦ ι$)

Let $I$ be a filtered category. We say a full subcategory $J$ is cofinal if for every object $A \in I$, there is an object $B \in J$ so that $\text{Hom}(A, B) \neq \emptyset$. Let $F : I \to \mathcal ...
love and light's user avatar
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1 answer
45 views

Two isomorphism about restriction function and idempotent embedding functor

I am reading the book “elements of the representation theory of associative algebras” and I have a problem about the proof of the theorem 6.8 in chapter 1. Let $A$ be a finite dimensional $k$-algebra, ...
fusheng's user avatar
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2 votes
0 answers
27 views

Convergence of analytic functors?

Let $\mathcal{C}$ be a symmetric monoidal cocomplete category. Let $F:\mathbb{P}^{op}\rightarrow \mathcal{C}$ be a functor, where $\mathbb{P}$ denotes the permutation category. Such a functor is ...
Margaret's user avatar
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0 answers
43 views

Every object of category $\mathcal{C}$ is compact in $Ind(\mathcal{C})$

Let $\mathrm{Ind}(\mathcal{C})$ be the Ind-completion. We can define it in two different (but similar) ways: as filtered colimits of representable presheaves and as the category of diagrams over ...
Kubrick's user avatar
  • 334
1 vote
1 answer
42 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 $X \in Obj(\mathcal C)$ ...
love and light's user avatar
3 votes
1 answer
118 views

Homotopy invariance of functors

I started following a course on Algebraic Topology without having any background in Category Theory, hence I am finding some difficulties in the exercises. Let $G:$ Top $\to$ hTop be the usual functor....
Alice in Wonderland's user avatar
4 votes
1 answer
96 views

Subcategory of functor category is complete

Let $D$ be a complete and cocomplete symmetric monoidal closed category (a Bénabou cosmos). Let $P$ be the permutation category. Consider the substitution product $\circ$ on the functor category $[P^{...
Margaret's user avatar
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1 vote
1 answer
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Proving bijectivity of the isomorphism when finding the right adjoint of functor F: Haus $\rightarrow$ to Top

Right adjoint of functor F: Haus $\rightarrow$ to Top Let $X$ be a topological space. We introduce a relation $\sim$ on X by declaring $x \sim y$ if for every continuous map $f : X \rightarrow H$ with ...
darkside's user avatar
  • 605
0 votes
1 answer
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Why is the functor from the lattice of open subsets of the reals to Set contravariant?

…our sheaf of continuous functions on the reals consists in specifying, for each open subset $U ⊆ R$, the set $C(U,R)$ of real continuous functions on $U$. Clearly, when $V ⊆ U$ is a smaller open ...
Julius H.'s user avatar
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1 answer
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The diagonal functor, but 'one level up'. Useful to show functoriality of the limit functor?

Given some index category $\mathcal{I}$ and a complete category $\mathcal{C}$, we consider the diagonal functor $\Delta : \mathcal{C} \to [\mathcal{I}, \mathcal{C}]$ to be the functor that sends all ...
Jos van Nieuwman's user avatar
4 votes
2 answers
133 views

How do I determine all the endomorphisms of the identity functor $\text{id}_{\text {Grp} }: \text {Grp}\rightarrow \text {Grp}$?

I am trying to determine all the endomorphisms of the identity functor $\text{id}_{\text {Grp} }: \text {Grp}\rightarrow \text {Grp}$ I know : -that the forgetful functor $F:\text {Grp}\rightarrow \...
darkside's user avatar
  • 605
0 votes
1 answer
111 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{...
darkside's user avatar
  • 605
1 vote
0 answers
55 views

How does a smooth functor guarantee the existence of another smooth vector bundle from a given one?

I am trying to understand the following definition of smooth functors on the category of finite dimensional vector spaces (in the context of smooth vector bundle theory): Let $\mathsf{Vec}$ be the ...
math-physicist's user avatar
3 votes
0 answers
79 views

Is this functor exact?

I would like to pose a question which is, from my point of view, surprisingly elementary, difficult and interesting at the same time. I hope that somebody could enlighten me. Let $k$ be a field and $A$...
Luca Francone's user avatar
0 votes
1 answer
95 views

Question 3.1.1 in AT

Here is the question I am trying to understand its solution: Show that $\operatorname{Ext}(H, G)$ is a contravariant functor of $H$ for fixed $G$, and a covariant functor of $G$ for fixed $H.$ Here is ...
Emptymind's user avatar
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1 answer
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Showing functor for composition of morphism two variable functor with one fixed variable?

The following question is taken from "Arrows, Structures and Functors the categorical imperative" by Arbib and Manes $\color{Green}{Background:}$ $\textbf{(1)}$ $\textbf{Definition:}$ A ...
Seth's user avatar
  • 3,219
4 votes
1 answer
688 views

Definition of functor

This is a question addressed to those familiar with category theory, since it concerns the preferability of two similar definitions, which I suppose is revealed only by further experience. In sum, the ...
Amanda Wealth's user avatar

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