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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59 views

Why I should not study embedded categories?

In my lecture course of representation theory of $K$-algebras, we embedded the category of $A$-modules (where $A$ is an associative unital $K$-algebra) into the category of $K<X_i|i\in I>$-...
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32 views

If F is a functor from $C_{1} \times C_{2}$ does this equlity hold: $F(f, h \circ g) =F(f,h) \circ F(f,g)$?

Lets say we have a functor $F:C_{1} \times C_{2} \rightarrow D$, where $C_{1},C_{2},D$ are catagories. Does this equation hold for funtors: $F(f, h \circ g) =F(f,h) \circ F(f,g)$, where $f,h,g$ are ...
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Additivization of functors in an abelian monoidal category

Crossposted on MathOverflow here. I'm having trouble with the proof of Lemma 2.9 in "Cohomology of Monoids in Monoidal Categories" by Baues, Jibladze, and Tonks, and I was wondering if someone could ...
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49 views

Adjointenes of unknown functors

I would like to understand what are here on the page $266$ the types of items in the adjoint euqation $$\mu^{Lim}_{\cal K}\vdash Lim\ \eta_{\cal K}^{Lim}.$$ They should be functors by the definition ...
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1answer
53 views

$G \to G'$ is a functor from $\text{Grp}$ to $\text{Grp}$

I'm not sure if this question has asked before, but I've just start to study category theory and I'm still learning what is a functor, so I have some specific questions about that. Ok, $\text{Grp}$ ...
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1answer
30 views

Prove that a functor is additive

Let $\mathcal{F}: Mod_A \rightarrow Mod_B$ be a covariant functor such that for all A-modules $M_1$ and $M_2$, the homomorphism $\mathcal{F}(M_1) \oplus \mathcal{F}(M_2) \rightarrow \mathcal{F}(M_1 \...
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1answer
26 views

Properties of an A-module

I must show that the following properties for an $A$-module $P$ are equivalent: 1) The functor $Hom(P,-)$ is exact. 2) There is an $A$-module $Q$ such that $P \oplus Q$ is free. 3) Every short ...
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A monic coalgebra morphism whose underlying $\text{Set}$ morphism is not injective

Let $F:\mathscr A\to \mathscr A$ be a functor. Consider the following category $\mathscr C$. The objects are arrows $A\to F(A)$. If $\alpha:A\to F(A)$ and $\beta:B\to F(B)$ are two objects, then a ...
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Categories without isomorphisms

A groupoid is a category in which every morphism is an isomorphism. I'm looking at a dual case: categories in which none morphism (except identites) is an isomorphism. These kind of categories are ...
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Functoriality of twisted de Rham Cohomology

Let $f: M\to N$ be a smooth map of finite-dimensional manifolds, and let $E\to N$ be a flat vector bundle over $N$. Consider the pullback bundle $f^*E\to M$ over $M$ and consider the twisted de Rham-...
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How to use projective cover to get a map right minimal?

Let $\Lambda$ be an artin algebra and $mod \Lambda$ be the category of finitely generated $\Lambda$-modules. Suppose $\mathcal{Y}$ is a subcategory of $mod\Lambda$ which is closed under extension and ...
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74 views

When a function on objects lifts to a functor?

Let $\mathbf{C}$ and $\mathbf{D}$ be two categories with classes of objects $\mathbf{C}_0$ and $\mathbf{D}_0$, respectively. Consider a function $f:\mathbf{C}_0\rightarrow \mathbf{D}_0$. When $f$ ...
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40 views

Tor is a covariant functor

Various books and many online notes and other posts such as Proving that $\operatorname{Tor}_n^R$ is a bifunctor give a rather touch-and-go treatment to Tor being a covariant functor. For example, in ...
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30 views

Example for a functor $F:Mod(A)\rightarrow Mod(B)$

I'm trying to find a simple example for a functor $F$ between modules, which is non linear and I don't really know how to approach this. Intuitively, I'm thinking matrix modules. A hint would be ...
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1answer
20 views

Functor preserves group objects

Let $C,D$ be finite product categories and $F:C\to D$ be a functor such that it preserves terminal objects and such that it preserves products then $F$ takes group objects of $C$ to group objects of $...
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48 views

Why is el(-)=$\int(-)$ a functor from functors to a slice cateory?

I am reading up a little on category theory. I am trying to solve Riehl's problem (Category Theory in Context) 2.4.vii on page 72. I believe that it should be quite easy, but it appears to me that ...
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Intuitive notion of functoriality in topological data analysis

For school, we have to give a presentation about topological data analysis and I am in charge of motivating why topological data analysis is cool and useful. Most of what I say is based on "Topology ...
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1answer
21 views

Colimit of Disjoint Metric Spaces vs Topological Coproduct

Let $\{X_i\}_{i=1}^{\infty}$ be a sequence of pairwise disjoint metric spaces. Here we use the convention that a metric space can assume infinite distance. Let Met be the category with metric spaces ...
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1answer
26 views

Associating to any vector space its $k$-linear dual and the resulting functor

From my understanding, a contravariant functor of a category $\mathcal{C}$ can be defined using the notion of opposite category, $\mathcal{C}^{op}$. Then for two categories $\mathcal{C}$ and $\mathcal{...
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2answers
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Can the exponential map (in Lie theory) be considered an example of a functor?

I've heard that the exponential map from the Lie algebra of a Lie group to the original Lie group can be considered as a functor of some sort, but I'm very new to all this and would appreciate some ...
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Functoriality in Modular Forms

So I'm still a little new to Modular Forms. I have started reading Lang, I have watched Keith Conrad's lectures and I have read his notes. I started to ask myself some questions. Let $\Gamma$ and $\...
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1answer
59 views

Interpreting a Group Action as a Functor

A group $G$ can be thought of as a groupoid $G$ with a single object. How is defining an action of $G$ on an object of a category $C$ the same thing as defining a functor $G \rightarrow C$. I know ...
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Product functor and diagonal functor

Let $C$ be a category and consider the product category $C \times C$. There is a diagonal functor associating to each object $X$ of $C$ the pair $(X,X)$ as an object of $C \times C$. On the other ...
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1answer
33 views

Is an equalizer really a special case of a limit?

Let $I$ be a category with $2$ objects $A,B$ and four morphisms $1_A, 1_B, u,v$ where $u,v \in Hom_I(u,v)$. Define a functor $F: I \to \mathcal{C}$ by $$F(A) = X, F(B) = Y, F(u) = f, F(v) = g$$ My ...
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82 views

Suppose every functor which preserves $\oplus$ arbitrarily is additive when restricted to fin. gen. proj. modules, then is $S$ stably finite?

(This is the culmination of my recent efforts to understand this problem, and it will be the last post of this type I think. I have a feeling that it might not allow for an answer that fits a forum ...
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1answer
47 views

Is there a functor $F$ of left modules preserving $\oplus$ by arbitrary isomorphism, but its restriction to fin. gen. proj. modules isn't additive?

Are there rings $R$, $S$ and a functor $F:{_R\textbf{Mod}}\to{_S\textbf{Mod}}$ such that For all left $R$-modules $M,N$, we have $F(M\oplus N)\cong F(M)\oplus F(N)$ via an arbitrary isomorphism, and ...
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1answer
41 views

Does a functor which is additive by arbitrary isomorphism restrict and corestrict to an actually additive functor?

(I appologize for asking a more and more refined version of the same problem for the third time in two days. This is just due to the fact that with every answer to a previous question, I realize that ...
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1answer
65 views

Is there a functor $F$ preserving finite direct sums but not split exact sequences, for which $F\mathbb{Z}$ is free and finitely generated?

Is there a functor $F$ from the category of abelian groups to itself, such that $F$ preserves finite direct sums (as well as the empty direct sum) up to (arbitrary) isomorphism $F$ doesn't in general ...
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39 views

The “Right” Concrete Category

One thinks of cardinality as being a property of groups, topological spaces and the objects of many other common categories. This is typically expressed through affixing to the category $\mathcal{C}$ ...
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27 views

Definition functor on empty morphism sets: what are the possibilities?

A functor $F: \mathcal{C} \to \mathcal{D}$ between two categories consists of the following data: A mapping $Ob \mathcal{C} \to Ob\mathcal{D}: A \mapsto F(A)$ For each pair of objects $A,B \in Ob\...
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26 views

Functor on a subcategory.

Quick sanity check here: Let $F: \mathcal{C} \to \mathcal{D}$ be a functor between two categories. Let $\mathcal{D}'$ be a subcategory of $\mathcal{D}$ such that $F(C) \in \mathcal{D}'$ for all $C \...
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2answers
47 views

Is the product bifunctor uniquely determined by how it acts on objects?

Saunders MacLane states that "The product objects provide, by $<a, b> \mapsto a \times b$, a bifunctor $C \times C \to C.$ " I know that this is not in fact a bifunctor, as the product is unique ...
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1answer
38 views

Lifting of a morphism in a hom-functor when hom-sets are empty

I'm having trouble in unserstanding a hom-functor. Suppose we have a hom-functor $Hom(X,$_$)$ for some Category $\mathcal{C}$. Suppose further that the hom-sets Hom$_{Set}(X,A)$ and Hom$_{Set}(X,B)$ ...
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1answer
38 views

Limits via universal arrows and functor categories

I would like to understand is some detail the connection between the 2 snippets taken from McLane's book CWM. Namely, I do not follow the connection between the functor $S$ and categories and arrows $...
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3answers
157 views

How to interpret this definition of adjoint functors?

Firstly consider the four definitions in the question: How to define rigorously [...]. Also consider the following definition: Definition: Let $C,D$ be two categories and $F,G:[C]\to [D]$ be two ...
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How to define rigorously the category of all functors?

Assume the NBG set theory. First I will introduce four definitions below. Definition 1: Let $A,B$ be two classes. A correspondence between $A$ and $B$ is a class $F\subseteq A\times B$ such that ...
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Existence of universal covering groupoids

I am not very experienced in this topic, so please excuse me if there are any stupid mistakes. Let $G$ be a groupoid. In “Calculus of fractions and Homotopy Theory” by Gabriel and Zisman it is shown, ...
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31 views

Why is it called the restriction functor?

I'm rather new to this subjects so please try to keep it simple. Let $f:A\rightarrow B$ be a ring homomorphism, and let $N$ be a $B$-module. Then $N$ is an $A$-module as follows: $a\cdot n:=f(a)\...
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Is this end-like property expressible in a nicer way?

I have recently stumpbled upon a situation that I am trying to describe categorically, and it seems both familiar and not quite exactly what it should be. After idealizing the situation and peeling ...
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Details of the criterion for representability of a functor of S-schemes

I've come across a problem that's made me look back over representabiilty of scheme functors and I'm having a lot of trouble piecing together some categorical details that I used to think I ...
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1answer
54 views

Fully faithful functors preserve and reflect isomorphisms

Here are my attempts to prove the following lemma: (a) Let $f:A\to A'$ be an isomorphism. Let $g:A'\to A$ be its inverse. Consider $$\alpha_{A,A'}: \mathscr A(A,A')\to \mathscr B(J(A),J(A'))$$ This ...
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108 views

Algebraic space

Let $S$ be the ring $\mathbb{Z}[1/2]$. Is the functor $$ \begin{array}{l|rcll} \Lambda : & S-\text{Algebra} & \longmapsto & \text{Set} \\ & R & \...
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1answer
112 views

Can functoriality be explained in the opposite way?

I have some confusion about functoriality in category theory. In general, functoriality means that functors must preserve composition of morphisms. That means Given a functor $F :C \to D$, for ...
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1answer
42 views

Showing that $H_A\simeq H_{A'}$ implies $A\simeq A'$

I'm trying to solve this exercise: So we are given that for all objects $B$, there is a natural isomorphism $$H_A(B)\simeq H_{A'}(B)$$ Let's write this isomorphism as $f\mapsto \bar f$. The ...
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1answer
227 views

Show that $\Gamma$, $\Lambda$, and the associated sheaf functor are all left exact.

This is Exercise II.6 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to the first few pages of this Approach0 search, it is new to MSE. The Details: The functors $\...
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. . . and what about $\Gamma$ in $\S II.5$ of Mac Lane and Moerdijk?

This is about $\S II.5$ of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]" and is a follow-up to this: Just what is Mac Lane & Moerdijk's $\Lambda$ from $\S II.5$? The ...
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Name for a “Tagged Functor”

I have a functor F generic over two types T and S (for Source), which is only functorial in ...
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1answer
55 views

Introductory exercise of category theory

A functor between two preorders is a function $T $ that is monotonic (i.e., $p\le p' $ implies $Tp\le Tp' $). This is an exercise from Categories for the working mathematician, (number 1.3.3) whose ...
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50 views

What is the name of this identity relating the monoidal products of finite sets and finite-dimensional vector spaces?

Note: This question is almost certainly a duplicate. Since I don't know the terminology involved, I couldn't find the original question. If someone can find the original question and link to it, then ...
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1answer
51 views

A sieve $S$ on $U$ in the category $\mathcal{O}(X)$ is principal iff the corresponding subfunctor $S\subset 1_U\cong{\rm Hom}(-,U)$ is a sheaf.

This is Exercise II.1 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to Approach0, this is new to MSE. The Details: On p. 36, ibid. . . . Definition 0: For an ...

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