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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Show naturality of $\infty$-natural transformation

Working with the model category of complete Segal spaces $\text{CSS}$, which has as its underlying category the category of simplicial presheaves on $\Delta$, one has a suitable internal hom in $\text{...
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1 vote
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Realizing the category of chain complexes in an additive category $A$ as a subcategory of $A$-valued presheaves on $\mathbb{Z}$?

Can the category of chain complexes in an additive category $A$ be realized as as subcategory of presheaves on $\mathbb{Z}$? Here I am thinking of $\mathbb{Z}$ as a poset category. I am asking a ...
0 votes
1 answer
45 views

What is the difference between functor composition and natural transformation?

If $F$ and $G$ are injective functors between the categories $C$ and $D$ $H$ is an endofunctor on the category $D$ such that $H∘F = G$ $η$ is a natural transformation from $F$ to $G$ Then for every ...
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Trianguled bifunctor $\mathcal{D}^{-}(A)^{op} \times \mathcal{D}^{+}(A) \to \mathcal{D}^{+}(A)$

Setting: Let $\mathcal{C(A)}$ be the category of complexes, $\mathcal{K(A)}$ the homotopy category, $\mathcal{D(A)}$ the derived category with respective essential images with $\mathcal{K^*(A)},\...
2 votes
0 answers
39 views

Left (right) derived functor construction: motivation

Suppose that $\mathcal{A}$ is a category with enough projective-injective. The construction of left or right derived functors I know is: for an object $A \in \mathcal{A}$ take a projective (resp. ...
0 votes
2 answers
59 views

Composition of morphism part of evaluation (bi)functor.

Before giving a lenghty introduction, I'd like to actually just ask one thing. We are given the object part Ev$_0$ of the evaluation functor $\mathcal{C} × [\mathcal{C}, \mathcal{D}] → \mathcal{D}$. I'...
1 vote
1 answer
25 views

Proving a formula involving Hom-set go colimit and constant functor

Let $I$ and $C$ be categories, assume $I$ is small and denote $\Delta$ the functor $C \to C^I$ that sends $Y\in C$ to the constant functor $I\to C$, i.e. $\Delta(Y)(i) = Y$ and $(i \to j) \mapsto \...
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30 views

equivalence between $K^b(\mathcal{P}) $ and $ D^{b}(\mathcal{P})$

In my homological algebra notes I found the proof that the homotopy category $K^{-}(\mathcal{P})$ is equivalent to the derived category $D^{-}(\mathcal{A})$ when $\mathcal{A}$ has enough projectives. ...
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0 answers
32 views

On the right cancellation in localization of category. Is it necessary?

I recently faced the concept of derived category, introduced as localization of homotopy category. I tried to verify of the axioms and I stucked in the following: In the definition of multiplicative ...
3 votes
0 answers
83 views

Showing a topological half-exact functor is topological exact .

The Problem is: Let, $G$ be a group, not necessarily abelian. Let $\mathcal{C}$ be the category of all based CW complexes. Show the functor $F \colon \mathcal{HoC}^{\mathrm{op}} \to \mathcal{Sets}$ ...
6 votes
3 answers
153 views

Does the abelianisation functor $\mathrm{Grp} → \mathrm{AbGrp}$ preserve composition?

I am to show there is a functor $F : \mathrm{Grp} → \mathrm{AbGrp}$. I have already checked that the assignments $F_0(G) := G_{\mathrm{ab}} := G/[G,G]$ and $F_1 (f \colon G → H) := \bar{f}: G_{\...
1 vote
1 answer
46 views

Left-inverse functor $\mathrm{Cat} \to \mathrm{Preord}$

As a follow-up to my previous question: Preorders as categories vs category of preorders, I’m trying to construct a functor $J \colon \mathrm{Cat} \to \mathrm{Preord}$ such that $J ∘ I = 1$. First ...
1 vote
2 answers
46 views

Alternative functor construction from universal morphisms

Let $G : \mathcal{D} \rightarrow \mathcal{C}$ be a functor. Suppose that for each object $X \in \mathcal{C}$, there exists a universal morphism $(F_X, \eta_X)$ from $X$ to $G$. The theory of adjoint ...
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2 votes
2 answers
92 views

Preorders as categories vs category of preorders

I understand how a preorder can be interpreted as a category (reflexivity translating into identity morphisms and transitivity to associativity). But now I want to understand the category of all ...
0 votes
2 answers
76 views

How can I understand this definition of a presheaf?

We have given the following definition of a presheaf: A preasheaf $F$ on a topological space is a functor $$F:\text{Op}(X)^{\text{op}}\rightarrow \text{Sets}$$ Where $\text{Op}$ is the category of ...
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2 votes
1 answer
45 views

Equivalent definitions of left exact functors

I am new studying categories and I have the following question. Given an additive functor $F$ between abelian categories, I have tried to prove that the following left exact functor definitions are ...
2 votes
1 answer
44 views

Uniqueness of a lift to the category of elements of a functor

This is exercise 2.4.viii of Riehl's "Category Theory in Context": Prove that for any $F:\rm{C} \to \rm{Set}$, the canonical forgetful functor $\prod:\int F\to \rm{C}$ has the following ...
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1 vote
1 answer
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Reference Request: Adjoint Functors and Universal Morphism

I currently work on non-category theoretical research but use some tools from category theory. I am not well-versed in category theory and also don't want to expand the preliminaries of my work too ...
0 votes
0 answers
37 views

Category and functor [duplicate]

Is there a functor Z: Grp $ \to $ Grp with the property that Z(G). is the centre of G for all groups G.
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2 answers
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Show that $F$ cannot be a extended to morphisms of category $\mathcal{Set}$

Let $F(X) = X \cap \mathbb{N}$. I have to show that $F$ cannot be extended to morphisms of category $\mathcal{Set}$. I wanted to assume it can and then find a contradiction with on of the properties ...
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Natural Isomorphism $(Y^Z)^X\cong Y^{Z\times X}$ in a cartesian closed category

Let $\mathcal{C}$ be a cartesian closed category. I'm working on a problem that asks me to show that for $X,Y,Z\in\text{ob}(\mathcal{C})$ there is a natural isomorphism $(Y^Z)^X\cong Y^{Z\times X}$. ...
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0 votes
1 answer
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Is it true that functors which are surjective on objects are obviously essentially surjective?

I am asking this as I have established a functor F between categories C and D such that F is faithful, full and surjective on objects. Can I say that F is an equivalence of categories? I think so but ...
-1 votes
1 answer
63 views

A morphism in $Hom (a,a)$ which is not the identity morphism

So, when defining a category, one is careful enough to define the identity of a object $a\in\text{Ob}(\mathcal C) $ as a particular element of the hom-class, $\text{id}_{a}\in\text{Hom}(a,a)$. That ...
1 vote
1 answer
47 views

Schur functors are pairwise non isomorphic

In Fulton-Harris Part I Weyl's construction there a characterization of some of the irreducible representation of $GL(V)$, with $V$ a finite complex vector space. In particular, Theorem $6.3$ point $(...
0 votes
1 answer
47 views

On linear functors on the abelian category of finitely generated modules over a Noetherian ring

Let $R$ be a commutative Noetherian ring, and let $\mod R$ denote the abelian category of finitely generated $R$-modules. Let $F:\mod R \to \mod R$ be an $R$-linear functor (https://ncatlab.org/nlab/...
2 votes
1 answer
51 views

How do you express that two commutative diagrams can be glued together along a common subgraph and retain commutativity?

I'm wondering about this, because I want to write software that lets you operate on CD's on the computer screen. I was wondering what existing mathematical tools are required to describe the titled ...
0 votes
1 answer
30 views

Constructing a well-defined category using repeated functor application

I'm trying to come up with a concise definition of a complicated category that arises from repeated application of functors to a "seed category" of objects. The following example isn't the ...
2 votes
2 answers
120 views

Does taking quotients commute with taking the group algebra?

I ll preface this by excusing myself because I am not quite sure how to ask my question. Basically I was messing around with group algebras $\mathbb{K}[G]$, I found it interesting that sub algebras ...
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5 votes
1 answer
345 views

Bifunctoriality stronger than functoriality in each variable?

Is it correct to say that bifunctoriality is a stronger condition than "functoriality in each variable"? More precisely, suppose that $\mathcal A, B, C$ are categories and I have $F \colon \...
-1 votes
1 answer
30 views

Finitely presentable objects in presheaves

How can be described finitely presentable objects in $${\mathbf{Set}^{\cal A}}^{op}$$ for $\cal A$ small ?
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cone injectivity in a presheaf category

I have a rather informal question: I would like to know how a small injectivity class in a presheaf category $${\mathbf{Set}^{\cal A}}^{op}$$ for $\cal A$ small formally can look like and also, ...
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1 vote
1 answer
57 views

How to understand in context of category theory: combination of contravariant functor and natural transformation

I have a category $\mathcal{C}$, a contravariant functor $D:\mathcal{C}\to\mathcal{C}$ (so that $\text{id}_{\mathcal{C}}$ and $D^2$ are covariant), and a natural transformation $\chi:\text{id}_{\...
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6 votes
1 answer
172 views

Adjoint functor theorem applied to a forgetful functor

Let $\mathbf{Cat}$ denote the category of small categories and $\mathbf{MCat}$ the category of small monoidal categories with monoidal functors. Consider the forgetful functor $\operatorname{U}:\...
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2 votes
1 answer
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Terminology: contravariant functor $F:\mathcal{C}\to\mathcal{C}$ for which $F^2$ is naturally isomorphic to the identity

I have a category $\mathcal{C}$ and a contravariant functor $F:\mathcal{C}\to\mathcal{C}$, so that $F^2$ and $\text{id}_{\mathcal{C}}$ are covariant functors $\mathcal{C}\to\mathcal{C}$. I also have a ...
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2 votes
0 answers
43 views

Extending natural transformation from a set of projective generators

$\mathcal{C}$, $\mathcal{D}$ be abelian categories. $\mathcal{C}$ is cocomplete, and has set $\mathcal{P}$ of projective generators. $F$, $G\colon \mathcal{C} \to \mathcal{D}$ are additive functors ...
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2 votes
1 answer
74 views

Short exact sequence of modules induces short exact sequence of cokernels

I have a commutative unital ring $A$, a short exact sequence $L\to M\to N$ of $A$-modules, a faithful exact contravariant functor $D:\text{Mod}_A\to\text{Mod}_A$ satisfying $$DM=0\iff M=0,$$ and ...
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0 votes
1 answer
110 views

The functor $\operatorname{Isom}_k(X,Y)$ is representable

Let $X,Y$ projective schemes over $k$. Consider the contravariant functor $$\operatorname{Isom}_k(X,Y): \left(\operatorname{Sch}/k\right) \to (\operatorname{Set})$$ $$ S\mapsto \operatorname{Isom}_S(S\...
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4 votes
0 answers
85 views

Is the functor $\operatorname{Spec}(R) \mapsto \operatorname{GL}_n(R) / R^*$ not a Zariski sheaf?

We call Zariski sheaf a contravariant functor $F: (\operatorname{Aff \ Sch}/k) \to (\operatorname{Grp})$ such that $F_X: (\operatorname{Op}(X) \cap (\operatorname{Aff \ Sch}/k)) \to (\operatorname{Grp}...
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2 votes
2 answers
132 views

MacLane’s coherence theorem and free monoidal categories

Original question The nLab gives one formulation of the coherence theorem for monoidal categories: Every diagram in a free monoidal category made up of associators and unitors commutes. It seems to ...
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1 vote
1 answer
49 views

Symmetric lax monoidal functor takes operads to operads

I am looking for a reference to the following theorem. If $F:\mathcal{C}\to\mathcal{D}$ is a symmetric lax monoidal functor between symmetric monoidal categories, then it induces a well-defined ...
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6 votes
0 answers
107 views

Root system as a category

I'm studying Lie Algebras from Humphreys's book. The definition of isomorphism of root systems seems to me to have some functor properties: if $\phi:(\Phi,E)\to(\Phi', E')$ is an isomorphism of ...
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2 votes
2 answers
173 views

How do I think of the Hom functor?

For an object $X$ in a category $C$, there is a functor $C(-\,,X)$ from $C^{\mathrm{op}}$ to Set that assigns to each object $Z$ the set $C(Z,X)$ and to each morphism $f: Y \to Z$ the pullback $f^*$ ...
0 votes
1 answer
74 views

Are algebraic groups functors exact?

Let $\mathbb{K}$ be a field (for example $\mathbb{K} = \mathbb{Q}$). An algebraic group $G$ can be thought of as a functor from $\mathbb{K}$-algebras to groups. Is $G$ exact, i.e. does it preserve ...
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3 votes
1 answer
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When can you cancel the identity functor in a natural isomorphism?

Suppose endofunctors $F, G : \mathbb C \to \mathbb C$ and the identity functor $1$. If we have an isomorphism $F \times 1 \cong G \times 1$, then under what conditions can we get $F \cong G$? In Set, ...
0 votes
1 answer
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Definition of product/coproduct of exact sequences

I am trying to understand Grothendieck's AB4(*) conditions https://stacks.math.columbia.edu/tag/079A but I don't know how product and coproduct of exact sequences are defined. My only guess is that ...
2 votes
1 answer
179 views

Unitors in star-autonomous categories

1.Context Let $(C, \otimes, I, a, l,r)$ be a monoidal category. Suppose $S: C^{op} \xrightarrow{\sim} C$ is an equivalence of categories with inverse $S’$. Assume that there are bijections $\phi_{X,Y,...
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1 vote
1 answer
83 views

Why is the preservation of identities a funtoriality axiom?

I'm a newbie at category theory and just started reading Emily Riehl's Category Theory in Context. I got to the definition of functors, which contains the following two axioms: if $F:C\to D$ is a ...
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2 votes
1 answer
51 views

Plan of proof about derived functors in general abelian category

I have to write a report about the derived functors of the inverse limit $\lim$ functor defined from the category of inverse systems (of modules, or maybe in some cases of cochain complexes). Now, the ...
0 votes
2 answers
70 views

What exactly do category theorists mean when they say: "prove there is a function with the following definition" in exercises to students?

$\newcommand{\A}{\mathscr{A}}\newcommand{\B}{\mathscr{B}}\newcommand{\C}{\mathscr{C}}$ This is reminiscent of this question I asked ages ago on projective limits of dynamical systems. Note that I have ...
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0 votes
0 answers
60 views

How do functors act on image objects?

Suppose $C,D$ are (additive) categories and $F:C\rightarrow D$ is a functor. I want to know when is $F(\alpha\cdot C)=F(\alpha)\cdot F(C)$ where $\alpha\cdot C$ denotes the image of $C$ under the ...
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