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.
614
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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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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 ...
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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 ...
- 3
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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)},\...
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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. ...
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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'...
- 1,924
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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 \...
- 607
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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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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 ...
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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 $F \colon \mathcal{HoC}^{\mathrm{op}} \to \mathcal{Sets}$ ...
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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_{\...
- 1,924
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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 ...
- 1,924
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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 ...
- 5,388
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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 ...
- 1,924
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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 ...
- 2,317
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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 ...
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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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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 ...
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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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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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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 ...
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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 ...
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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 $(...
- 5,324
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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/...
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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 ...
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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 ...
- 399
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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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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 \...
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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, ...
- 3,991
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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}_{\...
- 3,004
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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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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 ...
- 3,004
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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 ...
- 1,222
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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 ...
- 3,004
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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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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}...
- 3,214
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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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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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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
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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 $f^*$ ...
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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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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, ...
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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 ...
- 504
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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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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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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 ...
- 504
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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
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60
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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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