# 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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### 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'...
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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 ...
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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 ...
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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 ...
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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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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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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 ...