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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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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Hi... I just beginning with the study of categories so my question might seem elementary. [closed]

Please in want to know how to show that the functor $\mathcal{P}(\Sigma \times Id) $ weakly preserve pullbacks.
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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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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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The functor $\textbf{Hom}(-,W):\textbf{Vect}^{op}_k\rightarrow \textbf{Vect}_k$

I've recently started to study category through Leinster's Basic Category Theory. In example 1.2.12 it's said that $\textbf{Hom}(V,W)$ is a vector space. Then, Now fix a vector space $W$. Any linear ...
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Weighted limits

I have a very trivial question about the page 80 here: how this shape of $W$ $$W:2\to \mathbf{Set}$$ with $$\ast\sqcup\ast\to \ast$$ implies that the components of $$W\Rightarrow\cal{M}(m,f)$$ are ...
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Is there any category whose objects are semigroup acts and morphisms are semilinear morphism between acts?

I know that for a particular semigroup S all the S-acts form a category with S-act homomorphisms. My question is what happens if we do not fix the semigroup i.e. taking all semigroup acts (S,A) where ...
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$\operatorname{Ext}^1$ isomorphic to quotient ring

A duplicate of this question in MO: I'm reading this paper: Brochard, Iyengar and Khare: Wiles defect for modules and criteria for freeness. In lemma 4.5, there is an isomorphism $\operatorname{Ext}_A^...
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Group action on fibre functor

Let $C$ be a Tannakian category (ie. it is rigid tensor Abelian category where hom sets are $k$-vector spaces and there is a fibre functor $w$ from $C$ to category of vector spaces such that $w$ is a ...
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2 answers
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Attempt to Hausdorff-ize spaces

Introduction Let $\mathbf{2cT_x}$ denote the category of second-countable $T_x$ spaces. We all know there is a forgetful functor $\mathsf{F} : \mathbf{2cT_2} → \mathbf{2cT_0}$. I tried to find a ...
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Functor $F: \mathcal{C} \to G_1$-Sets is an equivalence of categories given a natural isomorphism to equivalence $H: \mathcal{C} \to G_2$-Sets

This is the argument I need in order to understand a proof of a certain theorem. I think that I'm mostly confused by the formalism, so I kindly ask for your help. Suppose we have a functor $F: \...
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$(X^Y)^Z\cong X^{(Y+Z)}$ or $(X^Y)^Z\cong X^{(Y\times Z)}$?

$\DeclareMathOperator\Hom{Hom}$I have the following exercise in my class of Category Theory: Prove that $\text{Hom}(Z,\Hom(Y,X))\cong \Hom(Y*Z, X)$ but I am not sure what $*$ is. I think that $*$ ...
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An endofunctor $F:\mathcal{C} \to \mathcal{C}$ does NOT specify any choices of morphisms $C \to F(C)$, correct?

Given an endofunctor $F:\mathcal{C} \to \mathcal{C}$, although we know that for every object $C \in Ob(C)$ the functor $F$ "sends" $C$ to $F(C)$ in some sense, the functor $F$ by itself does ...
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Do (special) natural transformations imply a commutative triangle of functors (but NOT vice versa)?

Given categories $\mathcal{C}, \mathcal{D}$, let functors $F,G: \mathcal{C} \to \mathcal{D}$ be such that for any objects $C_1, C_2 \in Ob(\mathcal{C})$, $F(C_1) = F(C_2) \implies G(C_1) = G(C_2)$, ...
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The category $\mathbf{Ban_1}$ equipped with a non-obvious functor $U_1: \mathbf{Ban_1} \to \mathbf{Set}$

Wikipedia says that For technical reasons, the category $\mathbf{Ban_1}$ of Banach spaces and linear contractions is often equipped not with the "obvious" forgetful functor but the functor $...
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3 votes
1 answer
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How do I prove if the functor is right exact or left exact?

Hello I have the following question: I need to show if the functor $\Bbb{Q}\otimes_R-$ from $\Bbb{Z}$-modules to $\Bbb{Q}$ vector spaces are left exact, right exact or even nothing. I somehow ...
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Show that the free monoid functor $ M: \text { Sets } \rightarrow \text { Mon } $ exists, in two different ways

I'm learning Category Theory by Steve Awodey's book. in the exercise 11 we have : Show that the free monoid functor $$ M: \text { Sets } \rightarrow \text { Mon } $$ exists, in two different ways: (...
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The functor that maps a (co-)monad to its (co-)Eilenberg-Moore category

I have noticed that the function that maps a monad $T : C \to C$ to the Eilenberg-Moore category $C^T$ can easily be extended into a functor $E_C$ from the category of monads $\textbf{Mnd}_C$ to $\...
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7 votes
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Is there a connection between free–forgetful adjunctions and tensor-hom adjunctions?

In the Wiki article on adjunction https://en.wikipedia.org/wiki/Adjoint_functors, there is a motivation section that talks about how adjunctions can be viewed as "Solutions to optimization ...
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Question about right derived functors and duality

UPDATE after some thinking (and without comments or answers) I want to ask the following: Say that I have a morphism $f:X\to Y$ of schemes and a induced morphism $f_*\colon \mathcal{S}(X)\to \mathcal{...
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5 votes
2 answers
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Graph with no nontrivial endomorphisms?

For context, I'm trying to determine whether there exists a full and faithful functor $F:\mathsf{Dgr}\to\mathsf{SimpGph}$ that "encodes" directed graphs as simple graphs. Right now I believe ...
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2 votes
3 answers
81 views

Functoriality of bifunctors: Joint functoriality equivalent to separate functoriality?

Since I haven't found anything related on MSE nor in Google, I'll post here this question. Let $\mathsf{C}, \mathsf{D}, \mathsf{E}$ be categories and consider an assignment $F:\mathsf{C}\times\mathsf{...
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Shift/translation functor in a triangulated category

I'm trying to get to grips with triangulated categories at the moment. According to Wikipedia, the shift/translation functor of a triangulated category $\mathcal{C}$ is an "additive automorphism (...
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$F:\mathcal{B}G \rightarrow Vect_{\mathbb{F}}$

Let $F$ be the functor $F:\mathcal{B}G \rightarrow Vect_{\mathbb{F}}$, where $Vect_{\mathbb{F}}$ is the category of vector spaces over a field $\mathbb{F}$. Prove that $\lim F \cong V^G$ is the fixed ...
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1 answer
104 views

What means " Tor functor commutes with arbitary direct sum in both vanishing"?

According to the following the video clip (at 39:11), $\operatorname{Tor}$ functor commutes with arbitrary direct sum in both vanishing. ...... (★), but I do not understand this property (I hope to ...
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2 votes
1 answer
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Does the (pseudo)functor that assigns a commutative monoid $M$ to the topos of $M\text{-Sets}$ preserve limits? [closed]

Let $\mathrm{CMon}$ be the category of commutative monoids, and $\mathrm{Topos}$ be the bicategory of (Grothendieck) toposes with geometric morphisms. Consider the (pseudo)functor $\mathrm{CMon}\to\...
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3 votes
1 answer
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Are the discrete and indiscrete functors the only sections of the forgetful functor from topological spaces to sets?

A section of the forgetful functor from topological spaces to sets assigns a particular topology $\tau_X$ on each set $X$ in such a way that every function $X \to Y$ becomes a continuous function $(X,\...
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0 answers
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Is there a notion of "contravariant enriched functors"?

Is there a concept of "contravariant enriched functors"? I'll add some context. Let $C$ and $D$ be categories enriched over a monoidal category $M$. An enriched functor from $C$ to $D$ is a ...
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1 vote
1 answer
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Adjunctions b/w constant diagram functor and limit/colimit functors for fixed index category

Let $\mathcal{C}$ be a locally small category and let $\mathcal{J}$ be a small category. Assume that $\mathcal{C}$ has all $\mathcal{J}$-shaped limits and colimits. Describe the unit and counit for ...
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1 answer
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Relation between right adjoints and counits

Assume that $F: \mathcal{C} \rightarrow \mathcal{D}$ is left adjoint to $G: \mathcal{D} \rightarrow \mathcal{C}$ with counit $\varepsilon: F G \Rightarrow$ id $_{\mathcal{D}}$ and unit $\eta$ : id $_{\...
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1 answer
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$F$ left adjoint to $G \iff F,G$ define a functor from $\textbf{Arr}(\textbf{X}\times\textbf{A}) \to 2\times 1$ square CDs in $\textbf{Set}$?

Let $\textbf{A, X}$ be categories and $F:\textbf{X} \to \textbf{A}$ and $G: \textbf{A} \to \textbf{X}$. Then there is a map that takes an object in $\text{Arr}(\textbf{X}\times\textbf{A})$ (the arrow ...
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4 votes
1 answer
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Constructing counit in adjoint functor theorem for total categories

The theorem I am referring to is, Let $C,$ $D$ be locally small categories. Assume $C$ is a total category (i.e. the Yoneda functor $Y : C \to \operatorname{PreSh}(C)$ has a left adjoint $Y^L$). Let $...
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Prove that additive functor preserves products and coproducts

Let $\cal A,B$ be additive categories and $F:\cal A\rightarrow B$ be an additive functor. Show that $F$ preserves products and coproducts. Since product and coproduct of a pair $A,B$ of objects in an ...
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3 votes
1 answer
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Forgetful functor from the slice category creates limits

I am reading "Category theory in context" and I'm having difficulties with proposition 3.3.8, top of page 92, pdf. Theorem: The forgetful functor $F : c/C \to C$ strictly creates all limits ...
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1 vote
1 answer
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A category of functors?

Let C be a category and I be a small category. Then we can consider the functor category whose objets are functors $F:I\to C$ and a morphism between two functors is a natural transformation. My ...
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3 votes
1 answer
114 views

Any two natural transformations between identity functors commute

Let $\mathcal{C}$ be a category, $id_\mathcal{C}:\mathcal{C} \to \mathcal{C}$ the identity functor. Prove that for any two natural transformations $\alpha, \beta : id_\mathcal{C} \Rightarrow id_\...
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2 votes
1 answer
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Adjunction in Abelian Categories

Let $C,D$ be arbitrary categories and $F:C\to D, G:D\to C$ functors. We say $F,G$ are adjoint if for every $X\in C, Y \in D$ there is an isomorphism of Hom-Sets between $Hom_D(FX,Y)\cong Hom_C(X,GY)$,...
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Why aren't sheaf categories locally small?

Given a scheme $S$ and reasonable topology, such as Zariski, étale, or fppf, is the category of sheaves ${Sh}(({Sch}/S)_{top})$ locally small? Or, if you like to assume that a category must be locally ...
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Consequences of the Globalization Theorem in Hirsch's Differential Topology

In Hirsch's book, there is a wonderful theorem 2.11: where a structure functor is simply a presheaf, and continuous means it is a sheaf (it has the gluing property). Nontrivial means there is at ...
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6 votes
1 answer
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Exercise 1.5.xi from Emily Riehl's "Category Theory in Context" on properties of some functors

I've been going through Emily Riehl's textbook on categories, and struggle with the exercise 1.5.xi. Consider the functors $Ab \to Grp$ (inclusion), $Ring \to Ab$ (forgetting the multiplication), $(-)...
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Definition of the cotensor in sSet

I am currently trying to understand the meaning of the cotensor in sSet. Suppose I have X,Y objects in sSet, the cotensor is denoted by $X^Y$. I found that for sSet the cotensor is a map of the form $$...
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If $A \in \text{R-MOD}$, then $\text{Hom}_R(A,-)$ is left exact.

This is a lemma we went over in our class. I have a few questions about it. If $0 \rightarrow B \stackrel{\varphi}{\rightarrow} B’ \stackrel{\psi}{\rightarrow} B’’ \rightarrow 0$ is an exact sequence, ...
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Let $F:\mathcal{A}\to\mathcal{B}$ be a fully faithful covariant functor. Then, if $F(f):F(A_1)\to F(A_2)$ is an isomorphism, so is $f:A_1\to A_2$.

I’d like to ask for checking of my attempt below. We want to find $g: A_2 \to A_1$ such that $f \circ g = 1_{A_2}$ and $g \circ f = 1_{A_1}$. So define $g: A_2 \to A_1$ to be a morphism such that $F(...
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Problem with functor category notation

I'm reading Mac Lane's category theory book. The notation $A^B$, $A$ and $B$ categories, is introduced for a category of functors $B\to A$ as objects and natural transformations of these functors as ...
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Reference for the category of functors from finite sets with surjections to abelian groups

In the talk of Jacob Lurie about Lie Algebras and Homotopy theory, he mention at the end about a category of functors that has the same derived category as the category as some category of universal ...
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1 answer
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Natural Transformation of Bifunctors

I had a hard time proving the statement: "a transformation between two bifunctors is natural if and only if it is a natural transformation in each of it's arguments". This is Proposition no. ...
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3 votes
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Notation in Katz-Mazur Arithmetic Moduli of Elliptic Curves

A moduli problem $\mathcal{P}$ is a contravariant functor $\mathbf{Ell}\to\mathbf{Set}$. The objects of $\mathbf{Ell}$ are arrows $E\to S$ from an elliptic curve $E$ to a varying base scheme $S$. The ...
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8 votes
1 answer
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On the monadicity of Cat

I heard that the category of small categories is monadic over the category of small graphs, the monad being the free-graph functor. I interpreted that statement as: the Eilenberg-Moore category of the ...
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