# 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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### 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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### 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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### 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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### 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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### $F$ 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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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 $... • 67 1 vote 1 answer 83 views ### Prove that additive functor preserves products and coproducts Let$\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$of objects in an ... • 1,031 3 votes 1 answer 80 views ### Forgetful functor from the slice category creates limits I am reading "Category theory in context" and I'm having difficulties with proposition 3.3.8, top of page 92, pdf. Theorem: The forgetful functor$F : c/C \to C$strictly creates all limits ... • 10k 1 vote 1 answer 76 views ### A category of functors? Let C be a category and I be a small category. Then we can consider the functor category whose objets are functors$F:I\to C$and a morphism between two functors is a natural transformation. My ... 3 votes 1 answer 114 views ### Any two natural transformations between identity functors commute Let$\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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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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### 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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### 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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### 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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