# 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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### A functor $G : \mathbf{Sets} \to {}_k\mathbf{Mod}$ with $G(X) = k^X$?

I am trying to solve question $1.5(i)$, on page $33$, from An Introduction to Homological Algebra - 2nd edition by Rotman. If $X$ is a set and $k$ is a field, define the vector space $k^X$ to be the ...
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### Size of the functor category

Let $\mathscr C$, $\mathscr D$ be categories. It's well-known that if both are small, then the functor category $\mathscr D^{\mathscr C}$ is also small. However, it seems to me that we can weaken ...
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### Why should the categories with 1 and 2 elements be considered "the same"? [duplicate]

In my course notes about the basics of category theory, the following is said. This is why a candidate definition of an equivalence of categories is not a good choice. Let $\mathcal{C}$ be the ...
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### What does it mean for a functor to be adjoint to the identity of another functor?

In "Transformation Groups and Algebraic K-Theory" by Wolfgang Lück he talks about the functor $J_{X}M:E_{X}\circ Res_{X}M\rightarrow M$, where $E_{X}:$MOD-$R[X]\rightarrow$MOD-$R\Gamma$ is ...
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### Is there an established notion of natural transformation for cofunctors?

A natural transformation $\eta:F\to G$ assumes that we have functors $F,G:C \to D$. But what if we have $F,G$ where either of $F,G$ is a cofunctor (i.e. contra-variant functor)? There are four ...
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### Simplicial presheaves present $\infty$-presheaves related question

I am trying to work out the details that every $\infty$-topos is presented by a model topos. By presented I mean it is the image under the homotopy coherent nerve. A model topos is a model category ...
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### The functor that preserves the direct sum is an additive functor

I am learning about additive categories on the Stacks Project, and I encountered some difficulties in proving that functor that preserves direct sums is an additive functor. There is such a ...
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### Model structure on Functors from Span to chain complexes

Given Span category ($1 \leftarrow 0 \rightarrow 2$) and the following model structure on chain complexes $Ch_{\mathbb{K}}$: Fibrations: level wise surjective. Cofibrations: level wise injective. ...
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### Frobenius Endomorphism does not preserve injectivity

Frobenius Endomorphism Let $R$ be a ring of prime characteristic $p>0$. The Frobenius endomorphism is the map $F: R\to R$ defined by $r\mapsto r^p$ for any $r\in R$. For any $R$-module $M$, we can ...
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### What is functorial isomorphism

I am learning triangle categories on the stacks project. In this definition, the author mentions functorial isomorphism $\xi_{X}: F(X[1]) \rightarrow F(X)[1]$. What is functorial isomorphism? Is this ...
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### Need Help To Prove Fullness of a Functor

I am working on proving the following proposition A functor $F:\mathscr{A} \rightarrow \mathscr{B}$ is an equivalence of categories if and only if it is faithfull, full and essentially surjective. ...
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### Name of an endo-"functor" but which doesn't change the source/target of morphisms?

An endofunctor maps objects $A$ to $F(A)$ and morphisms $m:A\to B$ to morphisms $F(m):F(A)\to F(B)$. Is there an established name for a different "endofunctor-like" class of objects (not ...
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### A sort of Day convolution without enrichment

Some time ago I was trying to define a monoidal structure on a functor category $[\mathcal{C},\mathcal{D}]$ between two monoidal categories $\mathcal{C}$ and $\mathcal{D}$, such that the monoid ...
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### Functoriality of Thom Space as Mapping Cone

One of the most general definitions of the Thom space associated to a real vector bundle $\xi\colon V \to X$ is as $\newcommand{\Th}{\operatorname{Th}} \Th(\xi) := C(V \setminus X \hookrightarrow V)$, ...
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### Forgetful functor $V: \underline{\mathbf{PSet}} \rightarrow \underline{\mathbf{Set}}$ is not full

This might be a trivial question, but I don't understand why the trivial functor $V: \underline{\mathbf{PSet}} \rightarrow \underline{\mathbf{Set}}$ is not full. ($\underline{\mathbf{PSet}}$ is the ...
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### Number of isomorphisms between categories

Let $\mathcal{C}_1$ be a category with objects $A=\{1,2,3\}$ and $B=\{4,5,6\}$. The number of isomorphisms between them is $3!=6$ and morphisms $27$. Can we extend this notion of number of ...
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### Cohomology Functor - Homotopic Morphisms Map to the Same Morphism? [duplicate]

For the sake of avoiding confusion, I provide the definition from Achar for this functor: Given an abelian category $\mathbf{A}$ with chain complex $A=((A_j),d_A^{j})$, then the cohomology functor ...
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### Constructing Shift Functor on Chain Complexes - Alternating Differential

I've got a question pertaining to the construction of the shift functor on the chain complex category Ch$(\mathbf{A})$ (where $\mathbf{A}$ denotes an additive category). It makes perfect sense to me ...
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### Functor $F\circ G$ with Natural Transformation to $1$ (Identity Functor)

This is a very broad question...but I'm looking for properties regarding a given map$^{1}$ $G:C\to D$ (where $C,D$ denotes categories) such that $F\circ G$ has a natural transformation to the identity ...
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### The pushout $F_\alpha(H)$ of $H \overset{\pi}{\longleftarrow} \mathbb{Z}^{\ast H} \overset{\alpha^{\ast H}}{\longrightarrow} G^{\ast H}$

$\require{AMScd}$ Definition 1: If $G$ is a group and $X$ is a set, define $G^{\ast X} = \ast_{x \in X} G$. This is functorial in both $G$ and $X$: If $G$ and $H$ are groups, $\varphi: G \to H$ is a ...
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### What does $Hom(h, B)$ mean in the contravariant functor?

The Wikipedia stated that Let C be a locally small category (i.e. a category for which hom-classes are actually sets and not proper classes). For all objects A and B in C we define two functors to ...
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### Functors from category of finite sets and bijections to the category of sets

[Tom Leinster, Exercise 1.3.31] A permutation of a set $X$ is a bijection $X \rightarrow X$. Write Sym$(X)$ for the set of permutations of $X$. A total order on a set $X$ is an order $\leq$ such that ...
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### Does a functor from one category to another imply homomorphism between automorphism groups.

I have two categories $C$ and $D$ and a functor $F: C\rightarrow D$. I select an object $A\in ob(C)$, is there guaranteed to be a homomorphism $Aut(A)\rightarrow Aut(F(A))$? It seems like there should ...
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### Comparing two definitions of cocartesian morphism

In the literature I've found two notions of "$\pi$-cocartesian morphism" in a category, and to my knowledge they're not equivalent. The first and I think most common one is the following : ...
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### Ordinary functor being E-indexed

I am learning parts of topos theory and it feels like I am missing something regarding indexed functors. This comes from the definition of a locally connected geometric morphism $f : E \rightarrow F$ ...
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### Equality of objects in a concrete category

A concrete category is a pair $(C,U)$ such that $U:C\to \mathbf{Set}$ is a faithful functor. It means that for any $Y,Z\in \mathrm{ob}(C)$, and a function between the sets $f:U(Y)\to U(Z)$ - if exists ...
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### Is there a functor between $\mathbf{Set}$ and $\mathbf{Set}/C$?

In my PhD thesis, I'm interpreting some of my code in Category Theory. I ended up modeling some stuff as a slice category $\mathbf{Set}/C$, and ended up with the question whether I could define a ...
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### Is "up to natural isomorphism" crucial?

Let $\mathcal{C}$ be a category. By diagram I mean a covariant functor $F\colon\mathcal{J}\to\mathcal{C}$ for some category $\mathcal{J}$. In this source it is said that a commutative diagram is a ...
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### Are (commutative) squares in some sense universal among edge-symmetric double categories?

Definitions: Given a category $\mathcal{C}$, and a double category $\mathbb{D}$, i.e. a category internal to $Cat$ with an “objects category” $\mathcal{D}_0$ and a “morphisms category” $\mathcal{D}_1$,...
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### Is there an idempotent e in a ring $R$ such that the functor $T_e$, induced by $e$, is not full?

Let $R$ be a ring and let $e$ be a non-zero idempotent in $R$. For each $R$-module $M$, define $T_e: M \rightarrow eM$, where $eM$ is the left $eRe$-module. For each pair of $R$-modules $M, N$ and ...
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### Among morphisms of morphisms, what makes commutative squares special?

Given two (1-)categories $\mathcal{C}, \mathcal{D}$, and given the 0-category (class) of funtors $\mathcal{C} \to \mathcal{D}$, denoted $Func(\mathcal{C} \to \mathcal{D})$, let's say we want to make ...