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.
753
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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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1
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Equivalence of the two functors $Alt^{k}(-*)$ and $(Alt^{k})^{*}$
I am finishing Vector Analysis of Klaus Jänich. I am stuck at chapter $12$ because I am confused about a notation. I hope some of you could untangle it for me.
Lemma
We can interpret each $\varphi \in ...
0
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1
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If $\mathcal{C}$ and $\mathcal{D}$ are abelian categories, and $F:\mathcal{C}\to \mathcal{D}$ is fully faithful, is $F$ exact?
If $\mathcal{C}$ and $\mathcal{D}$ are two abelian categories, and $F:\mathcal{C}\to \mathcal{D}$ is a fully faithful functor, can we conclude that $F$ is an exact functor?
By default, $F$ is an ...
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4
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Example of a wide fully-faithful functor that is not an isomorphism.
Given a functor $T:C \to D$ between two categories, we say that $T$ is wide if it is surjective on objects, full if all functions between Hom-sets $T_{x,y}:C(x,y) \to D(Tx,Ty)$ are surjective, and ...
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Does a functor which reflects limits also reflect cones?
Following Borceux's Categorical Algebra Definition 2.9.6:
Let $F: \mathcal{C}\to\mathcal{B}$ be a functor. $F$ reflects limits when, for every functor $G: \mathcal{D}\to\mathcal{A}$ with $\mathcal{D}$...
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functors preserve isomorphism of direct sum.
In this proof in the Stacks project, it is mentioned that the decomposition of identity morphism of direct sum:
... because the composition $F(A) \oplus F(B) \xrightarrow{\varphi} F(A \oplus B) \...
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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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1
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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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2
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1
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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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1
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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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4
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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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3
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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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35
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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 ...
- 1,021
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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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Trouble understanding special case of functor category
I'm reading Rotman's Advanced Modern Algebra: Part 1 and am slightly confused by an example showing a special case of the functor category:
If $\mathcal{A}$ is the (small) category with $\...
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3
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Serge Lang's Definition of a Free Group
Serge Lang says the following in his "Algebra":
We now consider the category $\mathfrak{C}$ whose objects are the maps of $S$ into groups. If $f:S\rightarrow G$ and $f':S\rightarrow G'$ are ...
4
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1
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Orbit functor is not co-representable
Let ${1}\neq H\le G$ be groups. Denote by $G\textit{-}\mathsf{Set}$ the category of sets with a $G$ action, with $G$-equivariant maps as morphisms. Let $(-)/H: G\textit{-}\mathsf{Set}\to \mathsf{Set}$ ...
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Examples of pseudofunctors into Cat
I'm trying to find (basic) examples of pseudofunctors into Cat, but the only example I keep coming across is the extension of scalars (from the functor $R\text{-Mod} \to S\text{-Mod}; M \mapsto M\...
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1
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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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0
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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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Finding a matrix for the functor $\Lambda^{p}f$ of functions from anti-symmectric tensors onto itself
Let $\mathcal{B}$ be an ordered basis for a vector space $V$ of dimension $n$. Denote the basis for the set of anti symmetric tensors ($\Lambda^{p}{V}$) by $\mathcal{B}^{p}$. For $n=3$, let $A=\left( ...
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What is the name of this functor's property?
Assume there is a functor $L$ from a category $C$ to a category $D$ which satisfies the following property: for any objects $X,Y,Z$ from $C$ and morphisms $f\colon X\to Y, g\colon X\to Z$ such that $L(...
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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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1
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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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1
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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 ...
- 1,468
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Subcategory of left-right projective bimodules
Let $\mathbb{K}$ be a fixed field. Take a finite-dimensional $\mathbb{K}$-algebra $A$ and look at the category $\text{mod}(A^e)$ of finite-dimensional $A$-bimodules. It has a full subcategory $\text{...
- 1,769
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Diagrams as visualizations of functors
Can this quiver $A\to B\to C$ be a diagram in some category? (Let's hope that now my question is "specific" so the community bot won't close it)
This is the background for me asking this: my ...
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2
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Quivers as the underlying structure of categories
The structure of a small category is a quiver - every vertex of the quiver represents an object with its identity morphism, and the edges of the quiver represent the non trivial morphisms (I know you ...
- 121
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2
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86
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(Co)Products are bifunctors, but are general (co)limits also functors?
In a category with all products or coproducts, the (co)product operation can be understood as a bifunctor. More generally let $\mathcal{C}$ be a category with all limits of shape $D$, where for ...
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0
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49
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When We say a diagram in a category is a functor, then how do we treat the category? Do we take it as 2-catgegory?
In a Category $\textbf{C}$, we say that an $\cal{I}$-diagram in a category is a functor $F: \cal{I} \to \textbf{C}$. In fact, Any object A can be viewed as a diagram by defining a functor $F: \cal{I} \...
1
vote
1
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Prove that $F$ is left adjoint to $G$ iff $G$ is right adjoint to $F$.
I started the definition of adjoint functor using universal morphisms. A functor $F:\mathcal C\to \mathcal D$ is a left adjoint functor if given any object $Y\in D$, there is an object $G(Y)$ in $\...
- 41
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Does an exact sequence of holonomic D-modules imply an exact sequence in the solution space?
Here is a probably quite basic question on holonomic D-modules, but I am only a physicist so please bear with me.
If I have the weyl algebra $D$ in $n$ variables, and an exact sequence of holonomic $D$...
- 41
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1
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98
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GL$_n$ schemes representation
Let $n \geq 0$ be a natural number. Consider the schemes
$$ \mathrm{GL}_n:=\operatorname{Spec}\left(\mathbb{Z}\left[X_{i, j} \mid 1 \leq i, j \leq n\right]_f\right),\quad \mu_m :=\operatorname{Spec}\...
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