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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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 ...
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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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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) \...
- 301
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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 ...
- 63
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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 ...
- 301
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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 ...
- 4,128
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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 ...
- 301
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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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- 315
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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 ...
- 13.4k
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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 ...
- 26.3k
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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 $\newcommand{\Th}{\operatorname{Th}} \Th(\xi) := C(V \setminus X \hookrightarrow V)$, ...
- 4,963
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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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