Questions tagged [hom-functor]

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Kernel of $\phi \colon $Hom$_R(\prod_{i \in I}A_i, B) \to \prod_{i \in I}\text{Hom}_R(A_i, B)$ is isomorphic to Hom$_R(\prod A_i/\sum^\oplus A_i, B)$.

Let $R$ be a ring, $\{A_i\}, B$ be $R$-modules. Let $\iota_i \colon A_i \to \prod_{i \in I} A_i$ be the canonical injection. Let $\phi \colon \text{Hom}_R(\prod_{i \in I}A_i, B) \to \prod_{i \in I}\...
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Show that if $M$ is projective then $\operatorname{Hom}_R(M,.)$ is an exact functor. [duplicate]

Is there any reference that contains this proof: Show that if $M$ is projective then $\operatorname{Hom}_R(M,.)$ is an exact functor. Or any help in the proof will be appreciated. We defined ...
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Hom, H$^n$ and tensor product

If $X$ is a scheme, $\mathcal{F}$ is a quasi-coherent sheaf and $L$ is an ample line bundle, why it holds $H^0(X, \mathcal{F} \otimes L^n) = Hom(L^{-n}, \mathcal{F})$?
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coefficient ring for cohomology modules

I am a bit confused about a notation I am seeing while reading a paper. So basically I have a continuous map between topological spaces $f:X\to Y$, and I know that this induces a map on cohomology $f^*...
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1answer
44 views

Homomorphism into a union of $R$-algebras.

All rings/algebras appearing in this question are assumed to be commutative with unity and noetherian. Let $R$ be a ring, let $A, B$ be $R$-algebras, and let $(B_i)_{i \in I}$ be a family of sub-$R$-...
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What is a clear, elegant proof that $\text{Hom}(\cdot, Y)$ is a right-exact functor in a category of modules?

If you search the site for this proof, you will find duplicates, however they are hard to understand. In other words they brush by the most critical points of the proof as if they were not worth ...
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Hom is left exact, preserves limit

I'm trying to understand this sentence (From https://en.m.wikipedia.org/wiki/Exact_functor) "A covariant (not necessarily additive) functor is left exact if and only if it turns finite limits into ...
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Lifting of a morphism in a hom-functor when hom-sets are empty

I'm having trouble in unserstanding a hom-functor. Suppose we have a hom-functor $\mathrm{Hom}(X,$_$)$ for some Category $\mathcal{C}$. Suppose further that the hom-sets $\mathrm{Hom}_{\mathbf{Set}}(X,...
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Showing that $H_A\simeq H_{A'}$ implies $A\simeq A'$ [duplicate]

I'm trying to solve this exercise: So we are given that for all objects $B$, there is a natural isomorphism $$H_A(B)\simeq H_{A'}(B)$$ Let's write this isomorphism as $f\mapsto \bar f$. The ...
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2answers
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What is the name of this identity relating the monoidal products of finite sets and finite-dimensional vector spaces?

Note: This question is almost certainly a duplicate. Since I don't know the terminology involved, I couldn't find the original question. If someone can find the original question and link to it, then ...
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There is a unique map $P \to \text{Hom}(A, \prod_i C_i)$ whenever $P \xrightarrow{p_i} \text{Hom}(A, C_i)$?

I'm trying to prove directly that $\text{Hom}(A, \prod_i C_i) \simeq \prod_i \text{Hom}(A, C_i)$ whenever $\prod_{i\in I} C_i$ is a product of a family of objects $C_i$ in a category $B$, where $A$ is ...
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Difference between functors $\text{Hom}_B(-,B)$ and $\text{Hom}_{\mathbb C}(-,\mathbb C)$

I am working with a commutative finite-dimensional $\mathbb C$-algebra $B$ and I am supposed to consider and prove a theorem about the hom-functors $$\text{Hom}_B(-,B)\quad\text{and}\quad \text{Hom}_{\...
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1answer
45 views

Exactness of a sequence in Abelian category

Suppose we have a sequence in an Abelian Category, which consists of two morphism,lets say $f$:X-->Y, and a morphism $g$:Y-->Z such that gof=0 .If for every object M of the category the functor Hom(M,-...
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Is there a reason why we call $\mathrm{Hom}(A, -)$ covariant and $\mathrm{Hom}(-,B)$ contravariant?

I have trouble remembering what it means to apply the covariant or contravariant Hom functor. For example, when I see $\mathrm{Hom}(A, -)$, I always forget if I am going to reverse arrows or not. ...
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reflective subcategory

Suppose that we have the following adjoint situation: $$\hom(G-,-)\cong\hom(-,F-),$$ where $F:\cal K\to L$ and $G:\cal L\to K$ are functors. So if $F$ is a right adjoint then it has a left adjoint $...
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Defining $Hom(f,W)$

Fix a field $k$ and a vector space $W$ over $k$. All other vector spaces will vector spaces over $k$ as well. Given a linear map $f:V\to V'$, there is a linear map $f^\ast:Hom(V',W)\to Hom(V,W)$ ...
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1answer
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Weibel exercise 1.1.4, taking $A = Z_n$…

Exercise 1.1.4 Show that $\{\text{Hom}_R(A, C_n)\}$ forms a chain complex of abelian groups for every $R$-module $A$ and every $R$-module chain complex $C_{\cdot}$. Taking $A = Z_n$, show that if $...
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Is there a way to show Yoneda equality visually instead of compositional algebraic symbols?

The proof that $u \in F(A)$ is sufficient to define a natural transformation $\alpha : \text{Hom}(A, \cdot) \Rightarrow F(\cdot)$ goes like this: Let $\alpha_X : g \in \text{Hom}(X, Y) \mapsto F(f)(u)...
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1answer
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The Yoneda lemma and a natural bijection

Let $S\colon\mathbf {Set}^{\cal A^{op}}\to \mathbf{ Set}$ be a functor. How does it follow from the Yoneda lemma that the following is a natural bijection: $\underline{\hom(A,-)\to SY \quad\quad\...
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How to see that $S(\sigma) = \text{Hom}_{\Delta}(\sigma, [0,1])$ maps to the category $\tilde{\Delta}^{\text{op}}$.

Let $\Delta$ be the simplicial category. Let $\tilde{\Delta}$ be the subcategory of non-empty totally ordered sets as objects and order-preserving maps that also preserve the smallest and largest ...
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decomposition of hom-functors in a self-enriched category

Let $\mathbb{C}$ be a self-enriched category (such as Set). The Functor $\mathbb{C}(X, \mathbb{C}(Y,\_))$ is the same than the composition of functors $\mathbb{C}(X,\_) \circ \mathbb{C}(Y,\_)$. In a ...
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1answer
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Currying in a locally small category with coproducts

While studying for category theory course I stumbled upon the following question taken from a previous exam: Let $\mathcal{D}$ be a locally small category with all coproducts. Show that for every ...
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Does an adjoint of an internal Hom functor of a prounital closed category define a tensor product?

A closed category is a category equipped with internal Hom functors along with a unit object. Now this answer shows that if $C$ is a closed category whose internal Hom functor has a left adjoint, ...
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Is “monoidal category enriched over itself” the same as “closed monoidal category”?

If $M$ is a monoidal category, an enriched category over $M$ is a category $C$ whose hom-sets are viewed as objects in $M$. And a monoidal category $M$ is said to be closed if the tensor product ...
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Isomorphism between Hom-space and $k$

Let $\mathrm{vect}$ be the category of vector spaces over $k$ and let $V \in \mathrm{vect}$. Can one say $\mathrm{Hom}(V, V)$ (the hom-space in the category $\mathrm{vect}$) is isomorphic to $k$? ...
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Existence of injective homomorphism

I'm interested in knowing if $\operatorname{Hom}(-,S^1)$ is exact, understanding $S^1$ as the group of modulo $1$ complex numbers, is right exact on the category of finite abelian groups, because that ...
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Nearly locally presentable categories

Here1, in the remark $2.3 (1)$ how from the fact that ${\cal K}(A,-)$ does not preserve coproducts it follows that ${\cal K}(A,-)$ sends special $\lambda$-directed colimits to $\lambda$-directed ...
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$\operatorname{Hom}$ functor is projective?

Consider $A$ a small category and consider $\operatorname{Hom}(-,X) \in \operatorname{HOM}(A^\text{op},Ab)$ the category of contravariant functors from $A$ to $Ab$. It is true that the $\operatorname{...
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1answer
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Fully faithful nerve ⟹ injective on objects?

If $i: C ⟶ D$ (where $C$ is a small subcategory of $D$) is the inclusion functor, and $N_i := {\rm Hom}(i(=), -)$ is the nerve of $i$, is it true that $N_i$ is injective on objects as soon as it is ...
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There is a natural morphism $\lim\limits_{\longleftarrow} \beta \to \lim\limits_{\longleftarrow} \beta \circ \varphi^{op}$?

This is from Categories & Sheaves by Kashiwara & Schapira. $$\text{Hom}_{\text{Set}}(X, \lim_{\leftarrow} \beta) \xrightarrow{\sim} \lim_{\leftarrow} \text{Hom}_{\text{Set}}(X, \beta) \tag{...
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1answer
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Preservation of limits by representable functors

Let $\beta : I^{op} \to \text{Set}$ be a functor and $X \in \text{Set}$. I want to show that there is a natural isomorphism $$ \text{Hom}_{\text{Set}}(X, \lim_{\longleftarrow} \beta) \xrightarrow{\...
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1answer
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Existence of morphisms in a free completion under directed colimits,$\lambda$-accessible category

Let $\cal K$ be a $\lambda$-accessible category with directed colimits and $\cal C$ be its representative full subcategory consiting of $\lambda$-presentable objects. Let $\cal L$ be free completion ...
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When can functors fail to be adjoints if their hom sets are bijective?

Suppose we have categories $\mathcal{C}$ and $\mathcal{D}$ with some appropriate adjectives like local smallness along with a pair of functors $F: \mathcal{C} \rightarrow \mathcal{D}$ and $G: \mathcal{...
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Limits and $Hom(-,Y)$-functor in abelian categories

Let $\cal C$ be an abelian category and let $(\{x_i\} , \{\phi_{ij}:x_i\rightarrow y_j\}_{i\leq j})$ be a direct system with direct limit $(\varinjlim x_i, \phi_i: x_i \rightarrow \varinjlim x_i)$. ...
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Why is $\text{Hom}_{\text{Set}}(S, \text{lim} F) \simeq \text{lim } \text{Hom}_{\text{Set}}(S, F(\cdot))$?

Every category theory reference says that the isomorphism in the title is a triviality. How so? $$ \lim F \equiv \text{Hom}_{\hat{C}}(\text{pt}, F) \in \text{Set} $$ so $$ \lim \text{Hom}_{\text{...
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Question on the map induced by hom-functor

Let $R$ be a commutative ring with unity and let $M$ be a finitely generated noetherian $R$-module. Suppose we are given an $R$-homomorphism: $$ \varphi: R^{(2)}\to R^{(2)} $$ Fixing a basis of $R^{(...
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2answers
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Direct sum and Hom

Let $R$ be a commutative ring with unity and let $M$ be a finitely generated noetherian R-module. Can someone tell me how the isomorphism $$ Hom_R(R\oplus R, M)\simeq Hom_R(R,M)\oplus Hom_R(R,M)\...
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1answer
163 views

Does a fully faithful functor apply to identity arrows?

In Saunders Mac Lane's Categories for the working mathematician one can read, when talking about fully faithful functors: [...], but this need not mean that the functor itself is an isomorphism ...
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Hom-like functor to a topos other than Set?

The hom-set functor is often given as the bifunctor: $$ \mathbf{Hom}(-,-) : \cal C^{op} \times \cal C \to \mathbf{Set}$$ (This is of course under the assumption that $\cal C$ is locally small.) Is ...
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1answer
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What does “$\text{Hom}(\space, G)$takes values in Groups” mean?

From Rotman's Algebraic Topology: When we say that $\text{Hom}(\space , G)$ takes values in Groups, then it follows, of course, that $\text{Hom}(X, G)$ is a group for every object $X$ and $g^*$ ...
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3answers
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Is there only one isomorphism $h_C(X) \simeq h_C(Y)$ if it exists?

I'm trying to prove that if $F \simeq h_C(X)$ or "$X$ represents the functor $F$", then $X$ is unique up to unique isomorphism. I already know that if $h_C(X) \simeq F \simeq h_C(Y)$ that $s: X \...
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Representability $h_C(X) \simeq F$ of functors determines rep. $X$ up to unique isomorphism (using Yoneda Lemma).

A functor $F \in C^{\wedge}$ from $C^{op}$ to $\text{Set}$ is representable if there is an isomoprhism $\varphi \in \text{Hom}_{C^{\wedge}}(h_C(X), F)$ for some $X \in C$, we can also write that as $...
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How does $h_C(f) : h_C(X) \to h_C(Y)$ an isomorphism imply $f$ is (Yoneda Lemma)?

This is from Categories & Sheaves by Kashiwara & Schapira. Let $C$ be a category, $f : X \to Y$ a morphism in $C$. Assume that for each $W \in C$, the morphism $\text{Hom}_C(W, X) \...
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2answers
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Yoneda Lemma (again)

Let $C^{\wedge} = \text{Fct}(C^{\text{op}}, \text{Set})$. Let $\text{h}_C : C \to C^{\wedge}, \ X \mapsto \text{Hom}_C(\cdot, X)$. Then the Yoneda lemma is: For $A \in C^{\wedge}$ and $X \in C$, $...
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1answer
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Yoneda lemma, wrong direction, weighted colimit

Here in the definition of weighted colimit, it seems to me that the r.h.s. $C(W⋅F,c)≅Set^{J^{op}}(W,C(F−,c))$ has a wrong direction for the application of the Yoneda lemma.In fact, the Yoneda lemma ...
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2answers
264 views

Hom functors preserve fibre products.

I'm using the definition of fibre product from the Stacks Project. How do I show that $h_{x \times_y z}(w) = h_{x}(w) \times_{h_y(w)} h_z(w)$ ? Is equality correct or should it be isomorphism? I'm ...