Questions tagged [hom-functor]

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1answer
38 views

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 $Hom(X,$_$)$ for some Category $\mathcal{C}$. Suppose further that the hom-sets Hom$_{Set}(X,A)$ and Hom$_{Set}(X,B)$ ...
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1answer
42 views

Showing that $H_A\simeq H_{A'}$ implies $A\simeq A'$

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
50 views

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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1answer
26 views

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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0answers
27 views

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
32 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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2answers
81 views

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. ...
0
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1answer
48 views

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 $...
2
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2answers
60 views

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
96 views

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 $...
2
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1answer
86 views

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
90 views

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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1answer
25 views

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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0answers
51 views

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 ...
2
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1answer
45 views

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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0answers
45 views

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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1answer
227 views

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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1answer
67 views

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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2answers
169 views

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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0answers
91 views

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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1answer
114 views

$\operatorname{Hom}$ 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
60 views

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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2answers
78 views

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
52 views

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
82 views

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 ...
5
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2answers
121 views

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{...
2
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1answer
562 views

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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3answers
90 views

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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0answers
77 views

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^{(...
1
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2answers
2k views

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)\...
1
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1answer
143 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 ...
2
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0answers
50 views

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 ...
1
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1answer
122 views

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
65 views

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 \...
2
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2answers
162 views

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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2answers
94 views

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
83 views

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
91 views

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
206 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 ...