# Questions tagged [hom-functor]

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### Weighted limits

I have a very trivial question about the page 80 here: how this shape of $W$ $$W:2\to \mathbf{Set}$$ with $$\ast\sqcup\ast\to \ast$$ implies that the components of $$W\Rightarrow\cal{M}(m,f)$$ are ...
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### $(X^Y)^Z\cong X^{(Y+Z)}$ or $(X^Y)^Z\cong X^{(Y\times Z)}$?

$\DeclareMathOperator\Hom{Hom}$I have the following exercise in my class of Category Theory: Prove that $\text{Hom}(Z,\Hom(Y,X))\cong \Hom(Y*Z, X)$ but I am not sure what $*$ is. I think that $*$ ...
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### $F$ left adjoint to $G \iff F,G$ define a functor from $\textbf{Arr}(\textbf{X}\times\textbf{A}) \to 2\times 1$ square CDs in $\textbf{Set}$?

Let $\textbf{A, X}$ be categories and $F:\textbf{X} \to \textbf{A}$ and $G: \textbf{A} \to \textbf{X}$. Then there is a map that takes an object in $\text{Arr}(\textbf{X}\times\textbf{A})$ (the arrow ...
1 vote
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### How do you derive the adjoint's "naturality" condition as seen in MacLane & Moerdijk's book "Sheaves in Geometry and Logic"?

It's tag (7) as pictured below. I also included the definition of adjoint that they use. I know that by definition of adjunction (using the natural homset isomorphism), we have two naturality ...
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### Examples of not-full functors between two categories in both directions?

A functor F:C→D from a category C to a category D is said to be full if, for each pair of objects, x,y ∈ C, the function, F:C(x,y)→D(F(x),F(y)) between hom sets is surjective. What I am unsure about ...
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1 vote
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### 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 ...
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 $\mathrm{Hom}_{\mathbf{Set}}(X,... 0 votes 1 answer 66 views ### Showing that$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 ... 1 vote 2 answers 82 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 ... 0 votes 1 answer 35 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 ... 0 votes 0 answers 33 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 $$\text{Hom}_B(-,B)\quad\text{and}\quad \text{Hom}_{\... 1 vote 1 answer 75 views ### Exactness of a sequence in Abelian category Suppose we have a sequence in an Abelian Category, which consists of two morphism,lets say f:X-->Y, and a morphism g:Y-->Z such that gof=0 .If for every object M of the category the functor Hom(M,-... 1 vote 2 answers 121 views ### Is there a reason why we call \mathrm{Hom}(A, -) covariant and \mathrm{Hom}(-,B) contravariant? I have trouble remembering what it means to apply the covariant or contravariant Hom functor. For example, when I see \mathrm{Hom}(A, -), I always forget if I am going to reverse arrows or not. ... 0 votes 1 answer 60 views ### reflective subcategory Suppose that we have the following adjoint situation:$$\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 votes 2 answers 68 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) ... 3 votes 1 answer 180 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 votes 1 answer 91 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)... 0 votes 1 answer 118 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\... 0 votes 1 answer 34 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 ... 0 votes 0 answers 59 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 votes 1 answer 57 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 ... 1 vote 0 answers 74 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, ... 7 votes 1 answer 691 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 ... 0 votes 1 answer 165 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? ... 6 votes 2 answers 529 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 ... 1 vote 0 answers 100 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 ... 0 votes 1 answer 209 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{... 1 vote 1 answer 67 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 ... 2 votes 2 answers 86 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{... 1 vote 1 answer 103 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 $$\text{Hom}_{\text{Set}}(X, \lim_{\longleftarrow} \beta) \xrightarrow{\... 0 votes 1 answer 105 views ### Existence of morphisms in a free completion under directed colimits,\lambda-accessible category Let \cal K be a \lambda-accessible category with directed colimits and \cal C be its representative full subcategory consiting of \lambda-presentable objects. Let \cal L be free completion ... 5 votes 2 answers 140 views ### When can functors fail to be adjoints if their hom sets are bijective? Suppose we have categories \mathcal{C} and \mathcal{D} with some appropriate adjectives like local smallness along with a pair of functors F: \mathcal{C} \rightarrow \mathcal{D} and G: \mathcal{... 5 votes 1 answer 1k views ### Limits and Hom(-,Y)-functor in abelian categories Let \cal C be an abelian category and let (\{x_i\} , \{\phi_{ij}:x_i\rightarrow y_j\}_{i\leq j}) be a direct system with direct limit (\varinjlim x_i, \phi_i: x_i \rightarrow \varinjlim x_i). ... 1 vote 3 answers 111 views ### Why is \text{Hom}_{\text{Set}}(S, \text{lim} F) \simeq \text{lim } \text{Hom}_{\text{Set}}(S, F(\cdot))? Every category theory reference says that the isomorphism in the title is a triviality. How so?$$ \lim F \equiv \text{Hom}_{\hat{C}}(\text{pt}, F) \in \text{Set} $$so$$ \lim \text{Hom}_{\text{... 1 vote 0 answers 102 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^{(... 