Questions tagged [hom-functor]
The hom-functor tag has no usage guidance.
63
questions
0
votes
1
answer
43
views
Representing object for a functor mapping a category to composable morphisms
If I have a functor $D_n : \mathsf{Cat} \to \mathsf{Set}$ that maps a category into the set of all $n$-tuples of composable morphisms,
$D_n(C) = A_1 \to A_2 \to A_3 \to \dots \to A_n$, what would its ...
- 11
1
vote
1
answer
44
views
the hom-functor is left exact: an equivalent condition
I would like to understand the equivalence of the 2 facts that hom is left exact in the sense that it carries short exact sequences to left exact sequences and that it preserves all small existing ...
- 3,959
5
votes
1
answer
354
views
Bifunctoriality stronger than functoriality in each variable?
Is it correct to say that bifunctoriality is a stronger condition than "functoriality in each variable"?
More precisely, suppose that $\mathcal A, B, C$ are categories and I have $F \colon \...
-1
votes
1
answer
37
views
Finitely presentable objects in presheaves
How can be described finitely presentable objects in $${\mathbf{Set}^{\cal A}}^{op}$$ for $\cal A$ small ?
- 3,959
0
votes
2
answers
67
views
Is the $Hom (-,k)$ functor exact? [duplicate]
$\DeclareMathOperator{\Hom}{Hom}$
Let $k$ be a field of characteristic $0$.
I want to know if $\Hom(-,k)$ is an exact functor from the category of abelian groups to itself. If it's true can you give a ...
- 544
1
vote
1
answer
80
views
Characterizing Representable / Hom Functors via Universal Property
I have been trying to think of a way to characterize the hom functors via universal property. I could not find any such thing elsewhere online. So I came up with a property, inspired by Yoneda lemma, ...
- 1,766
0
votes
1
answer
46
views
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 ...
- 3,959
0
votes
0
answers
62
views
$(X^Y)^Z\cong X^{(Y+Z)}$ 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 $*$ ...
- 439
0
votes
1
answer
53
views
$F$ 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 ...
- 19.9k
1
vote
2
answers
129
views
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 ...
- 19.9k
1
vote
1
answer
141
views
Definition of the cotensor in sSet
I am currently trying to understand the meaning of the cotensor in sSet. Suppose I have X,Y objects in sSet, the cotensor is denoted by $X^Y$. I found that for sSet the cotensor is a map of the form $$...
- 143
0
votes
1
answer
48
views
Categories in which every object is a hom-object?
A closed category basically contains its own hom-objects. Flipping this around, is there a word for categories in which every object is a hom-object for a pair of objects in the category? Or ...
- 115
2
votes
2
answers
395
views
Proof verification: Yoneda preserves limits
$\newcommand{cop}{C^{op}}$
$\newcommand{cSet}{\mathsf{Set}}$
$\newcommand{Hom}{\operatorname{Hom}}$
$\newcommand{\eval}{\operatorname{eval}}$
Let $J$ be a small category, and let $C$ be locally small. ...
- 8,156
1
vote
0
answers
183
views
Proof of derived tensor-hom adjunction
EDIT: this has now been cross-posted here.
As far as I know, for $R,S,V,W$ rings and $M$ an $(R,W)$-bimodule, $N$ an $(R,S)$-bimodule and $L$ an $(S,V)$-bimodule, we have an isomorphism in D($V$-lmod)
...
- 1,252
0
votes
0
answers
70
views
How to refer to image of homset under a (not full, not faithful) functor.
Consider a functor $F: C \rightarrow Set$. If $F$ is not full and not faithful, how can I talk about the image of $F$ on morphisms (a subset of $Hom_{Set}(F(c), F(c'))$) in relation to $Hom_C(c, c')$? ...
- 101
0
votes
1
answer
97
views
Determine if a functor has right/left adjoint
In $Set$ category let $P_X$ a functor defined as follows: given a set $Y$, $P_X(Y)=X\times Y$ and a function $f: Y \to Z $, $P_X(f)= 1_X \times f$. Determine if $P_X$ has left/right adjoint.
At first ...
- 490
2
votes
0
answers
72
views
Injectivity of derived Hom functor base change map
According to Tag 0E1V of Stacks, given a ring map $R \rightarrow R'$ and two $R$-modules $K,M$, there is a base change map
$$
R\mathop{\mathrm{Hom}}\nolimits _ R(K, M) \otimes _ R^\mathbf {L} R' \...
- 1,252
0
votes
1
answer
24
views
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 ...
- 1
0
votes
1
answer
88
views
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}\...
- 423
1
vote
1
answer
401
views
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 ...
user889267
1
vote
0
answers
60
views
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 $H^0(X, \mathcal{F} \otimes L^n) = Hom(L^{-n}, \mathcal{F})$?
- 31
1
vote
1
answer
76
views
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^*...
- 1,551
1
vote
1
answer
56
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-$R$-...
- 861
0
votes
1
answer
167
views
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 ...
- 19.9k
4
votes
1
answer
135
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 $\mathrm{Hom}(X,$_$)$ for some Category $\mathcal{C}$.
Suppose further that the hom-sets $\mathrm{Hom}_{\mathbf{Set}}(X,...
- 165
0
votes
1
answer
74
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 ...
- 11.6k
1
vote
2
answers
89
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 ...
- 1,228
0
votes
1
answer
37
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 ...
- 19.9k
0
votes
0
answers
36
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 $$\text{Hom}_B(-,B)\quad\text{and}\quad \text{Hom}_{\...
- 55
1
vote
1
answer
138
views
Exactness of a sequence in Abelian category
Suppose we have a sequence in an Abelian Category, which consists of two morphism,lets say $f$:X-->Y, and a morphism $g$:Y-->Z such that gof=0 .If for every object M of the category the functor Hom(M,-...
- 330
1
vote
2
answers
139
views
Is there a reason why we call $\mathrm{Hom}(A, -)$ covariant and $\mathrm{Hom}(-,B)$ contravariant?
I have trouble remembering what it means to apply the covariant or contravariant Hom functor.
For example, when I see $\mathrm{Hom}(A, -)$, I always forget if I am going to reverse arrows or not.
...
- 11.5k
0
votes
1
answer
72
views
reflective subcategory
Suppose that we have the following adjoint situation:
$$\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 $...
- 3,959
2
votes
2
answers
75
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)$ ...
- 11.6k
3
votes
1
answer
217
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 $...
- 19.9k
2
votes
1
answer
93
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)...
- 19.9k
0
votes
1
answer
127
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\...
- 3,959
0
votes
1
answer
36
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 ...
- 19.9k
0
votes
0
answers
72
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 ...
- 1,534
2
votes
1
answer
66
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 ...
- 579
2
votes
0
answers
84
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, ...
- 9,760
9
votes
1
answer
974
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 ...
- 9,760
0
votes
1
answer
206
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$?
...
- 391
6
votes
2
answers
576
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 ...
- 6,138
1
vote
0
answers
102
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 ...
- 3,959
0
votes
1
answer
280
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{...
- 783
1
vote
1
answer
84
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 ...
- 77
2
votes
2
answers
101
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{...
- 19.9k
1
vote
1
answer
161
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{\...
- 19.9k
0
votes
1
answer
117
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 ...
- 3,959
5
votes
2
answers
160
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{...
- 3,421