# Questions tagged [hom-functor]

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### 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 ...
1 vote
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### 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 ...
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1 vote
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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 ... 1 vote
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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 $H^0(X, \mathcal{F} \otimes L^n) = Hom(L^{-n}, \mathcal{F})$?
1 vote
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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 ...
1 vote
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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 ...
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 ...
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1 vote
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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{\...
### 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 ...
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{...