All Questions
Tagged with limits-colimits adjoint-functors
53
questions
0
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1
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80
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Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives
Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives. State and prove the dual result.
I have no idea on ...
- 3,800
2
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2
answers
105
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Showing that the diagonal functor $\Delta:\mathbb{C} \to \mathbb{C} \times \mathbb{C}$ having a right adjoint implies $\mathbb{C}$ having products.
I started brushing up on my understanding of adjunctions and came across this well-known fact (rephrased in my own words):
Let $\mathbb{C}$ be a category, and let $\Delta:\mathbb{C} \to \mathbb{C} \...
- 371
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1
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78
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Prove that the forgetful functor $U: \textbf{Ab} \to \textbf{Set}$ preserves all filtered colimits
I realize that my question is exactly the same as this post here. However, I tried finding the book that was mentioned, Borceux's Handbook of Category Theory I, but my efforts to find the book here ...
- 313
3
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37
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Dual Concept of a Well-Powered Category
I was studying the SAFT theorem (Special Adjoint Functor Theorem), using Leinster's Basic Category Theory and I had the homework to dualize it. So far I have the following,
Consider a category $\...
- 315
2
votes
1
answer
90
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Inclusion functor is final?
For $n\geq 1$ let $K_n$ be the partially ordered set of compact subsets of $\mathbb{R}^n$ ordered by inclusion. Let $B_n$ be the totally ordered set of closed balls in $\mathbb{R}^n$ centered at the ...
- 1,769
4
votes
5
answers
219
views
Extending an adjunction using colimits
I'd like a confirmation of a fact about colimits and adjunction; the motivation is that I think that this fact is used implicitly sometimes in Kerodon, but I've never seen it written out. $\require{...
- 904
1
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1
answer
77
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Example of non representable limits-preserving functor
Let $\mathcal{C}$ be a locally small category, and $F : \mathcal{C} \rightarrow \mathcal{Set}$ functor. Then :
$$F \ \text{has left adjoint} \implies F \ \text{is representable} \implies F \ \text{ ...
- 811
3
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1
answer
106
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When representables are adjoints
Let $\mathbf C$ be a complete category and let $U:\mathbf C\to\mathbf{Set}$ be a representable functor. Show that $U$ preserves limits.
In general representables preserve limits, but the hypothesis ...
- 1,649
3
votes
0
answers
79
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The difference between totally (large) cocontinuous functors and small cocontinuous functors
$\newcommand{\cat}{\mathbf}\newcommand{\op}{\mathrm{op}}\newcommand{\Hom}{\operatorname{Hom}}\newcommand{\cSet}{\cat{Set}}$A category $\cat C$ is total if the Yoneda embedding $\cat C→[\cat C^{\op},\...
- 2,061
1
vote
1
answer
192
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Adjunctions b/w constant diagram functor and limit/colimit functors for fixed index category
Let $\mathcal{C}$ be a locally small category and let $\mathcal{J}$ be a small category. Assume that $\mathcal{C}$ has all $\mathcal{J}$-shaped limits and colimits. Describe the unit and counit for ...
4
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1
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208
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Constructing counit in adjoint functor theorem for total categories
The theorem I am referring to is,
Let $C,$ $D$ be locally small categories. Assume $C$ is a total category (i.e. the Yoneda functor $Y : C \to \operatorname{PreSh}(C)$ has a left adjoint $Y^L$). Let $...
- 105
3
votes
0
answers
113
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Localization of cocomplete categories and right orthogonality: does the equivalence always hold?
In Handbook of Categorical Algebra, Volume 1: Basic category theory, Borceux proves the following (around theorem 5.4.7 page 198).
Definition. An object $x$ of a category $\newcommand{\cC}{\mathsf{C}}\...
- 2,061
2
votes
0
answers
103
views
Do open continuous maps/local homeomorphisms between locales possess adjoints?
Recently I started learning "theory of locales" (point-free topology) by my-self. While being a very beautiful, natural subject and parallel to point-set topology, some of its notions are ...
- 18.4k
2
votes
1
answer
125
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Exercise with cartesian closed category
Suppose that the category $\mathbf C$ is cartesian closed: I must prove that, chosen three objects $x$, $y$, and $z$, the object $(x\times y)^z$ is isomorphic to $x^z\times y^z$. My idea was to define ...
- 2,523
0
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0
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83
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Exercise about adjunction of preorders
I must prove that a functor $G:Y\to X$ that preserves limits has a left adjoint $F$, where $X$ and $Y$ are preorder categories with all limits (i.e. all products). I don't see how to define $F$ with ...
- 2,523