All Questions
Tagged with order-theory category-theory
149
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Categories in which there is a mono $A \to B$ iff there is an epi $B \to A$
Consider the property $P$ of a category $\mathcal{C}$ that for two objects $A$, $B$ in $\mathcal{C}$ there exists a monomorphism $A \to B$ iff there exists an epimorphism $B \to A$. Does the property $...
- 1,649
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simplicial category is generated by cofaces and codegeracies
As the title says, I'd like to understand whether the following proof of the well known fact that give $f \in \Delta([m],[n])$ weakly increasing is uniquely determined by being $f = \delta^{i_1}\circ \...
- 5,164
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Terminology for "transposition" of monomorphism to epimorphism in simplex category?
Recall that the simplex category $\Delta$ is dual to the category of intervals $\mathbb{I}$. By $\Delta$ I mean the category of finite ordinals $\mathbf{n} \in \omega$ with monotone functions between ...
- 804
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Generalisation of "discrete" and "indiscrete" left/right adjoint to the forgetful functor for general "ordered" Categories with monotone. morphisms.
The following is taken from "An introduction to Category Theory" by Harold Simmons
$\color{Green}{Background:}$
$\textbf{Example (Galois connection):}$ We modify the category of $\textbf{...
- 3,683
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94
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There exists a definition of order homomorphism?
Let be $(X,\preccurlyeq)$ and $(Y\curlyeqprec)$ partial ordered sets so that let be $f$ a function form $X$ to $Y$. So if $\rightthreetimes$ and $\leftthreetimes$ was an operation on $X$ and an ...
- 4,906
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30
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Terminology for non-empty suprema preserving function
Is there an established name for a map of complete lattices $f : L \to L'$ that preserves nonempty suprema? I.e. for all $U \subseteq L$ with $U \neq \emptyset$,
$$ f( \bigvee U) = \bigvee_{u \in U} f(...
- 469
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2
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58
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Distinguishing partitions of a set
This is another question based on "An Invitation to Applied Category Theory", this time based on exercise 1.12 (p. 9). I was able to instinctively answer the question but I don't know why ...
- 418
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1
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59
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Partial ordering and joining
This question is based on material from "An invitation to applied category theory" (ISBN 9781108711821) - I asked a closely related question before (Partial order of the Booleans true, false)...
- 418
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47
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What conditions can ensure the inverse limits of finite groups over finite posets preserve the exactness?
I asked a question about the commutativity of inverse limits before. Since I am interested in using inverse system to describe certain subgroups of a finite group, I always assume the groups and ...
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1
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83
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A monoid $M$ is $\omega$-presentable in the category $M$-$\mathbf{Set}$
I feel that this is true but I'm unable to prove it formally: a monoid $M$ is $\omega$-presentable in the category $M$-$\mathbf{Set}$. This is the category of $(X,\rho)$ where $\rho:M\times X\to X$ ...
- 3,978
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Binary relations from one poset to another defined as montone maps in Jean Bénabou course
all.
I have started reading "Distributors at Work" which is an introduction to distributors based on a course by Jean Bénabou: https://www2.mathematik.tu-darmstadt.de/~streicher/FIBR/DiWo....
- 73
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65
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Clarifications needed in an exercise about semilattice and abelian monoids in Arbib and Manes' text
The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes
Exercise: A $\textbf{semilattice}$ is a poset in which every finite subset has a ...
- 3,683
2
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109
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Coequalizers in the category of partially ordered sets
Let $X$ and $Y$ be posets and $f, g : X \to Y$ be monotonic functions. I wish to construct the coequalizer $(Z, \; h : Y \to Z)$ of $f$ and $g$.
My attempt
I first define the relation $R \subseteq Y*Y$...
- 1,548
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44
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Quick questions about showing that defining poset over a vector space is a category.
The following question is taken from $\textit{Arrows, Structures and Functors the categorical imperative}$ by Arbib and Manes
Exercise: Let $X$ be a vector space and let $\leq$ be the partial ...
- 3,683
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1
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What do you call a bijection that must preserve the order of its mappings?
A bijection is an injective and surjective function, as such:
But the following is also a bijective function:
This means there is a further restriction that could be added to the definition of a ...
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