5
votes
1answer
113 views

Two topologies over the same set which are carried to each other by a lattice isomorphism induce homeomorphic spaces?

This is a question that I've been turning in my head and haven't been able to come up with a way to proceed to either prove it or construct a counter-example: Let $X$ be a set and $\mathfrak{T}_X$ ...
3
votes
2answers
83 views

Topology inducing Order

We know by Munkres that any (total) ordering induces a topology and (luckily) for the real line that coincides with the euclidean topology. In fact, this construction can be carried over to any ...
4
votes
0answers
90 views

adjoints and duality

Let $X$ and $Y$ be posets seen as a category. Let $F:X \to Y$ and $G: Y \to X$ be an adjunction. That is, $(G \circ F) (x) \ge_X x$ and $(F \circ G) (y) \le_Y y$. Now, $F$ is additive and $G$ is ...
9
votes
2answers
102 views

Which of these constructions are left adjoints?

A groupoid can be regarded as a small category in which every arrow is an isomorphism A monoid can be regarded as a small category with only a single object A preorder can be regarded as a small ...
2
votes
0answers
66 views

Do subhomomorphisms / subfunctors have a standard name, and where can I learn more?

Sorry for all the mistakes in the original! I think they're mostly fixed now. Thank you for your patience. Part 1. If $A$ and $B$ are models in $\mathrm{Pos}$ of an algebraic signature $\sigma$, ...
1
vote
1answer
58 views

Adjoint of forgetful functor

I want to solve an exercise about adjoints (but I am not sure about the steps). Let $U:\textbf{CL}\rightarrow\textbf{Posets}$ be the forgetful functor. Here we have a functor from the complete ...
2
votes
1answer
57 views

Can the obvious “product” of complete atomistic Boolean algebras be realized as a categorial product?

Let $X$ and $Y$ denote sets, and $\eta_X,\eta_Y : X,Y \rightarrow X+Y$ denote the natural injections to the disjoint union. Then intuitively, the "product" of the Boolean algebras $2^X$ and $2^Y$ ...
2
votes
3answers
52 views

A notation for a morphism in a thin category

Consider a thin category with objects $A\leq B$. There exists a unique morphism $A\rightarrow B$. Is there a standard notation for this morphism (given $A$ and $B$)?
1
vote
1answer
50 views

Two definitions of a monovalued morphism

Let fix some dagger category every of Hom-sets of which is a complete lattice, and the dagger functor agrees with the lattice order. I define a morphism $f$ to be monovalued when $f\circ f^{-1}\le ...
9
votes
2answers
209 views

When does a left adjoint between Heyting algebras preserve 1?

This is an exercise from Johnstone's book Stone Spaces: Let $f: A \to B$ and $g: B \to A$ be order-preserving maps between complete Heyting algebras with $f$ left adjoint of $g$. Show that $f$ ...
3
votes
1answer
77 views

Where can I learn more about order-reflecting functions? And is the following result well-known?

I stumbled across a cute result about order-reflecting functions. A couple of questions: Is the following result well-known? I know lots of place to learn about order-preserving functions, but is ...
2
votes
0answers
58 views

Is there a categorial description of the Cantorian sum of posets?

Given two partially ordered sets $P$ and $Q$, we can form the product $P \times Q$ and the coproduct $P \oplus Q,$ and these have their predictable interpretations. However, we can also form what we ...
0
votes
0answers
69 views

Direct products in a partially ordered category

Consider a category, whose set of objects is a poset. Let $f=(f_0,f_1)$ is an indexed family of objects of this category. (Thus $f$ is a 2-indexed family, we can extend it to any index set in an ...
4
votes
0answers
75 views

Terminology for metric space with “anti-symmetric” distance

I'm interested in spaces that have a two-place function $d$ with non-negative real values, satisfying the following three conditions (for all $x$, $y$, $z$): $d(x, x) = 0$ $d(x, y) + d(y, z) \geq ...
8
votes
1answer
125 views

What kind of category is a cyclically ordered set?

Background: A preorder is a binary relation $\leq$ which is reflexive and transitive. We can write the transitive property as ${\leq}(a,b)\wedge{\leq}(b,c)\to{\leq}(a,c)$. There are additional axioms ...
4
votes
1answer
71 views

Presheaves over sieves and posets

I'm looking for a proof of the claim that given a poset P, the topos of presheaves over P is equivalent to the topos of presheaves over the complete Heyting algebra of sieves on elements of P. I found ...
2
votes
0answers
75 views

Are there any texts on order theory that treat it as decategorified category theory?

I read this post on nLab and now I want to learn some order theory. What are good texts that contain the above referenced topics, and are there any that are explicitly about order theory as ...
7
votes
2answers
103 views

What structure does the space of functions into $X$ (or the cartesian exponentiation of $X$) inherit from $X$?

When dealing with a space $X$, that posses a lot of structure (complete lattice, complete metric space, vector space), what can be said about the cartesian exponentiation $X^Y=\{f \mid f:Y\rightarrow ...
6
votes
1answer
92 views

A new(?) partial order on the set of continuous maps

Let $X,Y$ be topological spaces. Define a partial order on $\hom(Y,X)$ as follows: $f \leq g$ if $f^{-1}(U) \subseteq g^{-1}(U)$ for all open subsets $U \subseteq X$. Equivalently, $f(y)$ is a ...
1
vote
0answers
64 views

Relationships between zero morphisms and least morphisms

Zero morphism $0_{XY}$ is defined by the formulas $a\circ 0_{XY}=b\circ 0_{XY}$ and $0_{XY}\circ c= 0_{XY}\circ d$ for every morphisms $a$, $b$, $c$, $d$ of suitable sources and destinations. I ...
1
vote
0answers
33 views

Names of certain morphisms in Pos

Pos is the category of small posets and monotone maps. I call a morphism $f:\mathfrak{A}\rightarrow\mathfrak{B}$ of Pos monovalued iff it maps every atom of $\mathfrak{A}$ either into an atom of ...
-2
votes
1answer
119 views

Does an isomorphism induce an order isomorphism?

Let $\mathfrak{A}$ is a poset. For $a, b \in \mathfrak{A}$ we will denote $a \curlyvee b$ if only if there is a non-least element $c$ such that $c \leqslant a \wedge c \leqslant b$. Let ...
3
votes
1answer
125 views

Down-sets in posets and directed sets

Let P be a poset and let us say that a subset A of P is a down-set if: $$x \in A, y < x \implies y \in A.$$ A directed set is a poset P such that for every two elements, $a,b \in P$ we can find ...
5
votes
1answer
262 views

Need construction for coequalizer in $\mathbf{Poset}$

My question can be stated quickly: I would like to see a construction of the coequalizer of two arbitrary Poset morphisms (along with a proof of its correctness, of course). Thanks! (The ...
2
votes
2answers
90 views

Construction of a partial order on a quotient of a coproduct

[NB: Throughout this post, let the subscript $i$ range over the set $\unicode{x1D7DA} \equiv \{0, 1\}$.] Let $(Y, \leqslant)$ be a poset, and $X\subseteq Y$. Let $\iota_i$ be the canonical ...