4
votes
1answer
172 views

A reflective subcategory of the category of inverse semigroups.

The Question. I'm reading Lawson's "Inverse Semigroups: The Theory of Partial Symmetries" and I've hit something I don't understand. It's claimed on page 34 of my copy that The category of ...
2
votes
0answers
152 views

Profinite completion of a partial order

In Johnstone's Stone Spaces it is proved that the category of profinite partial orders is (equivalent to) the category of ordered Stone spaces (also called Priestley spaces) and that the obvious ...
0
votes
1answer
36 views

Left & right adjoints in the context of complete lattices.

This is a follow-up question from this question of mine. In the same paper as the one mentioned in my previous post, it's stated that In the context of complete lattices, a monotone map has a ...
1
vote
1answer
48 views

Left & right adjoints in the context of posets.

Definition 1: A function $\theta: X\to Y$ between posets is monotone if whenever $x\le y$, we have $\theta (x)\le\theta (y)$. Definition 2: For any pair $f:A\to B$ and $g:B\to A$ of monotone maps, we ...
3
votes
0answers
25 views

(Almost) co-free objects?

another naming question from me, which comes up because I try to use category theory as a compass in developing some (new or not?) order-theoretic notions. According to my book (The Joy of Cats, ...
4
votes
1answer
74 views

Do any familiar adjunctions arise from this construction?

I was pondering an answer that user Andreas Blass provided to an old question of mine and wondered if the following construction cropped up anywhere other than the example I will provide. For any set ...
5
votes
1answer
118 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
3answers
105 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
94 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 ...
11
votes
2answers
126 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
70 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
67 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
60 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
54 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
52 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
213 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
82 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 ...
0
votes
0answers
74 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 ...
5
votes
1answer
97 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
133 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
89 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 ...
3
votes
0answers
76 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
106 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
96 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
35 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
121 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
134 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
295 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
96 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 ...