Tagged Questions

6
votes
2answers
82 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 ...
5
votes
1answer
80 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 ...
2
votes
0answers
41 views

Is there any way to define morphisms between filters in order to get a category, one which its opposit category would be the category of ideals?

Its well known that filters and ideals are dual. I would like to see how to express this fact "Categorically". I would be very thankful if someone could help me with that.
1
vote
0answers
45 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
31 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 ...
-3
votes
1answer
87 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
59 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 ...
4
votes
1answer
149 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
77 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 ...