1
vote
1answer
111 views

Directed subposet of a poset containing the minimal elements

The following appears naturally in a certain context: Let $P$ be a partially ordered set which is bounded below in the sense that for each $x\in P$ there is a minimal element $m$ with $m\leq x$. Let ...
-2
votes
0answers
146 views

Does this “modeling relationship” occur in mathematics (Galois connections, relation algebra, category theory)? [on hold]

Be $R$ a subset of the cartesian product $S \times T$ of two sets, and $f: S \to T, \ g: T \to S$ two partial functions, such that $f$ and $g^{-1}$ are disjoint as sets of ordered pairs, and $R = (f ...
6
votes
1answer
189 views

The category of categories and adjunctions

What is known about the category that has small categories as objects and adjunctions as morphisms? Obviously, it has neither terminal nor initial objects. But what about other kinds of limits? Are ...
3
votes
1answer
145 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 ...
2
votes
0answers
55 views

A categorical analogue of Debreu's independent factors theorem

Background A major question in Decision Theory is that of the cardinal meaning of a utility function. That is, given a set $X$, a utility function $u:X\rightarrow \mathbb{R}$ represents the choices ...
7
votes
1answer
266 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 ...
6
votes
1answer
199 views

Terminology question for poset maps

Is there a standard name for order-preserving maps $f\colon P\to Q$ of posets with the property that the image of a lower set is a lower set, or equivalently if $q\leq f(p)$ then there exists $p'\leq ...
7
votes
1answer
295 views

What should the morphisms in the Category of Directed Sets be?

Directed sets are defined to be sets equipped with a preorder that admit (finitary) upper bounds e.g. pairs $(D, \preceq)$ such that $\forall p,q \in D$ there exists $r \in D$ such that $p \preceq r$ ...
9
votes
1answer
302 views

Is there a natural measurable structure on the $\sigma$-algebra of a measurable space?

Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space? Here is one natural ...
6
votes
1answer
499 views

A categorical characterization of the lexicographic order

In $Pos$ (the category of partial ordered sets and order preserving maps) there is the categorical product of two objects, but on the set product there is (naturally) also the lexicographic order. I ...
3
votes
2answers
557 views

Is it reasonable to define `poset homotopy' as a `natural transformation of posets'?

Pausing to speak metaphorically for a bit, I have always thought of partial orders as being something like a simplified version of topological spaces. (In the finite case at least, all topological ...
1
vote
0answers
298 views

Functoriality of a standard integral domain construction.

The evident forgetful functor from fields to integral domains has a left adjoint, namely the construction of the quotient field for a given integral domain. Another standard construction is taking the ...
3
votes
0answers
192 views

An elegant formulation for typed sets

Fix a poset $T$, which we'll think of as a set of "types," interpreting $a \leq b$ as "$a$ is more general than $b$." Construct a category of TSet as follows. Objects: Pairs ($X$, $\tau : X ...
3
votes
2answers
194 views

Finite categories and partial orders

I'm studying category theory for the first time in a very succint book for computer scientists (I'm not actually a computer scientist, I'm a physicist, but my interest in cat theory is related to ...