All Questions
Tagged with lattice-orders category-theory
59
questions
3
votes
1
answer
88
views
Is there a connection between the topology of $\mathbb{R}/\mathbb{Q}$ and the logic fundamental to Smooth Infinitesimal Analysis?
As I read through the discussion related to this question (Visualizing quotient groups: $\mathbb{R/Q}$) posted 11 years ago I am reminded of a line on page 20 in a book by J. L. Bell, A Primer of ...
5
votes
0
answers
70
views
Lattices/Topology and the Stone Duality
For some context I have some partial understanding of lattices and an intermediate understanding of topology.
I at some point in the past week started thinking about a funny way to view a topology on ...
- 61
1
vote
0
answers
84
views
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
0
votes
0
answers
30
views
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 $U \neq \emptyset$,
$$ f( \bigvee U) = \bigvee_{u \in U} f(...
- 469
2
votes
1
answer
40
views
Seeking related work to cocomplete category with "compatibility relation"
I am looking for related work that matches a set-up as follows:
Consider a cocomplete category $\mathsf X$ (i.e. a category with all colimits), and a compatibility relation $\ast$ that is a functor
$$
...
- 538
0
votes
1
answer
65
views
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
3
votes
1
answer
133
views
left-adjoint to join in a Heyting algebra
Define a Heyting algebra to be a bounded lattice $L$ with an operation $\to : L^{op}\times L \to L$ such that for any $x, a, b \in L$ we have $x\wedge a \leqslant b$ iff $x \leqslant a \to b$. ...
- 11k
3
votes
1
answer
116
views
Right unitor in the monoidal category of sup-lattices
In short: I am trying to find a right unitor for the monoidal category $\mathsf{SupLat}$ of sup-lattices.
1. Preliminaries
Let $L,M$ be sup-lattices.
Denote by $L^*$ the set $L$ with order $l \leq_{L^*...
- 3,068
7
votes
1
answer
179
views
Is the category of sup-lattices rigid?
In An Extension of the Galois Theory of Grothendieck Joyal and Tierney show that the category $\mathsf{SupLat}$ of sup-lattices is star-autonomous. By listing compact closed categories and $\mathsf{...
- 3,068
3
votes
1
answer
90
views
Is a Hetying algebra with coexponetials Boolean?
Suppose we have a Heyting algebra $\mathcal A$ with coexponentials. Specifically, for every $a, b : \mathcal A$ we have an object $b \backslash a$ with the properties that $b \le a \lor (b \backslash ...
- 2,225
2
votes
2
answers
220
views
Left adjoint to the inclusion of Boolean algebras into distributive lattices
Let $\mathbf{Boole}$ be the category of Boolean algebras.
Let $\mathbf{BDL}$ be the category of bounded distributive lattices.
There is a fully faithful functor ${\mathbf{Boole} \rightarrow \mathbf{...
- 279
1
vote
1
answer
148
views
What is a relation of category of complete sublattices to category of lattices?
What is a relation of category of complete sublattices to category of lattices?
I haven´t found much about a category of lattices, but I assume objects = lattices, morphisms = lattice homomorphisms.
...
- 2,452
0
votes
1
answer
66
views
Product structure in Frame [closed]
Is there a well known product structure for Frames? I.e. if $L$ and $L^{\prime}$ are frames, is there a product object in $\mathbf{Frm}$ category isomorphic to an object with underlying set the set ...
- 97
1
vote
1
answer
79
views
What are the subfunctors of $\operatorname{Hom}_{\Bbb Q}(q,-)$
Let $\Bbb Q$ be the linearly ordered set of rational numbers, and $q\in\Bbb Q$ . I think the functor $\operatorname{Hom}_{\Bbb Q}(q,-)$ is in bijection with $A=\{t : t\in \Bbb Q, t>q\}$. I want to ...
- 859
10
votes
1
answer
1k
views
How Do Heyting Algebras Relate To Logic?
My question is, broadly speaking, how are Heyting algebras related to logic ? It would be great if someone could answer this question without being too technical (or point to easy to read literature). ...
- 1,659