From the definition of directed topological space it follows that the unit circle with $2n$ circumference clockwise paths ($n\in\mathbb{N}$) is a d-space.

This d-space is “nonlocal” that is not determined by small fragments of the path.

“Regular” clockwise circle with $n$ circumference clockwise paths ($n\in\mathbb{N}$) is a d-space too. And this one is “local”.

I ask you to help me define “locality” or “nonlocality” of d-spaces. What is the definition and how is it called?

]]>i would like to ask about categorical properties( like cartesian closed category,colimit,limit etc.) of DCPO category ( objects : directed complete partial orders & morphism: scott continuous function).which material would you recommend me in order to find these properties with proof???

secondly, is DCPO→SET functor a topological functor (in sense of Brümmer)?? how can i search whether DCPO→SET is a topological functor or not??? ]]>

Hi,

I have seen lots of physics going up in recent months and so I thought I would share what I have been working on. The following is an attempt to make categorical structures look super primitive.

If we take a light switch to embody an entire category, we could take the light switch to be a set with two elements and the morphisms are all endofunctions. Let’s say, for fun, that we define the endofunctor for the monad as:

flip switch up $\rightarrow$ light turns on

flip switch down $\rightarrow$ light turns off

flip switch $\rightarrow$ light toggles

do nothing to switch $\rightarrow$ light does nothing

This looks like the identity endofunctor. Now, this endofunctor, in my mind, is deeply fundamental as it is used to test a causal relation between things like the light and the light-switch. The monad is nothing but the identity monad and so, I think, the algebra is nothing but an identity element. (I already asked at mathematics stack). One normally looks at this kind of thing as passing a signal from one system to another and this then goes up to information theory. If you have read my post correctly, though, you will see that I am trying to lift that whole idea up to where we talk only about morphisms and causal structure as opposed to systems of state and the information that encode them. It was a let down to find that the algebra was this trivial for such an important bit of behaviour, one that every physicist working in a lab will use every day.

Can anyone take this thinking and get the first non-trivial algebra (it should be TINY!!!) and keep the spirit of “behaviours in a laboratory”? The co-algebra is also interesting.

If anyone is wondering where this is coming from, consider the fact that one can construct a TQFT entirely within FDHilb by replacing the usual category of cobordisms with the internal category of comonoids. Thus, the background becomes the internal category of classical structures. The category of internal comonoids is defined with axioms that look like the copying and deleting of information. If you read this post carefully you will see that I am abstracting this idea to replace the category of internal comonoids with just comonads.

]]>Stub for quotient space.

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