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Hi,
I am new, and just put together a short post related to subobject classifiers of topoi. I am curious what connection subobject classifiers of topoi have to generalizations of the classical bit of information, and whether anyone knows of any work on constructing a topos in which the quantum bit is a subject classifier. Any thoughts, or references to research articles, are most appreciated. Thanks in advance for any knowledge you may provide!
Sincerely,
Edwin Schasteen
This sort of discussion probably works better on the Forum itself than as a post in the Lab. (But if the discussion generates results, then those could go in the Lab.) I would encourage you to copy and past your post from the Lab to the Forum here, where I expect that you’ll get a better response.
Hi Edwin Schasteen,
the question whether there might be a topos whose internal logic is related to quantum logic is an old one, but people are still trying out various possibilities.
One idea that has found some attention is that of Bohr toposes. However, their subobject classifiers are not related to the qbits that you are after, but consist of collections of “classical contexts” in which a given quantum observable has a given value. You might enjoy the comments on quantum logic in the introduction of this article.
Then concerning your writeup: I was a bit surprised after the first sentences to find the rest be a discussion of the case that the subobject classifier is trivial. It is trivial precisely only in the trivial topos.
Hi TobyBartels,
Thank you for your response! I will copy the post below as suggested.
Urs,
Thank you for the links. I started reading both the article on Quantum logic, and the link to Bohr toposes. Also, thanks for reviewing my writeup. I was not sure if the trivial subobject classifier is trivial only in the trivial topos: that is useful to know. Much of the article then would be valid only for the trivial topos, and that is not very interesting.
I am somewhat new to category theory, and topos theory. My training is in Applied Mathematics, and I had no formal exposure to category theory in my coursework, but the following work by Dr. Olivia Caramello convinced me that category theory, and Topos Theory, may be of use in Applied Mathematics. I think that category theory is sort of underutilized in my field, so my overall (long term) goal is to explore the uses of Topos Theory in the development of techniques in Applied Mathematics to solve various problems related to that field. Because of the various recent works using topos theory to attempt to make headway toward a theory of quantum gravity, I figured that would be a good area to explore. In any case, right now I am in the process of learning about Sheaves and the Presheaf Topos by bouncing back and forth between the section on Sheaves and Presheaf Toposes in Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Mac Lane and Moerdijk.
Your mention that the subobject classifier is trivial only the trivial topos did bring up an idea: but I do not know whether this idea makes any sense or if it will lead to anything useful.
If we consider the topological space the real line $R$ with open sets given by the open intervals $(a,b)$ with $a$ and $b$ real numbers with $a \leq b$ then consider the map $\lim_{t \to 0}(t*a,t*b)=\empty$. For any $t$>$0$, with $t$ fixed, if we were to consider $Sh(R,t)$ the category of Sheaves on $R$ at the time $t$, then according to Saunders and Mac Lane [1], this would form a topos. I think the topos would be non-trivial if $t$ is fixed and $t$>$0$. If we were to take $\lim_{t \to 0} Sh(t*a,t*b)$ for each bounded open subset of $R$, I am wondering if this would constitute a functor from the $Sh(R)$ the topos Sheaves on R to the trivial Topos? The reason I am interested, is this seems like it might be a very oversimplified model of the Big Bang singularity where the spatial part of the universe is modeled as the one-dimensional real line.
A slightly different thought related to sieves: one thing mentioned in [1], is that sieves are downward closed sets, and [2] mentioned that the notion of a filter is opposite, or the dual of, the notion of a sieve. If we have a physical system that exhibits two different theories valid on two different length scales, wouldn’t a sieve be useless since downward closure implies that the open sets in a sieve may be made arbitrarily small; whereas, the behaviour of interest may only be valid down to a certain finite length scale? Since a filter is an upward closed set, might filters be useful for studying systems with a smallest length scale?
I am definitely new at category theory, so if I make any obvious novice errors, please feel free to let me know, as I am very open to, and appreciate, feedback. Again, I appreciate the links above, and will continue to work through them to gain some more insight!
Sincerely,
Edwin Schasteen
References:
[1] Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Mac Lane and Moerdijk
[2] Sites and Sheaves Mainline 1, by mimrir via the following youtube video
I am attaching the post from the nlab here.
In the topos Sets, the sub object classifier $\Omega$ is any two element set according to [1], and which, according to [2], also is a classical bit of information. Since the classical bit underlies classical physics according to [2] and the classical bit is a sub object classifier in the topos Sets [1], then it seems natural to ask whether the quantum bit is a sub object classifier in some suitable topos. I do not know the answer to this question, but if the answer is yes, then sub object classifiers of topoi would be very worthy of intense (and I think consistent with the overall $n$POV) study in order to try to get a deeper understanding of the nature of information, in its various forms.
I am not sure if any of this is useful, or belongs here, and so if this is out of place, I apologize.
The following is an observation noted about sub object classifiers of a Topos.
Let $\Eta$ be a topos, $\Omega$, $1$ the sub object classifier and terminal object in $\Eta$, respectively. Let $truth$:$1 \to \Omega$ be the universal monic from $1$ to $\Omega$ in $\Eta$. Let $\delta$ be the unique morphism from $\Omega \to 1$ in $\Eta$. Then since $1$ is the terminal object in $\Eta$, $\delta \circ truth$ is equal to the identity morphism on $1$. If $truth \circ \delta$ were equal to the identity on the sub classifier $\Omega$, then $\delta \circ truth$ together with $truth \circ \delta$ would define an isomorphism from $\Omega \to 1$ in $\Eta$. Thus, the only thing that keeps the sub object classifier from being isomorphic to the terminal object $\Eta$ is the failure of $truth \circ \delta$ to be equal to the identity morphism on $\Omega$.
Theorem: Let $\Eta$ be a topos, then the sub object classifier $\Omega$ of $\Eta$ is isomorphic to the terminal object $1$ in $\Eta$ iff the unique arrow $\delta$:$\Omega \to 1$ in $\Eta$ is a monic.
Proof: If $\delta$:$\Omega \to 1$ in $\Eta$ is an isomorphism, then by [3], $\delta$ is both monic and epic, and hence $\delta$:$\Omega \to 1$ is a mono $\Eta$. The proof of the converse is given in Theorem 1 of [4].
For a locally small topos $\Eta$, the failure of $\delta$:$\Omega \to 1$ in $\Eta$ to be monic requires there to exist at least one object A in $\Eta$ such that $\Eta$ $(A,\Omega)$ has at least two distinct elements. For otherwise, $\Omega \in \Eta$ would be a terminal object (since there always exists at least one arrow from any object $A$ in $\Eta$ to $\Omega$ given by $truth \circ \delta_{A}$, where $\delta_{A}$ is the unique arrow from $A$ to the terminal object 1 in $\Eta$ and $truth$ is the universal monic from the terminal object 1 to the sub object classifier $\Omega$ in $\Eta$ ), and since terminal objects are unique up to unique isomorphism [1], this would result in $\Omega$ being isomorphic to 1 in $\Eta$.
In Sets, the failure of the unique arrow from the two element set to the one element set in Sets encodes the notion that a two element set is too large to be a subset of a one element set. Or, if we think of the two element set as one bit of information, and the one element set as a state space with a single state in it, then the failure of the unique arrow from the two element set to the one element set in Sets encodes the notion that for a single classical bit cannot (or is somehow to big to) encode a single classical state, as one classical bit encodes exactly two distinct states. So, at least in Sets, the unique arrow from the two element set to the one element set not being monic encodes the notion that the sub object classifier in Sets encodes the distinction between two distinct objects, the distinct objects being the elements of the two element set (and distinguishes objects in Sets in a way that the singleton cannot). Do toposes containing a sub object classifier that is not isomorphic to terminal object in the respective topoi also encode the notion of that sub object classifier being able to distinguish distinct objects in the respective topoi?
Any thoughts, or modifications and improvements to this article are most welcome, and appreciated!
*[1] Saunders Mac Lane & Ieke Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory.
*[2] Professor Leonard Susskind, Understanding Modern Physics Video(Video Lecture).
*[3] Dr Bartosz Klin, (pdf).
*[4] Edwin Schasteen, Mathematical Conflict Theory Development: A Topos Theory Perspective (Hubpages).
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