Questions tagged [category-theory]

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Is there an operadic structure to classical mechanics?

Consider a closed system of several particles in motion with a certain eom (equation of motion). We can replace each of these particles with another closed system and produce another classical ...
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3 answers
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What is the name of this construction of quantum things?

Let $E$ be a set, and let $\mathcal{F}$ be a set of maps $E \rightarrow \mathbb{R}$ that "carries all the information on $E$", that is, the map $x \in E \mapsto (f(x))_{f \in \mathcal{F}}$ ...
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What is the current status of 2d and 4d Yang-Mills as a functorial QFT?

I am looking for the current status of YM, in specific 2d and 4d Yang-Mills, described as a functor. As far as I understand the issue resides in defining the propagator functorially but I have not ...
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3 votes
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Excitations of the string-net Hamiltonian

A quite general 2D topological order can be constructed through the string-net theory. Here, if the input data is some braided fusion category $C$ (i.e. the $F$-symbols), the elementary excitations ...
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Excitations and representation categories

Apologies if this question is a little vague — I'm fuzzy on the details, which is why I ask. In quantum mechanics, we commonly encounter the idea that types of particle-like charges or excitations ...
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Topological Quantum Field Theory with Symmetries and Knot Quandles

It is well known that Chern-Simons theory provides an intrinsically three dimensional way to compute knot invariants like the Jones Polynomial. 3D TQFTs also have an algebraic description in terms of ...
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Physical interpretation of TFTs

1. Defining TFTs Let $n$ be a positive integer and $\mathbb k$ be a field. In my lecture I was introduced to TFTs using the following definition going back to Atiyah (around 1988): A $n$-dimensional, ...
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Hopf algebras vs Fusion categories for topological order

Disclaimer: Before I begin with the question I want to warn that some people would argue that it is a math question and not a physics question. However, it finds it origins in the study of topological ...
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Can branes in string theory be described by different systems of mathematics or logic?

Mathematically, in string theory, branes can be described using the notion of a category and the mathematical category theory says that logic can change from one category to another. We can build ...
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Where does Category theory appear in physics? [closed]

I've been curious about the concept of categories in physics and tried finding some more information about the subject. There are lots of textbooks about category theory, some from mathematicians and ...
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Anyons under weaker assumptions?

I. Duality assumption In Anyons in an exactly solved model and beyond p.74, Kitaev says, "We will see that for theories with particle-antiparticle duality, condition 3 can be dispensed with.&...
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SUSY Loop diagrams from a categorical viewpoint

In the paper "A Prehistory of $n$-Categorical Physics" J. Baez and A. Lauda give an account of the use of category theory throughout physics. In section “Penrose (1971)” starting from page 25 they ...
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How are local observables encoded in this formulation of quantum field theory?

I've recently begun trying to understand a formulation of quantum field theory as a functor from a category of spacetimes-with-boundaries (bordisms) to a category of Hilbert spaces, as reviewed in ...
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Math of anyons: Quantum dimension of 1 implies abelian charge

This question originates from the following statement in Bonderson's thesis: Link to Thesis page 16 or pdf-page 23: The quantum dimension $d_a$ of an anyon of charge $a$ satisfies $d_a \geq 1$ with ...
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The category of physical instuments [closed]

A physical instrument is used to measure physical systems and return the values for those measurements in classical data. I was wondering if anyone here has been looking at instruments in terms of ...
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Is every fusion category with unitary $S$ a unitary modular tensor category (UMTC)?

A tensor category whose solutions to the pentagon/hexagon equations are unitary and whose braiding is nondegenerate is called a unitary modular tensor category (UMTC). When trying to find UMTCs, ...
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Experiments are monads (I think), but are they comonads too?

Quantum theory is equally an epistemic (ie about information) and ontic theory (see "reality of the wavefunction" on Google Scholar). My question is about a theory that aligns with this, ie, ...
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Open conjectures on the Fukaya category

This is ported from mathoverflow question https://mathoverflow.net/q/263693/. Can somebody give examples to open conjectures on the behavior of the $Fuk(M,ω)$ (that's the "mathematical A-model ...
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Alternative quantization of quantum electrodynamics?

A Quantum field theory is determined, if a Hilbert space Basis with Operators acting on it (such that one element of an Hilbert space is also an element of the same Hilbert space if an Operator acting ...
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On Seiberg-Witten theory in 3d and 4d

According to Gukov et al. in this 2017 paper Seiberg-Witten theory in 4d categorifies Seiberg-Witten theory in 3d. In what sense is this phrase mentioned? I know what the process of categorification ...
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3 answers
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Grassmann numbers & supermanifolds

I'm asking this question because I'm currently trying to learn about Super Symmetry but I'm having trouble understanding the concept of super-space and super-manifold. I read that in super-spaces you ...
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7 votes
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Higher category theory, renormalization, and non-perturbative QFTs

I'm (vaguely) aware of certain uses of higher category theory in attempts to mathematically understand quantum field theories -- for example, Lurie's work on eTQFTs, the recent-ish book by Paugam, and ...
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Reference Requests: The Turaev-Viro model and Topological QFT's

I've been inspired by the string-net condensation paper by Levin and Wen (see here) to learn about the Turaev-Viro model and its relationship to topological QFTs. Any references that could be helpful ...
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Looking for the Category of Quantum Systems

I've recently written a comment in a different forum to a question about how to implement infinite potentials in quantum mechanics, that is things like a non-relativistic particle in an infinite ...
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Braided Monoidal categories for quantum gravity

We have seen that symmetric monoidal categories capture quantum theory and quantum computing quite well. Do braided monoidal categories capture any aspects of gravitation, or are they used in quantum ...
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9 votes
2 answers
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Why are topological phases described by modular tensor categories?

After some reading, I have an inuitive idea what topological phases of matter are. But where is the connection to modular tensor categories? Is there fundamental literature where this is covered? ...
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2 answers
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Implementing Category Theory in General Relativity

I was thinking if it may be possible to implement category theory in general relativity. I don't mean writing simply in terms of categories, but actual fundamental ideas (i.e. physics of the theory ...
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1 vote
1 answer
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When will Einstein's theories become laws? [closed]

Einstein theories , specifically relativity, have been fascinating us for around 100 years yet with all the real and actual evidence of its validity we still consider it a "theory"..... How much more ...
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What is the relationship between a brane, a manifold, and a space?

I've read many ways to define manifold; one way is to define it as a type of mathematical space (a type of topological space to be exact). All of the definitions that I've seen for brane, on the ...
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7 votes
2 answers
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Gauge theory for mathematicians?

I'm looking for a textbook or set of lecture notes on gauge theory for mathematicians that assumes only minimal background in physics. I'd prefer a text that uses more sophisticated mathematical ...
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6j symbols with Majorana indices

The Levin-Wen model is a Hamiltonian formulation of Turaev-Viro (2 + 1)d TQFTs. It can be constructed from a unitary fusion category $\mathcal{C}$, which can be equivalently defined using $6j$ symbols:...
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4 votes
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Rigorous definition of superselection sector/quasiparticle type in anyon systems

The systems I have in mind are for example Kitaev's toric code model (arXiv:quant-ph/9707021) and Kitaev's honeycomb model (arXiv:cond-mat/0506438). What I'm looking for is a mathematically rigorous ...
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Does model theory in mathematics have any usefulness in the modeling of physical systems?

The term Model Theory in mathematics seems to have a somewhat precise definition here. Reading through that reference you'll largely see discussions strictly relating to mathematical concepts, but one ...
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Dilemma: Fusion space from a direct sum of anyons or NOT

In Preskill's note, 9.1.2 in page 44, concerning the fusion space, it states that: The fusion rules of the model specify the possible values of the total charge $c$ when the constituents have ...
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2 votes
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What insights does category theory offer in terms of grand unified theories?

What insights does category theory offer in terms of grand unified theories? Any references to books or papers that give categorical descriptions of any of the common grand unified theories would be ...
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Topological theta term as a topological quantum field theory?

It is well known that the theta term $\int d^4x\frac{\theta}{4\pi}Tr[F\wedge F]=\int d^4x\frac{\theta}{4\pi}\epsilon_{\mu\nu\sigma\lambda}Tr[F^{\mu\nu}F^{\sigma\lambda}]$ is a topological term, ...
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What is a modular tensor category / functor?

I have reads several answers here about this notion, especially regarding topological order, see e.g. this answer, but this notion sounds completely new for me. Also, I found nothing really helpful on ...
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About category theory and physics [duplicate]

Could the ideas of category theory be applied to Physics, maybe simplifying how algebraic topology and sheaf theory and other hard-to-explain subjects are used in physics?
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6 votes
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target category of extended field theory

For a topological field theory to be a true “extension” of an Atiyah-Segal theory, the top two levels of its target (ie its $(n-1)^{\text{st}}$ loop space) must look like $\text{Vect}$. What other (...
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11 votes
3 answers
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TQFT associates a category to a manifold

Any 3d TQFT (topological-quantum-field-theory) associates a number to a closed oriented 3-manifold, a vector space to a Riemann surface, a category to a circle, and a 2-category to a point. This is ...
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17 votes
3 answers
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about the Atiyah-Segal axioms on topological quantum field theory

Trying to go through the page on Topological quantum field theory - The original Atiyah-Segal axioms - "Let $\Lambda$ be a commutative ring with 1, Atiyah originally proposed the axioms of a ...
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7 votes
1 answer
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Are identity types interpreted physically in an infinity-topos formulation of equations of motion?

In reference to Urs Schreibers paper/book on foundations of field theory Differential cohomology in a cohesive infinity-topos I wonder: are identity types there used "only" for the computations, or ...
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1 answer
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Etale bundles and sheaves

Before answering, please see our policy on resource recommendation questions. Please try to give substantial answers that detail the style, content, and prerequisites of the book or paper (or ...
14 votes
1 answer
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How algebraic geometry and motives appears in physics?

First, I'm not a physicist so I have just a little background in physics. I have been reading some noncommutative geometry books and papers (Connes, Rosenberg, Kontsevich etc) and a lot of high ...
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Quantization as a functor [duplicate]

Can anyone give an mathematical elaboration of the following statement: Quantization is a functor carrying the category of Hilbert space and linear maps to that of Symplectic manifolds satisfying ...
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1 vote
2 answers
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Studying the logical structure of physics as a mathematical object per se? [closed]

I was wondering is there a branch of mathematical physics which studies the underlying logical structure of physics as a mathematical object per se? Let me explain what I mean by that. I'm ...
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5 votes
2 answers
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Does the mathematics of physics require impure set theory?

Suppose for the sake of this question that all mathematics is ultimately reducible to set theory in such a way that the only mathematical objects there really are, are sets. Now, there is a common ...
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Can quantum field theory be seen as an epistemic restriction on (quantum) causal structure

Suppose we take Vicary's quantum harmonic oscilator as a kind of "toy quantum field theory". Next, take the category of internal comonoids to not represent the background causal structure. We ...
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2 votes
1 answer
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The (co)algebra for the (co)monad of a light switch

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 ...
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7 votes
2 answers
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Could motives aid in the study of the Navier-Stokes equations?

Recently, mathematicians and theoretical physicists have been studying Quantum Field Theory (and renormalization in particular) by means of abstract geometrical objects called motives. Amongst these ...
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