# Tagged Questions

Various structures are studied in category theory using properties of objects and morphisms between them. Many constructions are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category ...

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### What exactly are the meaning of the followings in the definition of a category?

In Awodey's Category Theory a category is defined as follows. A category consists of the following data, Objects: $A, B, C,\ldots$ Arrows: $f,g,h,\ldots$ For each arrow $f$ there are ...
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### category-theory, right group action

Let $G$ be a group. We observe the category $(Set)_G$ of right group actions. a) Let $X\times G\to X$ and $Y\times G\to Y$ be two transitive right group actions with $x\in X$ and $y\in Y$. Find a ...
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### Corollaries of the Yoneda Lemma in Analysis?

I am looking for some simple examples of how the Yoneda Lemma can be applied in analysis and probability theory and related fields. A simple candidate example that I can think of and somewhat ...
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### Universal property, localization of rings and modules, and initial element in a category

Sorry for the confusing title. I just started learning category theory and am very confused about the concept "universal property". I am not even sure whether my "proof" is a proof or is just a ...
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### How to construct a G-extension of a category C?

Note: I'm a physicist so this will be phrased somewhat in physics language. Suppose we have a unitary modular tensor category $\mathcal{C}$. In physics language, we can think of $\mathcal{C}$ as ...
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### Frobenius-Perron dimension on a fusion category

Let $C$ be a fusion category with simple objects $V_i\in I$. The fusion rule is $V_i\otimes V_j \cong N_{i,j}^k V_k$. The Frobenius-Perron dimension of a simple object $V_i$, $\mathrm{FPdim}(i)$, is ...
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### Is Tambara-Yamagami category admits a braiding when G is a nonabelian group?

Tambara-Yamagami category is a fusion category which its simple objects are elements of a group and one element out of group. i.e : $$simple\;objects = G \cup \{m\}$$ The fusion rule of this ...
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### Three different definitions of Modular Tensor Categories

I have found three different definitions of Modular Tensor Categories. I want to know if anybody can give a sketch of proof for their equivalences (some parts are easy of course) A Modular Tensor ...
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### When is an object in a linear or abelian category simple? Or: How should I define fusion categories?

I'm confused about the notion of simple objects. Now ncatlab says that an object is simple in an abelian category if it only has itself and 0 as subobjects. On another page, it says that the simple ...
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### Constructing a coreflection functor from its components

Let $\mathbf{A}$ be a coreflective subcategory of $\mathbf{B}$ and for all $B$, $A_B\xrightarrow{c_B}B$ an $\mathbf{A}$-coreflection. This is $\forall$ $\mathbf{B}$-objects $B$. I claim that there ...
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### Universal object

I'm trying to define the concept of universal object in category theory, but I failed to find a universal way to do it. Maybe someone can help ? THe idea is quite simple. Let $\mathcal{C}$ be a ...
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### Show that for each of the following graphs G there exists up to isomorphism precisely one category A with G(A) = G.

I was working through the exercises in Abstract and Concrete Categories: The Joy of Cats (http://katmat.math.uni-bremen.de/acc/acc.pdf) and I was stuck on exercise 3A.(d). It seems to me that the ...
Let $\mathsf{C}$ be a category. We call $\mathsf{C}$ binormal if it has a null object, has all equilizers and coequilizers, all monomorphisms are kernels and all epimorphisms are cokernels (whereby a ...
In the HoTT book the type of functions $A\to C$ construction is described first and the product type $A\times B$ construction later, using function types in its definition. So my obvious naive ...