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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11 views

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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2answers
27 views

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 ...
6
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2answers
170 views

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 ...
1
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1answer
45 views

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 ...
3
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1answer
28 views

Left adjoint for the “strings category” functor

Let $\mathbf{Cat}$ be the category of small categories and let $\mathbf{sCat}$ denote the category of simplicial objects in $\mathbf{Cat}$. We have a functor $$ \text{str}\colon \mathbf{Cat}\...
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70 views
+50

representation of a group and its center

Let $G$ be a finite group and let $Z(G)$ be its center. Let $C=\mathrm{Rep}(G)$ be the category of finite dimensional representation of $G$. Let $D$ be the fusion subcategory of $C$ generated by $V \...
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2answers
146 views

What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?

For any topological space $X$, define a functor $\times X:\mathbf{Top}\to\mathbf{Top}$ by $Y\mapsto Y\times X$ (and acting on the hom-sets in the natural way). I know that if $X$ is locally compact, ...
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3answers
263 views

What is the product and coproduct of Morphism category (Arrow category)?

Given a category C, its morphism category D means a category that has 1) "morphisms of C" as its objects 2) "pairs (f,g) s.t. the diagram (square) commutes" as its morphisms. The above definition ...
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3answers
18 views

Lift a morphism onto the category theoretical product

For three sets $X, Y, Z$, we can lift a morphism $f:Y\to Z$ onto the cartesian product with $X$, such that we get a morphism $\bar f:X\times Y\to X\times Z$ (obviously $\bar f(x,y)=(x,f(y))$). How ...
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1answer
20 views

Finite product exists implies finite coproduce exist.

Let $C$ be a category such that the law composition of morphisms is bilinear, and there exists a zero object $0$, and the products exists for arbitrary finite sets of objects of $C$. Then the ...
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0answers
46 views

Chain Homotopy in abelian category

When dealing with complexes of modules or groups, the following lemma is pretty easy: If $f,g :E\rightarrow E'$ are homotopic, i.e. $f-g=d'h+hd$ for some h, then $f,g$ induce the same homomorphism ...
2
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1answer
34 views

A question about the definition of adjoint functors

Aluffi's "Chapter 0", on pg. 492, says the following: Let $C$ and $D$ be categories, and let $\mathcal{F}:C\to D, \mathcal{G}:D\to C$ be functors. We say that $\mathcal{F}$ and $\mathcal{G}$ are ...
1
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1answer
16 views

homology theory for $C^*$-algebras, map is natural wrt morphisms of short ecact sequences

I want to assure me if I understand the part of the exactness axiom that "$\delta$ is natural" corretly and if not, then my question is: what does it mean? The setting is the following definition: ...
2
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0answers
55 views

Homotopy Colimit of Truncations

Let $\mathcal{A}$ be an additive category with countable coproducts. I am just starting to learn about homotopy colimits and I am struggling with the following example that I am very interested in ...
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1answer
49 views

Meaning of functorial

It's known that for a short exact sequence of complexes, $0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0$, it associates a homology sequences $...\rightarrow H(E')\rightarrow H(E)\rightarrow ...
2
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1answer
45 views

A question about monoidal categories

I am learning about monoidal categories and I am a bit confused about the following: Suppose $(A,\otimes,I)$ is a monoidal category. What can be said about the opposite $A^{\text{op}}$? Is it ...
2
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2answers
38 views

About the minimal equivalence relation identifying some points.

I am solving a problem where I have a set $X$ together with a subset of elements that I want to identify. To do this I consider the minimal equivalence relation identifying these points. I have a ...
4
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2answers
139 views

How can you actually do universal algebra with monads?

Instead of digging deep into "classical" universal algebra, it seems more interesting or fruitful to look at universal algebra categorically. This should be doable with monads, since every category of ...
2
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0answers
28 views

Show that the law of the excluded middle does not hold in a BCCC

I want to show that the law of the excluded middle do not hold in a bicartesian closed category (BCCC), interpreted as follows: In general, there need not be a morphism $1 \to A + 0^A$ for $A \in \...
2
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2answers
79 views

$\text{Ext}_R^n(\bigoplus_{i\in I} M_i, N) = \prod_{i\in I}\text{Ext}_R^n(M_i,N)$

Let $(M_i)_{i\in I}$ be a collection of $R$-moduls. Show that for all $N\in \text{Ob}(_R\text{Mod})$ is $$ \text{Ext}_R^n(\bigoplus_{i\in I} M_i, N) = \prod_{i\in I}\text{Ext}_R^n(M_i,N). $$ My ...
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0answers
77 views

Category whose objects are subsets of A with (some) morphisms as subset or superset proofs [on hold]

In the process of trying to solve some other problem I found myself constructing the following category, which seems a little baroque but quite interesting. I'm wondering if this example is known and ...
2
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1answer
32 views

Recursive definition in kernel of a morphism.

I find the following definition in Serge Lang's Algebra on p 133: Given a morphism $f:E\rightarrow F$ in an additive category, we define a kernel of $f$ to be a morphism $E'\rightarrow E$ such ...
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1answer
35 views

Completing a Category definition with Nodes as sets and Arrows as a triples of a Set and two functions

I would like to study a category that: Objects: are (finite) Sets. Arrows: are triples of the form $(A, src:A\rightarrow B,trg:A\rightarrow C)$, such that A, B, C are sets and src and trg are ...
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1answer
79 views

Categorical semantics explained – what is an interpretation?

I’ve been really having a hard time trying to understand categorical semantics. In fact, I am confused to the point I am afraid I don't know how to ask this question! I’ve been reading textbooks like ...
2
votes
1answer
67 views

Graphic intuition for generalizing to weighted limits

One of the ways to define a limit of a functor $F:\mathsf C\longrightarrow\mathsf D$ is a representation of $\mathsf{Nat}(\Delta-,F)$. Along the journey of generalization to the enriched setting, one ...
7
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1answer
72 views

Is there a category theory notion of the image of an axiom or predicate under a functor?

Let me first state that I am a category theory novice so your patience is appreciated. I might be making some very basic conceptional mistakes and I might just need a simpler language to do what I ...
4
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3answers
343 views

Covariance, contravariance and all that jazz

For the love of God, can someone explain to me the difference between functors of the form $\mathcal{C}^{\text{op}}\to \mathcal{D}$, $\mathcal{C}\to \mathcal{D}^{\text{op}}$ and $\mathcal{C}^{\text{op}...
2
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0answers
24 views

How to calculate braiding eigenvalues in a fusion category?

Statements like this are found in published articles: The context: Assume $\mathcal{C}$ is a complex fusion category (i.e. complex linear, finitely semisimple, monoidal, with duals, with simple ...
2
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0answers
99 views
+100

Question on the coskeleton functor $\mathbf{coskel}'$ for pointed simplicial sets

There is an abstract construction of the coskeleton functor for simplicial set as follows: Fix an integer $n\geq 0$ and take the inclusion of the full subcategory $i_n\colon\Delta_{\leq n}\...
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2answers
33 views

Composition of a unique arrow with the inverse of another

Suppose we have the arrows $u:T \rightarrow Q$, $v:T \rightarrow P$ and $f:P \rightarrow Q$. Furthermore, suppose $u$ is unique and $f$ is iso. I understand that we can say that $v = u;f^{-1}$, but ...
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0answers
33 views

Existence of an illegitimate conglomerate of isomorphisms of $\mathbf{Set}$ onto itself

Definitions: A conglomerate $\mathbf{K}$ is legitimate if there exists a class $\mathbf{C}$, and a surjection $f:\mathbf{C}\to\mathbf{K}$, or alternatively, if there exists an injection $f:\mathbf{K}\...
7
votes
3answers
150 views

Where to study $2$-category theory?

Is there any place where I can read about $2$-categories? I am looking for a proper treatment - there is a section in Borceux's Handbook of Categorical Algebra, but it only sketches some parts of the ...
2
votes
2answers
75 views

nonequivalence of category of Groups and category of Pointed Groups

Am I correct in thinking that the category pGrp, whose objects are pairs $(G,g)$ where $G$ is a group, $g \in G$ and $$\hom\left((G,g),(H,h) \right) = \{ \varphi: G \rightarrow H \hspace{1mm} \big\...
3
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0answers
51 views

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 ...
2
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0answers
53 views

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 ...
4
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0answers
68 views

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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0answers
25 views

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 ...
2
votes
1answer
81 views

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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2answers
50 views

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 ...
3
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1answer
165 views

Yoneda Lemma for 2-categories - lax version

Is there a sort of lax Yoneda Lemma for 2-categories? Here is what I seem to have proven (although I have not checked all the details): If $\mathcal{C}$ is a (weak) 2-category, $A$ is an object of $\...
2
votes
1answer
65 views

Covering by open subfunctors and epimorphisms of sheaves.

I am trying to learn about the functor of points approach to algebraic geometry. Given the category of locally ringed spaces $GSp$ (geometric spaces) we have a functor $$\mathcal{G}: GSp \to Set^{...
4
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1answer
36 views

Let $\mathbf{A}$ be a subcategory of $\mathbf{B}$, s.t. $\forall$ $\mathbf{A}$-objects $A$, $A\xrightarrow{id}A$ is an $\mathbf{A}$-reflection

I wish to prove that this implies that $\forall$ $\mathbf{A}$-objects $A$, any $\mathbf{A}$-reflection $A\xrightarrow{r_A}A^{*}$ is an $\mathbf{A}$-isomorphism. What I have managed to show, without ...
0
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1answer
77 views

What is the dual category of topological spaces? [duplicate]

What is the dual category of topological spaces $Top$? I know that the order theoretic dual of a topological space is a closed set system rather than an open set system. However, this doesn't answer ...
4
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1answer
43 views

Hamiltonian Mechanics and the Symplectic Category

Are canonical transformations (in the sense of Hamiltonian mechanics) morphisms for a certain category? They seem to fit the archetypal description of morphisms being "structure-preserving maps". ...
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2answers
59 views

Is every finite category identifiable with a directed multigraph? (and vice versa?) [duplicate]

What seems implicit in this talk on youtube, is the claim that every directed multigraph (with loops) can be identified with a finite category and vice versa, if we consider the paths of the directed ...
4
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3answers
177 views

Category of pointed sets and category of sets are not equivalent

Let $\mathbf{Set}$ denote the category of sets, with $\mathrm{hom}(X,Y)=Y^X$, and let $\mathbf{pSet}$ denote the category of pointed sets, with objects of the form $(X,x),\, x\in X$, and $\mathrm{hom}(...
1
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1answer
346 views

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 ...
2
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1answer
56 views

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 ...
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0answers
35 views

Does a binormal category always admit an additive structure?

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 ...
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vote
3answers
112 views

In category theory: Do we need products to define exponentials?

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 ...