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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22 views

Cylinder with bases collapsed to a point.

The problem, although arising from some deeper facts, is quite simple. I would like to visualise the quotient space $A$ given by the cylinder $I\times S^{1}$ ($S^{1}$ is the circle in $\mathbb{R}^{2}$ ...
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1answer
25 views

Arrow between endofunctors over a symmetric monoidal category.

Consider an arrow between the categories of endofunctors over two symmetric monoidal (SM) categories $\mathcal{C}$ and $\mathcal{D}$ $$a:End(\mathcal{C}) \rightarrow End(\mathcal{D})$$ It is a ...
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24 views

exercise involving exactness

I have been stuck on this exercise for a little while. We are in abelian category. How do I show that $0 \rightarrow A \rightarrow B$ is exact if and only if $f: A \rightarrow B$ is a monomorphism? ...
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2answers
60 views

Which constructions on a category are still interesting for a groupoid?

By a groupoid, I mean a (small) category in which every morphism is an isomorphism. It looks to me that constructions on a category like the opposite ("dual") category or products and (co-)limits ...
3
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1answer
73 views

Is a homomorphism expected to be a (structure-preserving) map?

Is a homomorphism a special type of morphism, namely a structure-preserving map? For a morphism (of a category), it is clear that we can't always expect that a morphism is necessarily a ...
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38 views

Category of Presheaves on a small category $C$ is locally cartesian closed

I'm trying to fill in the details of the proof and need the following result: $Set^{C^{op}}/P\simeq Set^{D^{op}}$, where $D$ is the category of elements of $P$. The objects are pairs $(x,C)$ with $C ∈ ...
2
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1answer
45 views

equivalence in category

First gives some definitions, and then the property that I am confused. $A$, $B$ are both $R$-module, and $C$, $D$ an (additive) abelian group, consider the category $M(A,B)$ whose objects are all ...
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45 views

Why is $\mathsf{HTAG}$ (Hausdorff, Topological, Abelian Groups) preabelian?

The category of Hausdorff topological abelian groups are commonly cited as an example of a category which is preabelian, but not abelian. I think one reason that is is not abelian comes from the ...
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48 views

Generators and relations as a functor [on hold]

Make “generators and relations” into a functor. What is its left adjoint? [Bergman] How could one do this?
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2answers
29 views

What is a zero morphism in an abelian category

I am trying to familiarize myself with some basic category theory and I am getting confused with what a $0$-morphism is. If we are in category of say $k$-vector spaces then I am guessing ...
2
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1answer
17 views

Behaviour of $\operatorname{Ext}$ with left exact sequences.

Maybe is a trivial question but I am not so good in derived functors. Assume we are in the category of abelian groups and we have an exact sequence $$0\longrightarrow A\longrightarrow B\longrightarrow ...
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1answer
47 views

Is the category of these particularly nice spaces cartesian closed?

Is the category of Hausdorff, compactly generated, locally path-connected, semi-locally 1-connected spaces (and continuous maps between them) cartesian closed? If not, in what ways does it fail to be? ...
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87 views

Foundations book using category theory?

I'm about to embark on a PhD in mathematical biology. My major is in computer science. I would like to acquire a more rigorous understanding of math, which I am going to need to tackle some research ...
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1answer
60 views

Topological construct

I just started working with some category theory and I would like to understand the link between what I am studying now and what I know about topological spaces. By definition, a construct (in our ...
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0answers
28 views

~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
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22 views

Underlying functor of tensor product in a closed and symmetric monoidal category.

I will follow, for terminology and notation, G. M. Kelly, Basic Concepts of Enriched Category Theory. For sake of a self-contained exposition, I will try to write here all the needed concepts. Let ...
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1answer
82 views

Is there a category of categories?

My question is quite simple, I would like to know if we can define the category of the categories, unlike Cat which is the category of the small categories. By the way, are there any particular reason ...
3
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1answer
94 views

Properties Shared by Equivalent Categories

If two categories are equivalent, then if one has products, then so does the other. The proof of this is easy enough so I'm guessing the same result holds for exponentials but I am having trouble ...
2
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1answer
57 views

The idea of “generators” for arbitrary categories

Given a partial order $\langle X, \leq \rangle$ and a subset $I \subseteq X$ it is common to consider $I$ as the generators of the set $\{ x \in X: i \leq x \textrm{ for some }i \in I \}$ (i.e., the ...
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2answers
49 views

Derivatives on Functors

I'm not even sure if this question makes pedantic sense but is there any way to rigorously define the notion of taking the derivative of a functor?
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1answer
23 views

Query on a simple exercise involving representations of functors.

I am trying to prove the following fact Let functors $K,K':\mathcal{D}^{\text{op}}\to\mathbf{Set}$ have representations $(r,\phi)$ and $(r',\phi')$ respectively. Prove that to each natural ...
2
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1answer
35 views

If the functor on presheaf categories given by precomposition by F is ff, is F full? faithful?

Suppose that $F: A \rightarrow B$ is a functor, and $- \circ F: \widehat{B} \rightarrow \widehat{A}$ is the functor on the presheaf categories induced by precomposition. If $- \circ F$ is full and ...
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32 views

Module and bimodule categories equivalent to a 2-functor

Let $A$ be a tensor category (i.e. monoidal category). A (right) module category of $A$ is a category $M$ with a coherent action $\mu\colon M\otimes A\to M$. Denote $BA$ be the one-object bicategory ...
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38 views

Existence of a natural transformation

Let $G$ be a group, $S$ a set and $\chi$ an action of $G$ on $S$. This defines a functor $F$ from the group as a single object category to the category of sets. Let $\phi$ be an automorphism of $G$, ...
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1answer
166 views

Algebraic topology and homotopy in category theory

I've heard many times that for an algebraic topologist two spaces which are homotopy equivalent are essentially the same. But when the topological space is contractible then it is equivalent to a ...
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1answer
24 views

The functor of monoids

I'm studying this book on introductory level category theory and I couldn't solve this exercise: In the first part I've been thinking about the monoid homomorphisms $F: S\to T$ and regarding of ...
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1answer
26 views

Initial F-algebra and its isomorphic arrow proof

A very stupid question from a programmer here. There is a theorem in Pierce's "Basic category theory for computer scientists" that sounds as follows: Let $K$ be a category, and $F$ be an endofunctor ...
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1answer
38 views

Why are global elements so special?

I'm starting to study category theory and I don't understand these remarks in this book: I have the following questions: Why does we study the global elements $x:1\to M$, since as we see in ...
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0answers
43 views

Exact functors in the category of left R-modules - “Fun for the whole family”

The following question has proved troublesome and prompted some deeper questions which I will elaborate on. Our definition of a left exact functor is one which takes exact sequences: $$0 \rightarrow ...
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21 views

functors on Zero-Object in $_RMod$-category

If I have a functor $F:_RMod\rightarrow _SMod$ that is between the category of $R$-modules to the category of $S$-modules. Can I show that it must be the case that $F(0)=0$. I know that $_RMod$ and ...
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1answer
61 views

Do we lose everything, if the natural transformations in a monad are exactly inverse?

I'm currently explaining monads $$T:{\bf C}\to{\bf C},\hspace{1cm}\eta:1_{\bf C}\to T,\hspace{1cm}\mu:T\circ T\to T,$$ to my brain and the "only" tricky thing are really the identity relations. I ...
2
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2answers
89 views

Is the categorical product for projective spaces essentially the tensor product?

I wonder whether the categorical product of two projective spaces is essentially given by the tensor product of the underlying vector spaces. Is this at least true for projective Hilbert spaces? One ...
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1answer
49 views

Inequivalent Model Categories

A Model Category is, informally, a category where a "reasonable" notion of homotopy can be developed. I'm curious to know when two model categories are considered equivalent to each other. Thanks for ...
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41 views

$\mathbf{Cat}$ the category of the categories is a category

I'm studying this book and I'm trying to prove this assertion the author made: The identity functor is the identity in this category, i.e., for each category $C$, $Id_C:C\to C$ is the identity ...
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1answer
28 views

Are there differences between total functions, epimorphic functions and surjective functions?

I've read three definitions which seems to point to the same idea. I've read about epimorphic functions in Mazzola's Comprehensive Mathematics for Computer Scientists - in this book, he treats it as ...
3
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1answer
19 views

CABool not cartesian closed

I was reading a proof I missed in class that the category $CABool$ (of complete atomic Boolean algebras) is not cartesian closed by showing first that $CABool$ is equivalent to the category ...
2
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3answers
210 views

Definition of groups in a more abstract way

I'm trying to understand the definition of group objects in categories, this is an extract of Paolo Aluffi's book: QUESTIONS Can I say that $e(1)$ is the identity in our group $G$ we have just ...
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3answers
112 views

How to recognize adjointness?

Reading math has gradually expanded my understanding of the word "symmetry", so that now I can recognize symmetries that I would have not noticed before, and without having them pointed out to me. I ...
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2answers
64 views

Preservation of Limit by Hom: Naturality Question.

$\DeclareMathOperator{\Hom}{Hom}$Let $D_{i}$ be a diagram in a category $C$,with d the limit. We have (1) $\lim \Hom_{C}(X,D_{i}) \cong \Hom_{C}(X,d)$. Whenever I see this result used, it is always ...
3
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3answers
247 views

Is every set a pointed set?

My question is quite simple, It seems every non-empty set is a pointed set, only we have to do is choice some element to be the distinguished element, am I right? I'm looking for non-empty sets which ...
3
votes
1answer
39 views

Pull-backs of diagrams of groups with free product.

Until recently I calculated only pull-back of diagrams of finite groups. Now I am trying to calculate the pull-back of diagram of groups when the groups are free products of other groups. It seems ...
2
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1answer
33 views

Are commuting squares preserved by taking adjoints?

Suppose $A\rightarrow A'$ in category C and $B\rightarrow B'$ in category D, and you have a commutative square, the top and bottom arrows of which are $FA\rightarrow B$ and $FA'\rightarrow B'$, ...
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44 views

Push-out of product of push-out diagrams

Let $\pi(U\cup V)$ be the push-out of the diagram $\pi(U)\leftarrow \pi(U\cap V) \rightarrow \pi(V)$ that appears when we apply Vam-Kampen Theorem to the open sets $U,V$ in a topological space $X$. ...
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44 views

The Categories $\mathrm{Set}\times\mathrm{Set}$ and $\mathrm{Set}+\mathrm{Set}$

I'm thinking about the following problem. In $\mathrm{Cat}$ I can form the product $\mathrm{Set}\times\mathrm{Set}$. Elements are tuples, say $(A,X)$. I think that inner products and coproducts are ...
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0answers
27 views

Dependent Product Functor [duplicate]

I am trying to finish the proof that in category C with Cartesian closed slices, the dependent product functor is the right adjoint of the pullback, so that C is locally Cartesian closed. The proof is ...
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30 views

Elementary embeddings, elementary substructures,category of sets

I would like to characterize elementary embeddings AND elementary substructures in the category of sets and functions, Set. Not only characterize, but also justify this characterization.
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1answer
41 views

Why is a braided left autonomous category also right autonomous?

In Selinger, P. A survey of graphical languages for monoidal categories (New Structures for Physics, Springer, 2011, 813, 289-233), it is stated that: Lemma 4.17 ([23, Prop. 7.2]). A braided ...
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1answer
33 views

Why do just continuous maps are morphisms in the category of topological spaces

My question is quite simple, I would like to know why the maps (not being necessarily continuous) can't be a morphism in the category of the topological spaces, since they satisfy the properties to be ...
5
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2answers
72 views

Is there a logical interpretation for equalizer and co-equalizer?

I know the logical equivalent to several universal constructions. For example product $\times$ is $\land$ and co-product $+$ is $\lor$. The associated arrows are projection and inclusion. The ...
2
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1answer
39 views

Weak enriched Yoneda lemma

I am reading http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf and on pg. 21, for the proof of the weak enriched Yoneda lemma starting from "Next, if α is defined by (1.47)..." he wants to show ...