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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Example of Faithful, but not Full Functor

I have a couple examples of functors that were given to me in class that are faithful, but not full. However, I'd like an actual proof of these facts in case I have to explain myself on an exam. ...
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1answer
53 views

Lifting Functors to Adjoints

There are well-understood theorems that give sufficient conditions for a functor $R: D\to C$ to have a left adjoint. For example, $R$ should preserve limits and $D$ should have nice categorical ...
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1answer
55 views

Cone of an adjunction

I came across this sentence "...let $\varepsilon: GG^\vee \to \operatorname{Id}$ be the counit of adjunction and $Z$ its cone." I thought that cones were constructions on functors. $\varepsilon$, ...
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31 views

Projective objects of chain category

I am tempted to think that the projective objects in the chain category $\text{Ch}(\mathcal C)$ for $\mathcal C$ abelian are exactly the complexes $P_\bullet$ for which each $P_i$ is projective. Is ...
3
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1answer
87 views

Is there a coproduct in the category of path connected spaces?

Well, first of all, does the coproduct exist in the category of path-connected spaces, and if not how would you prove it? If it does exist what is it and how do you find it? The usual coproduct of ...
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36 views

When a monoidal category is equivalent to its center

The notion of the center of a monoidal category categorifies that of the center of a monoid. Similarly, the notion of a braided monoidal category is a categorification of that of a commutative monoid. ...
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36 views

Unit of Left Kan Extension in Coend Formula

We have the following result:if $K:M\rightarrow C;\ T:M\rightarrow A$, and if the copower $[Tm',c]\cdot Tm$ exists, then $T$ has a Left Kan Extension given by $Lc=\int ^m[Km,c]\cdot Tm$ whenever the ...
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49 views

Snake Lemma: “natural” or “functorial” [duplicate]

There are some really good discussions of the meaning/ambiguity of the term "natural" here and here. One thing I didn't quite get from those answers: when we say the Snake Lemma, for instance, is ...
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70 views

Does equivalence of categories preserve equivalence relations?

Let $F:\mathscr{A}\longrightarrow \mathscr{B}$ be (a part of) an equivalence of categories. My questions are: 1) Does $F$ preserve equivalence relations? 2) Does $F$ reflect equivalence relations? ...
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1answer
52 views

Yoneda's Lemma in Vakil's notes

Vakil's Notes in the exercise 1.3Y, what does 1.3.10.2 commutes with the maps 1.3.10.1 mean? I can't see any relation between 1.3.10.1 and 1.3.10.2.
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45 views

About a Corollary of Yoneda's Lemma

I am reading Assem-Simson-Skowronski's book "Elements of The Representation Theory of Associative Algebras". I do not understand a Corollary 6.2, (IV. 6.2, Functorial Aproach to almost split). It says ...
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1answer
68 views

How to show $\hat{\mathbb{Z}}/n\hat{\mathbb{Z}}\cong \mathbb{Z}/n\mathbb{Z}$

I'm trying to do exercise 1.9 from the following PDF: http://websites.math.leidenuniv.nl/algebra/Lenstra-Profinite.pdf RELATED: Elements in $\hat{\mathbb{Z}}$, the profinite completion of the ...
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83 views

Category-theoretic properties of cardinals

Let $\kappa$ be a cardinal, let $\mathbf{H}_\kappa$ be the set of hereditarily $\kappa$-small sets, and let $\mathbf{Set}_{< \kappa}$ be the full subcategory of $\mathbf{Set}$ corresponding to ...
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0answers
31 views

Proving exact functor preserves kernels

I'm trying to show that if $F$ is a functor on abelian categories that preserves exact sequences, then it also preserves kernels. I have started with $0 \to \text{ker}(f) \overset{k}{\to} A ...
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50 views

Some propreties about $ \mathfrak{Coh}_X $ and $ \mathfrak{QCoh}_X $.

I would like to know : why is the category $ \mathfrak{Coh}_X $ of coherent scheaves the smallest abelian category containing line bundles ? Why is the category $ \mathfrak{QCoh}_X $ of quasi ...
2
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56 views

Over $SET$ category, a relation R there is a contractible pair of morphisms $f,g$ if and only if $[x]$ has one element $x_{*}$ such…

If I am over $SET$ category , I want to prove that given a relation R there is a contractible pair of morphisms $(f,g)$ if and only if every equivalence class $[x]$ has one element $x_{*}$ such ...
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1answer
21 views

Fiber product with diagonal morphism [duplicate]

Stacks tag 01KR states that the diagram of schemes is "by general category theory" "a fibre product diagram". I tried to show this using the universal property, but didn't obtain anything useful. ...
3
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1answer
98 views

Monoids where $\operatorname{Hom}(M,M) \cong M$

What are some examples of monoids where $\operatorname{End}(M) \cong M$? Is there a nice characterization of such monoids? E.g., they will necessarily have a zero element, since $\forall x (x \mapsto ...
2
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1answer
140 views

Why is Grp not an Abelian Category?

As I understand it, the category of groups (not just abelian groups) satisfies all of the definitions of an abelian category. It has all kernels/cokernels as well as products/coproducts. Further the ...
2
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1answer
33 views

Subfunctors of product-preserving functors

Let $\mathscr{C}$ be a category with finite products. Let $F:\mathscr{C}\longrightarrow \operatorname{Set}$ be a finite product-preserving functor. Let $G:\mathscr{C}\longrightarrow ...
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1answer
44 views

Covering sieves in a Grothendieck topology

I'm trying to get my head around some of the basics of Grothendieck topologies. Let $(\mathcal{C}, J)$ be a site, let $U$ be an object of $\mathcal{C}$ and let $J(U)$ be the set of covering sieves on ...
2
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0answers
50 views

When can we use a monic morphism to copy an algebraic structure?

Let $(T,\mu, \eta)$ be a monad over the category $\textbf A$ , let $(A,a)$ be a $T$- algebra and $m: B\rightarrow A$ be monic. Prove a morphism $b:TB\rightarrow B$ is the structure for an algebra ...
2
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1answer
74 views

Two skeleton of one category are isomorphic

I have the next doubt about this problem: If $K_{1}$ and $K_{2}$ are two skeleton of the category $A$ i have to show that these two skeleton are isomorphic. Because $K_{i}$ is a skeleton we have ...
3
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47 views

What are universal abstract $\sigma$-algebras on $\sigma$-frames?

In this paper, the authors make the following definitions: An (abstract) $\sigma$-algebra is a boolean algebra with countable joins. A $\sigma$-frame is a bounded lattice with countable joins, where ...
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1answer
33 views

A question on concrete category

This following is excerpted from Category Theory by S. Awodey. "Theorem 1.6. Every category C with a set of arrows is isomorphic to one in which the objects are sets and the arrows are functions." ...
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37 views

Can every monad give rise to a monad transformer?

Can every monad give rise to a monad transformer? In the paper Calculating monad transformers with category theory by Oleksandr Manzyuk, one finds a construction of monad transformers as ...
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41 views

An abelian category such that all objects are injective

The problem is 'Let C be an abelian category such that all objects in C are injective. Prove that all abjects are projective.' If C has enough projectives, then the 'Ext' functor can be defined. Thus ...
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1answer
41 views

Distinguished category of groups, is it abelian?

Let $\mathcal{A}$ denote the following category. The objects of $\mathcal{A}$ are the pairs $(H,G)$ where $G$ is an abelian group and $H$ is a subgroup of $G$. The morphisms between $(H,G)$ and ...
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1answer
40 views

Strange endomorphism that translates in a functor one (?)

Let $C$ be a category composed by three pieces: $C_{-2}, C_0$ and $C_2$, where $C_{-2}=C_2=\mathbb{K}-$modules and $C_0 = \frac{\mathbb{K}[x]}{(x^2)}$-modules. Let $E$ be the functor that acts this ...
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1answer
62 views

Category theory: do other examples of “resplendent” properties exist?

Call a predicate $P$ defined on categories resplendent iff it satisfies the following condition: for all categories $\mathbf{D}$, if $P(\mathbf{D}),$ then for all categories $\mathbf{C}$, we have ...
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1answer
50 views

Is there a weaker version of Yoneda?

Let $A,B$ be objects of a locally small category $\mathcal{C}$ such that $\operatorname{Hom}(D,A)\hookrightarrow \operatorname{Hom}(D,B)$ for every object $D$ in a natural way. In what situations can ...
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1answer
84 views

What is a projective object in $\rm Set$?

What property of a set in $\sf{ZF}$ is equivalent to its being a projective object in the category $\rm Set$? Since all sets are projective assuming $\sf AC$ my guess is that it is equivalent to ...
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2answers
35 views

Prove that every discrete poset is projective.

Is this proof correct (assuming the Axiom of Choice)? $e:P\longrightarrow Q$ epic, then $e$ is a surjection (Proof that the epis among posets are the surjections). For any arrow, $f:A \longrightarrow ...
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1answer
44 views

“A Category Object in another Category”?

In a seminar I heard things like a groupoid in the category of vector bundles, a group in the category (..). Since I don't know much category theory I'm wondering if there is a general theory of ...
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1answer
35 views

Why are lax functors from the terminal $2$-category the same as monads?

I use boldface (e.g. $\mathbf{C},\mathbf{D}$) for categories and fraktur (e.g. $\mathfrak{B}$ for bicategories.) According to this blog post at nLab, with some of the notation changed a little: ...
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1answer
91 views

Finding simple worked exercises for Category Theory

I am in the process of learning Category Theory with the purpose of being able to create a game that will help explain it to others in a simple way. I have read many texts and articles about it. While ...
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1answer
31 views

Relationship between functions given by same rule, but different domains.

We have a function $1_{x}: P(X) \rightarrow \mathbb{R}$ given by $ 1_{x}(E) := \begin{cases} \hfill 1 \hfill & \text{ if } x \in E\\ \hfill 0 \hfill & \text{ if } x \not \in ...
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1answer
26 views

Skeleton of finitely generated modules

If $A$ is a pid, then let $C$ be the category of finitely generated $A$-modules. Is there a skeleton of $C$ that can be described explicitly? I imagine that the structure theorem can be used to do ...
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0answers
30 views

What does it mean for a topos to be “generated” by some kind of objects?

Is there a universal definition of what that phrase should mean? Suppose we are considering objects of a topos with a particular property, call them $P$-objects (I have in mind the case where ...
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2answers
52 views

Given a continous $f:A\mapsto B$, $f(A)$ dense in $B$, $f$ is an epimophism [closed]

I know how to prove that $f$ is epimorphism in the Haus category, I just want to give a counterexample to show that this is not true in the Top category. Thanks
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1answer
60 views

Does the Hom-functor $H( _, A)$ take limits to colimits?

Let Ab be the category of abelian groups, and $A$ a finitely generated abelian group. One can deduce the two Hom-functors $Hom(A,\:\_ \;): Ab \to Ab$ and $Hom(\:\_ \;,A): Ab^{op} \to Ab$, both of ...
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1answer
50 views

Proving the magic pullback square using Yoneda

In these two questions, it is mentioned that easy proofs of "magic square" identities can be given using the Yoneda lemma to reduce to the case of sets. Can someone explain exactly how to do this? ...
4
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1answer
75 views

Multi-pullbacks and the relative chinese remainder theorem

Let $I,J$ be two-sided ideals of a ring $R$. In this question I asked for an "automatic" proof of the fact the natural map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism (a direct ...
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1answer
69 views

Category Theory: Naturality and Notation

I'm confused by some Category Theory notation but I give the whole question I'm interested in solving below for the sake of context. Also, I want to verify my understanding of the proper approach to ...
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1answer
64 views

Free Product of Groups with Presentations

There is a highly believable theorem: Let $A, B$ be disjoint sets of generators and let $F(A), F(B)$ be the corresponding free groups. Let $R_1 \subset F(A)$, $R_2 \subset F(B)$ be sets of relations ...
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1answer
46 views

Given an adjunction $\langle F,G,\eta, \epsilon\rangle$ with either $F$ or $G$ full, prove that $G\epsilon$ is invertible

I have this doubt with this problem: "Given an adjunction $\langle F,G,\eta, \epsilon\rangle$ with either $F$ or $G$ full, prove that $G\epsilon:GFG\rightarrow G$ is invertible with inverse ...
6
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1answer
93 views

Relative chinese remainder theorem and the lattice of ideals

Let $R$ be a ring with two-sided ideals $I,J$. The proof of the "absolute" chinese remainder theorem revolves around the fact that if $I,J$ cover $R$ in the lattice of ideals, i.e $I+J=R$, then the ...
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1answer
30 views

Show that Every set is projective in the category Set assuming the axiom of choice. [closed]

How do we prove the statement that every objects in the category Set is projective?
4
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1answer
43 views

Left adjoint of a strange functor

I'm looking at the category (call it $\mathcal{C}$) of colimit preserving functors from $R$-mod to $Vect$, where $R$ is some commutative $k$-algebra. There is an obvious functor $$ \mathcal{F} : ...
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1answer
24 views

Free Finite Rank Abelian Group category is not balanced

I'd like to prove(by finding a counterexample or whatever) that Free Finite Rank Abelian Group category is not balanced, i.e., a bimorphism doesnt imply isomorphism. Thanks