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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1answer
86 views

Image under a left adjoint functor

Suppose $\psi: \mathbf{Groups} \rightarrow \mathbf{Sets}$ is a left adjoint functor. How would I go about evaluating $\psi(\mathbb{Z})$? Since $\psi$ is left adjoint, let $\psi$ be left adjoint to a ...
2
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1answer
59 views

How to prove uniqueness of *wannabe* final object in a slice category?

I am beginning to study category theory, and I think I need your help to find my way in this sea of uncertainty (!). I have the following problem (n. $5.11$ from Aluffi's Algebra: Chapter $0$). Let ...
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0answers
24 views

Colimits in the 2-category of partial functions (which is locally posetal)

I am interested in the category $\mathcal{Pfn}$ of partial functions (using sets as objects), which is well-known to be bicomplete. And it is also known that one can consider $\mathcal{Pfn}$ as a ...
0
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0answers
17 views

Does Extremal Mono imply Split Mono in (Epi, Regular Mono)-factorization categories?

I am trying to prove that if $m\colon A\to B$ is an extremal mono in a category with (epi, regular mono)-factorizations, i.e., each arrow factorizes as a epi followed by a regular mono, then $m$ is a ...
4
votes
1answer
137 views

Direct sum and direct product of infinitely many abelian groups are not isomorphic

Let $I$ be an infinite set, and for each $i$ let $A_i$ be an abelian group with order $o(A_i) \ge 2$. Prove that the direct product $\prod A_i$ and the direct sum (coproduct) $\bigoplus A_i$ are ...
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0answers
29 views

Aluffi: submodule $\Longleftrightarrow$ cokernel?

Aluffi makes the following brief statement, in the context of modules: "The last sentence of Proposition 6.2 simply reiterates the slogan submodule $\Longleftrightarrow$ kernel and its mirror ...
1
vote
1answer
67 views

If $\Phi: \mathbf{Vec} \rightarrow \mathbf{Vec}$ with $\Phi(V) = V^{\ast\ast}$ and $f: V \rightarrow W$, what is $\Phi(f)$?

Let $\Phi$ be an endofunctor of the category of vector spaces over a field which sends a vector space to its double dual. Let $V$ and $W$ be 2 vector spaces and let $f: V \rightarrow W$ be a morphism ...
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0answers
29 views

Axioms of Abelian Category [duplicate]

I know that the one of the axioms of abelian categories is that the induced morphism $ \text{coker}(\ker f ) \longrightarrow \ker ( \text{coker} f ) $ for any morphism $ f $ is an isomorphism. ...
3
votes
1answer
104 views

Morita contexts without tears

My question is: Has anybody seen Morita contexts introduced as it is done below? I first intended this as an answer to the question "Reference request: Morita contexts" by Bey, but then decided to ...
0
votes
1answer
35 views

What are the faces of $Id_{A}$?

I've been reading Awodey's Category Theory and I've seen the definition of the Identity function in it. As it's definition was always a function $f:A\to A$ I used to assume that the identity function ...
0
votes
1answer
58 views

Dinatural transformation,constant functor,hom functor

Let $U,V$ be functors between categories $C$ and $X$ and let $Y\in Set$. Why a dinatural transformation $Y\xrightarrow{\cdot \cdot}\hom_X(U-,V-)$ is a function which assigns to $y\in Y$ a natural ...
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1answer
141 views

Category theory? Logic? Anyone experienced this like me? [closed]

Mathematics is not logic, but if one uses Zorn's lemma and stuff he should accept logical impact on mathematics. I'm the one who cares a lot about logics. It seems like Category theory is inevitable ...
2
votes
1answer
63 views

Understanding the significance of a functor being full/faithful, especially with adjoints

I'm working through "Basic Category Theory" by Tom Leinster and am trying to get clarity on how to reason about things... one thing I'm not sure about is how to think about what a functor being ...
0
votes
1answer
54 views

CWM book,ends,category theory,natural transformation

I have a problem in the MacLane's book Categories for the working mathematician. On page 223,the chapter on Ends,he has two functors $U,V:C\rightarrow X$ and defines a dinatural transformation $\tau:Y ...
2
votes
2answers
65 views

Trying to understand significance of monoid as a one object category

So I generally understand the idea of a monoid from set theory, but I'm trying to understand better the mapping to category theory. http://en.wikipedia.org/wiki/Monoid#Relation_to_category_theory I ...
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0answers
60 views

Is the pullback of a *not necessarily continuous* open map along a continuous map open?

The pullback of an open map in Top is open. We could consider more generally the pullback in Set along a continuous function $g : A \to B$ of a function $f: C \to B$ which take opens to opens, but is ...
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5answers
73 views

Identifyng objects in a category

Is there a general way of identifying objects in a category to produce a new category? Something like a quotient by a relation on objects. How would the morphisms behave?
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0answers
25 views

Lang's Algebra: Chapter 1 Question 52,fiber coproducts

STATEMENT: Show that push-outs exist in the category of abelian groups. In this case the fiber coproducts of two homomorphism $f,g$ as above is denoted by $X\oplus_Z Y$. Show that it is the factor ...
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1answer
55 views

Weighted colimits,hom-functor,Usage of Yoneda lemma

I have a question about weighted colimits. Let $D:E\rightarrow Set$ be a diagram,and $\phi:E^{op} \rightarrow Set$ a weight. $\phi*D \in Set$ is defined by this iso (i.e. a bijection),natural in $X$, ...
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1answer
25 views

Mac Lane‘s proof of coherence theorem for symmetric monoidal categories.

In [CWM, Ch. XI, §1], Mac Lane prove the coherence theorem for symmetric monoidal categories by assuming the strictness. Thus we have a $n-$ary tensor functor $T$, and the theorem states that any two ...
2
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1answer
64 views

How would a category theorist describe Green's relations?

In Semigroup Theory, Green's relations are everywhere. Their equivalence classes, for instance, on a given semigroup $S$ can tell one a lot about the structure of $S$. There is some trivial sense in ...
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0answers
41 views

Yoneda lemma for enriched categories

Let $\mathcal{M}$ be a monoidal category. Is there a generalization of the Yoneda lemma to categories enriched over $\mathcal{M}$? In the specific case I need, $\mathcal{M}$ would be the category ...
3
votes
1answer
71 views

Equivalent Conditions for left-adjoint-left-inverse

I am doing an exercise in MacLane, in which three equivalent conditions for left-adjoint-left-inverses are to be proved. I have done the cycle of implications except for one point. Let ...
6
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1answer
68 views

Lacking properties of the category of smooth manifolds

According to Wikipedia "the category of smooth manifolds with smooth maps lacks certain desirable properties"(http://en.wikipedia.org/wiki/Differentiable_manifold#Generalizations). What are these ...
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0answers
36 views

Lang Fiber Products

STATEMENT: Let $\mathcal{C}$ be a category.A product in $\mathcal{C}_z$ is called the fiber product of $f$ and $g$ in $\mathcal{C}$ and is denoted by $X\times_zY$, together with its natural morphisms ...
5
votes
1answer
77 views

Aluffi, Exercise 2.12, regarding the cokernel in $\sf{Ring}$ of $\mathbb{Z} \hookrightarrow \mathbb{Q}$

I am working in Aluffi's Algebra: Chapter $0$ textbook, and Chapter 3, Exercise 2.12 asks one to determine the cokernel in $\sf{Ring}$ of the imbedding $i \colon \mathbb{Z} \hookrightarrow ...
2
votes
1answer
47 views

Computation of adjoint functors (sheafification)

In a (complete) category, limits can be "computed" assuming one knows how to compute products and equalisers. I have seen it mentioned that adjoint functors can be found using certain ...
3
votes
1answer
62 views

Equivalence of categories $\Pi_1:\mathbf{K^1_{CW,*}}\to \mathbf{Grp}$

Let $K(G,1)$ denote the Eilenberg-Maclane spaces with fundamental groups isomorphic to $G$. Consider the category $\mathbf{K^1_{CW,*}}$ where objects are pointed $K(G,1)$ CW complex, and morphisms ...
2
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0answers
46 views

A question regarding Yoneda's lemma.

Suppose you have two objects $A$ and $A'$ in a category $\mathfrak{C}$, and morphisms $i_C:\text{Mor}(C,A)\to\text{Mor}(C,A')$ for any object $C\in\mathfrak{C}$. Show that the $i_C$ are induced ...
3
votes
1answer
50 views

How does Yoneda lemma give that the natural isomorphism $\operatorname{hom}(A,-)\cong\operatorname{hom}(B,-)$ implies $A\cong B$?

I'm trying to work out an element free proof of the associativity of the tensor product, that $$ (M\otimes_A N)\otimes_B P\cong M\otimes_A (N\otimes_B P). $$ Since $\operatorname{hom}$ and $\otimes$ ...
0
votes
1answer
38 views

“Identity-free” definition of an isomorphism in a semigroupoid / semicategory

I am looking for a way to define "Isomorphism" in a semigroupoid (or semicategory), that is a "category", which does not necessarily have identities. To be more specific I am looking for a way to ...
3
votes
2answers
63 views

Determining final and initial object in a certain category

I am reading Paolo Aluffi's greatly entertaining book "Algebra: Chapter $0$" and I got stuck on some excercises dealing with universal properties. Let $C$ be a category, and let $A$ and $B$ be two ...
0
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1answer
32 views

Lang Universal Objects

STATEMENT: Let $\mathcal{C}$ be a category. An object $P$ of $\mathcal{C}$ is called universally attracting if there exists a unique morphism of each object of $\mathcal{C}$ into $P$, and is called ...
3
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1answer
56 views

Lang Category Theory

STATEMENT: Let $A,B$ beobjects of a category $\mathcal{A}$. Let Iso$(A,B)$ be the set of isomorphisms of Awith B. Then the group Aut$(B)$ opoerates on Iso(A,B) by composition; namely, if ...
4
votes
1answer
38 views

examples of additive categories which have morphism that has no kernel and morphism has no cokernels.

can you tell me examples of additive categories which have morphism that has no kernel and morphism has no cokernels. if you tell me reference which provide this kind of examples it will be ...
2
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0answers
34 views

Representable functors, why are they important? [closed]

why it is so important to a functor to be representable? what properties have a representable functor? can you give examples that arise naturally in other theories? examples of any reasonable functor ...
4
votes
3answers
180 views

Reference textbook developing NBG set theory

I'm starting Borceux "Handbook of Categorical Algebra". It starts with a brief discussion of the logical foundations of category theory. He describes two approaches: 1.defining universes and 2. With ...
3
votes
0answers
30 views

What's the difference between a cartesian monoidal category and a semicartesian monoidal category?

According to ncatlab: In a semicartesian monoidal category, any tensor product of objects $x \otimes y$ comes equipped with morphisms $$ p_x : x \otimes y \to x $$ $$ p_y : x \otimes y \to ...
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1answer
35 views

If the category $J$ has an initial object $s$, prove that every functor $F: J \rightarrow C$ to any category $C$ has a limit, namely $F(s)$.

Exercise 3.4.3 in MacLane's Categories for the Working Mathematician. I am able to show that $F(s)$ is a cone, but unable to show that is universal among other cones with base $F$. How do I establish ...
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1answer
62 views

Showing that the direct product does not satisfy the universal property of the direct sum

I feel intuitively that for $\prod_{i\in N}\mathbb{Z}$, as a $\mathbb{Z}$−module, and $\phi_i:\mathbb{Z}\to\mathbb{Z}$ the identity map, more than one homomorphism $\phi:\prod_{i\in ...
0
votes
1answer
41 views

Coproduct in category of pointed spaces

Let $X,Y,Z \in \mathbf{Top}_*$ be pointed spaces with basepoints $x_0,y_0$ and $z_0$. Then the wedge-sum $X\vee Y = X \sqcup Y / (x_0 \sim y_0)$ is a coproduct of $X$ and $Y$. Especially given pointed ...
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1answer
48 views

What is the 'type' of a natural transformation

Let $C,D$ be categories with objects $O_C,O_D$ and morphisms $M_C:O_C\times O_C\to Type_0$, $M_D:O_D\times O_D\to Type_0$. Let $F,G:C\to D$ be functors. A natural transformation $\eta$ associates to ...
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vote
1answer
44 views

Example of a commutative square without a map between antidiagonal objects?

In an abelian category, can there be a commutative diagram of (vertical/horizontal) exact sequences $$ \require{AMScd} \begin{CD} 0 @>>> N @>>> M\\ @. @VVV @VVV \\ 0 @>>> X ...
3
votes
1answer
56 views

split epimorphism,equalizer,universal property

Let $g:B\to A$ be a split epimorphism with $f:A\to B$, $g\circ f=\operatorname{id}_A$. Why is $g$ a coequalizer of $f\circ g$ and $\operatorname{id}_B$? Commutativity is clear,but the universal ...
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0answers
75 views

conditions for an “exponential”

[More information in EDIT 2] If one defines an operation $\odot: V\times V\rightarrow V$ between the elements of a linear vector space, what properties should this operation have in order for a ...
2
votes
1answer
52 views

In additive category product and co-product over finite family of objects are isomorphism.

Let C be a additive category.Show that the co-product and product over finite family of objects are isomorphism.
5
votes
2answers
187 views

Two point topological space

Is there a standard name for the two point space with precisely one singleton being the only nontrivial open set? What are its most noteworthy categorical properties?
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1answer
47 views

Showing that a CCC with a zero object is the trivial category

Let $\mathcal{C}$ be a cartesian closed and assume that $0\cong 1$ (its initial object is the same as its terminal object). I want a detalied proof of the answer given here: ...
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0answers
79 views

Projective covers of graded modules.

I want to prove that there exist projective covers in the category of graded modules over an algebra. I am fairly new to "this" kind of mathematic and don't really know where to start: I found the ...
3
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0answers
58 views

Monoids, Semigroups, and a Reflective Subcategory.

The following is reflective in the category $\mathbf{Sem}$ of semigroups and their homomorphisms: we add a new neutral element to each semigroup, even monoids; then the reflections are identity ...