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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A filtered poset and a filtered diagram (category)

Let $X$ be a poset. Statement: A subset $Y\subseteq X$ is filtered if an only if there exists a filtered diagram (category) $D$ with a functor $D\rightarrow X$ such that the image of $D$ is $Y$. How ...
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82 views

Uniqueness of Duals in a Monoidal Category

Given a monoidal category ${\cal C}$, and $X \in {\cal C}$, we define a left dual of $X$ to be an object $X^*$ such that there exist morphisms $\epsilon:X^* \otimes X \to I$, and $\eta:I \to X \otimes ...
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794 views

Why are there so many universal properties in math?

I don't really understand why there are so many universal properties in math or why they all need to be highlighted. For example, I'm studying some Algebra right now. I have found three universal ...
3
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1answer
337 views

Forgetful functor from R-modules to abelian groups?

I am trying to see, if the forgetful functor from $\mathbb{Z}[X]$-modules to abelian groups is exact and in case it is not exact, is it left or right exact. In general, i understand the definition of ...
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54 views

Recommendation on setting the reference axis for mathematical objects

(I don't know what the title should be for this post, please change it if you have a better title. Also tags) In many situations, there arises cases that one mathematical structure embeds into ...
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2answers
208 views

(Are there) subtleties in the definition of 'biproduct'

I always thought that a biproduct of two objects $A_1,A_2$ in some category $\mathcal{C}$ is an object $P$ with two maps $p_i:P\to A_i$ making it a product and two maps $j_i:A_i\to P$ making it a ...
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63 views

factorization of a unit of an adjunction

Let $F$ be a left adjoint functor to $V$. Factor $X \to VK$ through the adjunction unit $$ X \to VFX \to VK, $$ where the first map is $\eta_X$, the second map is $V$ of the adjoint map $FX \to K$, ...
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269 views

The ring of idempotents

Let $R$ be a commutative ring. Then its ring of idempotents $I(R)$ consists of the idempotent elements of $R$, with the same multiplication as in $R$, but with the new addition $x \oplus y := ...
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183 views

question about monoidal structure of a 2-category

Consider an extension of the 1-category of vector spaces and linear maps down to a 2-category $\mathcal{C}$ whose objects are $k$-linear categories. What is the symmetric monoidal structure on the ...
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119 views

The Verdier Quotient

In A.Neeman's book and D.Murfet's notes I have been reading about the construction of the Verdier quotient of a triangulated category, $\mathscr{T}$, by some triangulated subcategory $\mathscr{C}$. In ...
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94 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 ...
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95 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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38 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 ...
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766 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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1answer
62 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 ...
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1answer
70 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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43 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. ...
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164 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 ...
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1answer
43 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 ...
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1answer
111 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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206 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 ...
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2answers
331 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 ...
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1answer
75 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 ...
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414 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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165 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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108 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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94 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
133 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 ...
3
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1answer
98 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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1answer
195 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 ...
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174 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 ...
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167 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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1answer
86 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 ...
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1answer
141 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 ...
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1answer
108 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
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1answer
83 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 ...
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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
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1answer
83 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 $\otimes$ ...
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1answer
83 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 ...
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113 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 ...
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1answer
81 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
76 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 ...
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97 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 ...
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395 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 ...
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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
47 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
83 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 ...
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1answer
72 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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61 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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1answer
62 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 ...