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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25 views

Words in the Category of Sets

I was wondering about free objects in different categories and the "words" in those categories. I think I have a generally good grasp on the idea, but I started to think about stranger free objects ...
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1answer
15 views

are the sections of a sheaf always locally constant functions?

Given a sheaf $F:OX^{op}\rightarrow Set$ on a topological space $X$, we have the stalks $F_x:=Stk_x F:=colim_{x \in U}FU$. Then given sections $s,t \in FU$ over $U$, we have if $s_x=t_x$ then $s=t$; ...
0
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1answer
14 views

Is there a left adjoint to the inclusion of discrete (op)fibrations over $X$ into $\mathbf{Cat}/X$?

This would be intended to be like the adjoint to the inclusion of $Sub(X)$, the subsets of a set $X$ into $ \mathbf{Set}/X $, namely taking the image of a function--except "one level higher".
1
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1answer
24 views

Help understanding subobject classifiers

I'm reading Aluffi - Algebra Chapter 0. In problem 3.10, he describes subobject classifiers: for some category $C$, a subobject classifier is an $\Omega$ with the property that for all $A$ in ...
2
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1answer
21 views

Colored operads as finitely essentially algebraic theory.

I call a planar operad what is also called planar (multi-)coloured operad or multicategory and symmetric operad a symmetric multicategory or symmetric (multi-)colored operad. I have two questions ...
2
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1answer
42 views

hom(C) in category theory

I know in the basic definition of a category you have the class hom(C) of morphisms between objects in the category C. What never seems to be clear from textbook definitions is this: Are the members ...
1
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1answer
48 views

Categories and the direct image functor

I've just started learning about sheaves and yesterday I had a look at MacLane/Moerdijk's book "Sheaves in Geometry and Logic". I've discovered something in this book on which I'd like some ...
2
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1answer
140 views

Is this simple drawing a category?

As far as I can tell this is a category but I am not 100% sure. Objects are A, B, C. Arrows are f and g. Could someone please confirm that this drawing is indeed a category? So no arrows need to ...
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0answers
36 views

Diagrams characterizing ring characteristic, and in particular field characteristic 1?

For a given prime $p$, what is the diagram for saying (e.g. in some category where morphisms are field homomorphisms) that a field has characteristic $p$? Is there a diagram, or the shape of what ...
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0answers
48 views

Is it true that $V$ and $V^*$ are naturally isomorphic as finite vector spaces if $V$ is equipped with an inner product?

This is a homework question from my differentiable manifolds class: In general we know that if $\dim V<\infty$ then $V$ and $V^*$ are isomorphic because any two vector spaces with the same ...
3
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1answer
52 views

What categorical limits and colimits does $\pi_1$ preserve?

$\pi_1$ is a functor from the category of pointed topological spaces to the category of groups. It's clear that this functor preserves products, since $\pi_1(X\times Y)=\pi_1(X)\times\pi_1(Y)$, and a ...
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1answer
40 views

Dual notion of the subspace topology

Let $X$ and $Y$ be sets with $\iota:X\to Y$ an injection. If $Y$ is a topological space, we define the subspace topology on $X$ as the initial topology induced by this diagram. Analogously, if $X$ ...
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2answers
77 views

Does the functor that preserves limit always have a left adjoint?

If $F: Sets \to Sets$ is a functor that preserves limits, is it true that $F$ always has a left adjoint?
2
votes
2answers
46 views

Epimorphism that is not surjective in the category of Torsion Free Abelian Groups

In reading about cokernels (relating to a homework question I have) I came across the following: https://www.dpmms.cam.ac.uk/~jg352/pdf/CTSheet4-2013.pdf I specifically wondered about question 5a. ...
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votes
0answers
80 views

A contravariant functor taking colimits to limits is representable.

If $F$ is a contravariant functor from $Sets$ to $Sets$. And for any functor $H: I \to Sets$ that has a colimit $C$ we have $F(C)$ is a limit for $F \circ H: I ^{op} \to Sets $. Show that $F$ is ...
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0answers
29 views

What is the trivial module functor?

In Weibel's book on homological algebra, he mentions the trivial G-Module on page 160. By this, does he mean the the functor $\mathcal{F}: \text{G-Mod} \to \text{G-Mod}$ by making $G$ act trivially on ...
1
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1answer
36 views

Proof that the tensor product is the coproduct in the category of R-algebras

Given the category of commutative R- or k-Algebras, it is often mentioned that the coproduct is the same as the tensor product. I'm interested in the proof of this statement. One idea would be to ...
0
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2answers
36 views

How many isomorphisms from a set to itself

I am reading through Spivak's "Category Theory for Scientists" and one of the exercises is to find the number of Isomorphisms from a set X to itself. My attempt: If I am not mistaken, if |X|= n where ...
3
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2answers
56 views

Are $e : 1 \rightarrow X$ and $\mathrm{id}_X : X \rightarrow X$ the only operations of $\mathsf{Grp}$ that are homomorphisms?

Let $\mathsf{Grp}$ denote the Lawvere theory of groups. (For concreteness, let us say that $\mathsf{Grp}$ is presented by the generators $c : X \times X \rightarrow X$ $e : 1 \rightarrow X$ $i : X ...
2
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2answers
95 views

Introductory example(s) of a functor that is full but not faithful

What is your favourite example to offer real beginners of a functor which is full but not faithful?
2
votes
1answer
247 views

bijective homomorphisms between non isomorphic posets, example , explanation needed

Could someone please give a super simple example of a "bijective homomorphism between a non isomorphic poset"? I don't understand the sentence marked by green in the picture taken from Awodey's book. ...
1
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1answer
70 views

Why only two binary operations?

Ring theory considers things with 2 operations and category theory 101 talks about products and coproducts. I maybe understand why binary operations are more common to look at that trinary, ...
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1answer
39 views

Forcing a functor to map to a given set

Let $(X, \leq)$ be a poset and $J$ a small category. Let $S$ be a subset of $X$. Viewing $(X, \leq)$ as a category, does there exists a functor $F: J \rightarrow (X, \leq)$ such that $\{F(j)\}_{j \in ...
1
vote
1answer
30 views

Equivalence of categories is an equivalence relation

Suppose that $B \xrightarrow{F}C$ and $C \xrightarrow{G}D$ are equivalences of categories. I want to show that $G \circ F$ is an equivalence. (This becomes easy if I use that a functor is an ...
4
votes
2answers
129 views

What is the opposite category of $Set$?

In $Set$ the initial object is the empty set, and it has an unique morphism to each other object, namely $f=\emptyset$. However I find it difficult to think about the category ${Set}^{op}$, is there ...
2
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2answers
55 views

Connection between monads and posets?

I understand this is broad, but could any elucidate and/or direct me on which structures, areas, and objects to study to get a deep understanding of the relationship between monads and partially ...
2
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1answer
67 views

Constructing a colimit

Consider a functor $F: \mathcal{I} \rightarrow \textrm{Sets}$ where $\mathcal{I}$ is small. Then the colimit of $F$ is given by $\amalg_{i \in \textrm{Ob}(\mathcal{I})}F(i)/{\sim}$ where $\sim$ is the ...
2
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1answer
37 views

Relationships between initial/terminal objects and initial/terminal morphisms (if any) in the same category.

The definition of initial and terminal objects given here http://en.wikipedia.org/wiki/Initial_and_terminal_objects makes sense to me. The definition of initial and terminal morphisms given here ...
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1answer
148 views
+50

Functoriality of fundamental group

I'm trying to prove the following statement: If $\mathcal{C}_1$ is the category of path connected topological spaces and $\mathcal{C}_2$ is the category of groups, then the mapping $\mathcal{C}_1 \ni ...
2
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1answer
61 views

Yoneda Lemma: definition of Yoneda functors

In stating the Yoneda Lemma, my category theory book (Kashiwara's Categories and Sheaves) makes the following definition: Let $\mathcal{C}$ be a $\mathcal{U}$-category, where $\mathcal{U}$ is ...
4
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2answers
64 views

Examples of natural isomorphisms

I am a beginning Category Theory student, and a intermediate Algebra student. Could someone provide me with some examples of natural isomorphisms (in Category Theory) besides the natural isomorphism ...
2
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1answer
105 views

Yoneda implies $\text{Hom}(X,Z)\cong \text{Hom(}Y,Z)\Rightarrow X\cong Y$??

I came across a result on the page Theorems implied by Yoneda's lemma? which said that Yoneda's Lemma implies the isomorphism of the title; namely, if we have $\text{Hom}(X,Z)\cong ...
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0answers
53 views

Is the inverse limit of simplicial maps between finite directed graphs also a graph?

I think I have an intuitive understanding of why the following statement might be true, but I am not sure how to go about proving it. It's also possible my intuitive understanding is wrong and the ...
2
votes
0answers
58 views

Adjoints with a Parameter

I am tryng to prove a theorem in McLane without looking at his proof and I am stuck on one point. Suppose $F:X\times P\rightarrow A$ is a bifunctor, that for each $p\in P $, the functor ...
2
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1answer
51 views

Hurewicz model structure and cofibrantly generated model categories

Is it an open problem if $\mathbf{TOP}$ with Hurewicz (Strøm) model structure is cofibrantly generated?
0
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0answers
29 views

kan extension,(co)ends,natural bijection

Let $F:C\rightarrow E$,$K:C\rightarrow D$ be functors. How do I get left Kan extension of $F$ along $K$; $\operatorname{Lan}_K F:D\rightarrow E$ and $\eta:F\Rightarrow \operatorname{Lan}_KF$. $K$ ...
4
votes
2answers
71 views

The category of vector fields on smooth manifolds

In my differential geometry lecture today we learnt about the push-forward of a vector field by a diffeomorphism. I know some basic category theory and I noticed a functor popping up. Here's what I've ...
2
votes
2answers
61 views

A connected groupoid is equivalent to a groupoid with one object

This is same as saying a connected groupoid is equivalent to a group. But I have no idea how to construct such a group, can't even get started. Any help is appreciated.
6
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1answer
74 views

Natural isomorphisms of the forgetful functor

Let $U: \mathbf{Groups} \rightarrow \mathbf{Sets}$ be the forgetful functor. Must every natural transformation $\eta: U \rightarrow U$ be a natural isomorphism?
1
vote
1answer
30 views

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 ...
3
votes
1answer
40 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 ...
10
votes
1answer
469 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
votes
1answer
166 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 ...
3
votes
1answer
42 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 ...
0
votes
1answer
53 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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1answer
56 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 $FX \to K$, ...
6
votes
1answer
71 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 := ...
2
votes
1answer
53 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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vote
1answer
40 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 ...
3
votes
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 ...