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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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2
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2answers
44 views

Why does the arrow notation of categorical limit go from right to left?

Whilst studying some category theory, I was blindly using the notation $\lim_\leftarrow D$ for the limit of a diagram $D$ in some category $\mathcal{C}$ (as they are notated in MacLane, Awodey, my ...
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1answer
17 views

Representability is preserved by composition with functors admitting an adjoint

Exercise 1.9 on page 31 of Kashiwara's Categories and Sheaves asks: Let $F: C \to C'$ be an equivalence of categories and let $G$ be a quasi-inverse. Let $H: C \to \text{Set}$ be a representable ...
8
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0answers
84 views

How much set theory does the category of sets remember?

Question. Let $M$ be a model of enough set theory. Then we can form a category $\mathbf{Set}_M$ whose objects are the elements of $M$ and whose morphisms are the functions in $M$. To what extent is ...
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1answer
34 views

Maps between direct limits and functoriality of $f^{-1}:Shv(Y) \rightarrow Shv(X)$ and $f_{*}:Shv(X) \rightarrow Shv(Y)$

My question rises from exercise 1.18 in chapter 2 of Hartshorne. Given a continuous function $f:X \rightarrow Y$, one has to show there is a natural bijection between ...
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1answer
40 views

Unit and counit are close to being inverses

Suppose we have adjoint functors $C \underset{R}{\overset{L}{\rightleftarrows}} D$ with adjunction $$\theta: \text{Hom}_D(L(\cdotp), \cdotp) \xrightarrow{\sim} \text{Hom}_C(\cdotp, R(\cdotp)).$$ Let ...
2
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1answer
34 views

Show the following is a functor.

Suppose $p: Y \to X$ and $p': Y' \to X$ are covering maps, and let $\phi: Y \to Y'$ be a homeomorphism such that $p'\phi=p$. Show that the functors $p^{-1}$ and $(p')^{-1}$, from $\Pi_1(X)$ to the ...
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1answer
29 views

Is there a nice characterization of those categories in which $a \subseteq b$ implies $a \leq b$?

Given arrows $a:A \rightarrow Y$ and $b : B \rightarrow Y$, define that: $a \subseteq b$ iff for all $Z$ and all $g,g' : Y \rightarrow Z$, we have: $$gb=g'b \rightarrow ga=g'a$$ $a \leq b$ iff ...
1
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1answer
48 views

Functor between two categories

Let $G$ be a finite group. Let $C$ be the category with objects subgroups of $G$ and morphisms between two subgroups $H,H'$ be $ \{ g \in G \mid g H g^{-1} \subset H ' \}$. Let $D$ be the category ...
0
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0answers
43 views

Algebraic topology and functors.

We are asked to show that if $p : X \to Y$ is a covering map we can generalize the group action of the fundamental group on $p^{-1}(x)$, to show that $p^{-1}$ from the category $\Pi(X)$ to the ...
1
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1answer
37 views

Derive the unit from the adjunction

In learning about adjoint functors $C \underset{R}{\overset{L}{\rightleftarrows}} D$, we learn about an isomorphism of bifunctors $$\text{Hom}_D(L(\cdotp), \cdotp) \simeq \text{Hom}_C(\cdotp, ...
3
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1answer
47 views

Is there an opposite Yoneda Lemma?

For a functor $F: \mathcal{C^{\mathrm{op}}} \to \mathrm{Set}$ and an object $A \in \mathcal{C}$, we have $\mathrm{Nat}(\mathcal{C}(-, A), F) \cong FA$, which is the Yoneda Lemma. (There is a co-Yoneda ...
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2answers
24 views

Question about composition of categories

I'm reading the following to get a high-level overview of categories. http://en.wikibooks.org/wiki/Haskell/Category_theory In this, they have this image: and the author makes this claim: So ...
3
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1answer
192 views

Is there any category theoretic proof for independence of Continuum Hypothesis?

Both of set theory and category theory could be a foundation for mathematics. Many set theoretic arguments could be translated to a category theoretic argument and vice versa. Question: Is there ...
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0answers
20 views

Universal property of the commutator quotient group as a universal arrow [on hold]

Express the universal property of the commutator quotient group as a universal arrow for some functor $F$. ( $f \colon G \to A$ is a homomorphism of $G$ into an abelian group $A$ and $f$ factors ...
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0answers
26 views

Equivalence of categories [on hold]

The homework question. Let C be the category whose object class is the set { Rn, n >= 0 }, and whose morphisms are R-linear transformations. Let D be the category whose objects are finite ...
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1answer
21 views

Functor from Nor-N to Grp [on hold]

This is a homework question. Find a functor G from Nor-N into Groups such that the transformation defined by n: G -> G/N is a natural transformation from F to G. F is the inclusion functor from Nor-N ...
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1answer
58 views

Tannaka reconstruction: reference request

What is a classical and perhaps even original reference for the following result, often called Tannaka reconstruction? Let $G$ be a group and $R$ be a commutative ring in which $0,1$ are the only ...
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0answers
33 views

Direct limit with non-injective maps

Suppose I take a direct limit in the category of groups, or the category of R-modules, or similar. Let $I$ denote the index set, $A_i$ the objects and $f_{ij}: A_i \to A_j$ the structure maps of the ...
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0answers
28 views

Strong (trivial) cofibration in Lurie's HTT

in Lurie's book HTT in annexe A, proof of Proposition A.2.8.2 page 824, he mentions that a map is a "strong (trivial) cofibration" but I didn't succeed to find the definition of this notion that seems ...
1
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1answer
25 views

Example of a homomorphism with a right or left inverse function that its right or left inverse is not a homomorphism

If $f: A \to B$ is an injective or surjective homomorphism, then $f$ has a left or right inverse map respectively. The question is whether the right or left inverse function of $f$ is itself a ...
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1answer
46 views

Weaker, but not thin

Call a discrete category one which is equivalent to a category where all arrows are identities. In a discrete category there can still be non-identity arrows, but they have to be isomorphisms. If ...
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0answers
48 views

Resemblance between product and homotopy

The notion of product $X\times X$ for an object $X$ of a category $C$ resembles the notion of homotopy between two continuous functions. Indeed the relevant diagrams look the same: ...
7
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3answers
194 views

What's a good motivating example for the concept of a slice category?

What nice example can one give a beginner to really motivate the idea of a slice category, before they've met the more general notion of a comma category? There's the toy example of a poset category ...
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0answers
37 views

How to prove that a particular (sub-)category has a projective generator.

Suppose that $\mathcal{C}$ is an abelian $k$-linear category ($k$ a field) in which every object is of finite length and every $k$-vector space $\text{Hom}(X,Y)$ is finite dimensional. How does one ...
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0answers
27 views

presentable vs inductive categories

Let $\kappa$ be a regular cardinal, and let $D$ be a small $\infty$-category which admits $\kappa$-small colimits. Then is the canonical map $D \to Ind_\kappa(D)$ an equivalence, and if not why not? I ...
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1answer
61 views

Is a subobject classifier logically equivalent to set-inclusion?

Can one subsume the notion of set-inclusion $\subseteq$ and $\subset$ with the notion of a subobject classifier, expressed via an injective morphism $\hookrightarrow$? Specifically: Are the ...
2
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1answer
46 views

Are adjoint functors between additive categories additive?

The question is: Prove that an equivalence between two additive categories is an additive functor. By an additive category I mean a category with zero object, that every Hom-set of morphisms is a ...
0
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1answer
55 views

Definition of a coproduct and its universal property - connection?

I have a problem connecting the definition of a coproduct with its often mentionend universal property. Let's start with the definition (just for two objects): Let $A_1$ and $A_2$ be objects of a ...
3
votes
1answer
54 views

Group objects in category of $\mathcal{Set}$ are groups - How to prove it?

Reading about group objects in categories, it's a fact that a group object is in the category of $\mathcal{Set}$ just a common group. I am trying to give an actual proof of this, but I'm a bit ...
9
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2answers
298 views
+50

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 ...
0
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1answer
27 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 $s=t$; ...
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1answer
30 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".
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1answer
29 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 ...
4
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2answers
43 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
54 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
56 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
145 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
40 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
53 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 ...
4
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2answers
83 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
42 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
82 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
48 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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0answers
84 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
31 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 ...
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1answer
46 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
39 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
59 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
97 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?
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1answer
254 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. ...