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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92 views

What is the common preimage (in $Z$) and the equivalence relation for Pushouts

Here it says: Suppose that $X$, $Y$, and $Z$ as above are sets, and that $f : Z → X$ and $g : Z → Y$ are set functions. The pushout of $f$ and $g$ is the disjoint union of $X$ and $Y$, where elements ...
2
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1answer
61 views

graph of the compostion of morphisms category-theoretically

My question is about a certain category-theoretic statement really but since I came to it trying to prove something about non-reduced schemes, I'll state it in this language. Let $M$ be a scheme ...
7
votes
1answer
234 views

The infinite Direct Sum in the category Ring

If you don't have strong personal feelings about it already, most of you have at least witnessed the opposing factions on how we should define a ring and, by extension, how we should define a ring ...
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0answers
113 views

Flabby sheaf comonad

If one Googles sufficiently hard one finds the statement that Roger Godement gives the first example of a comonad, used to compute flabby resolutions of sheaves, in his monograph "Topologie algébrique ...
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27 views

Explicit fibrant replacement

Do you know an explicit fibrant replacement in the injective model structure on a functor category (I'm essentially interested in the case of presheaves of groupoids) ? Best
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1answer
45 views

Can anyone outline the steps for constructing general colimits?

Specifically, in a cocomplete category how can one construct general colimits via other colimits such as initial objects,Coproducts and Coequalizers? (I would prefer not very heavy mathematical ...
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2answers
459 views

Is the product of $\sigma$-algebras a tensor product in some sense?

In the wikipedia page for product measures it says: Let $(X_{1},\Sigma _{1})$ and $(X_{2},\Sigma _{2})$ be two measurable spaces, that is, $\Sigma _{1}$ and $\Sigma _{2}$ are sigma algebras on ...
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2answers
148 views

Why are left/right adjoint functors not called up/down?

I am studying category theory and I recently learned about adjoint pairs of functors. It seems to me that they are called left and right adjoints because if we have categories $\mathcal{C}$ and ...
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1answer
63 views

Moving tensor products inside homs

Suppose that $(\mathcal C, \otimes, I)$ is a closed symmetric monoidal category with $\hom(A,B)$ the hom-sets and $[A,B]$ the internal hom (where $[A,-]$ is right adjoint to $-\otimes A$). Is there ...
2
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1answer
67 views

Prove a categorical statement

In the answer to Direct products in subcategories it is said: If $\mathcal{D}$ is a full subcategory of $\mathcal{C}$ and $A \times_{\mathcal{C}} B$ is (isomorphic to) an object of $\mathcal{D}$, ...
5
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1answer
106 views

Monoidal categories and tensor products

Does the multiplication $-\square-$ biendofunctor in a Monoidal category, $\mathfrak{C}$ necessarily commute with coproducts? This is true in some familiar categories, such as $_RMod$, $Grp$ $CRings$ ...
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110 views

Are coproduct exact functors?

Are coproducts left exact or right exact functors in general? Let k be a commutative ring (unital assosiative). Specifically in the category of k-algebras is the tensor exact. (This is not the case ...
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99 views

Proofing that if f has a right inverse then f is an epimorphism. But it is not true for the converse.

Let $A$ and $B$ be objects of a category $C$, and let $f$ belong to $\mathrm{Hom}\,_C(A,B)$ be a morphism. Prove that if $f$ has a right inverse, then $f$ is an epimorphism. Show that the converse ...
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0answers
44 views

Size of the collection of morphisms of a category

Suppose we use Grothendieck'universes, at least 2 (named U and W). U has elements called (small-) Sets and its subcollections are called Classes. W has elements called Classes and its subcollections ...
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1answer
64 views

Duals of a finite dimensional eveloping coalgebra

Let $C^e$ be the enveloping $k$-coalgebra of a $k$-coalgebra $C$ and denote by ${C^e}^{\star}:=\mathrm{Hom}\,_{k}(C^e,k)$. Then is ${C^e}^{\star} \cong {C^{\star}}^e$?
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2answers
105 views

Do two definitions for kernels match?

Suppose that we work in Ab, the category of abelian groups. Consider a map $f : A \rightarrow B$ and let $\ker(f) = \{a \in A : f(a) = 0\}$. Now suppose that one can find a map $k : K \rightarrow A$ ...
10
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1answer
200 views

Why an integral symbol for the category of elements of a presheaf?

Let $\mathbf C$ be a category and $P \colon \mathbf C^{\rm op} \to \mathbf{Set}$ a presheaf. One can associate to $P$ the category of elements of $P$ (also called Grothendieck construction over $P$), ...
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2answers
68 views

Commuting two pullbacks

I have stumbled upon some interesting exercise whilst reading the "Category Theory for Scientists" book. Below is the universal property of fiber products: By using the universal property, I can ...
7
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1answer
115 views

Localization of an additive category which is no longer additive

Is there a nice example of an additive category $C$ and a family of morphisms $S\subset Mor(C)$ such that $C[S^{-1}]$ is no longer additive? I know that in general localization of categories ...
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0answers
98 views

Injective model structure

I equip the category of presheaves $[\mathcal{D}^{op},\text{Gpd}]$ with the injective model structure ($\mathcal{D}$ is just any small category). In this structure, weak equivalences and cofibrations ...
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5answers
2k views

Abelianization of free group is the free abelian group

How does one prove that if $X$ is a set, then the abelianization of the free group $FX$ on $X$ is the free abelian group on $X$?
2
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1answer
51 views

Are functions between sets of global elements of objects isomorphic to the hom-set between these objects?

Let $C$ a (perhaps well-pointed?) category with terminal object $\mathbf{1}$, so for objects $A,B$ we have the sets of global elements $G_A, G_B$ (i.e. hom-sets off $\mathbf{1}$) of $$ g_A \colon ...
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1answer
211 views

Are Monoids a category inside a category?

Looking at the definition of Monoids, it looks like they are an object inside a category with one object. I have also noticed that they have operations like composition and identity which must be ...
3
votes
1answer
97 views

Difference between the simplicial nerve and the nerve of a simplicial category

In Jacob Lurie's Higher Topos Theory book, he defines the following notion of a simplicial nerve: Definition 1.1.5.5. Let $\mathcal{C}$ be a simplicial category. The simplicial nerve ...
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2answers
70 views

Adjoint functors and exactness.

If $F, G$ are adjoint functors between two abelian categories, then if $G$ is exact would that imply $F$ is also? If not what assumptions need be made on $F$ or $G$ for this to hold?
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50 views

Presheaf Clarification

so I am reading through the Wikipedia article on sheaves, specifically the part on expressing the notion of a presheaf in terms of category theory. The article states that to define a presheaf on a ...
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1answer
73 views

1-1 correspondence between nuclei and regular monomorphisms of a locale

I am having a little trouble with Theorem 2.3 in Professor Johnstone's book on "Stone Spaces". The theorem depends on a Lemma (which I am not struggling with; I only include it for context) which ...
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0answers
57 views

Closed Monoidal Structures On The Category Of Complete Topological Vector Spaces

Context: The category of Banach spaces, with the projective tensor product is a closed monoidal category. Question 1: Is there a tensor product on the category of complete topological vector spaces, ...
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1answer
75 views

What is the category of internal locales in a topos equivalent to?

I (think) have heard in a conference, in passing, the sentence ''there is an equivalence between internal locales in a topos $\mathbb{S}$ and localic $\mathbb{S}$-topoi''. Is this true in any sense? ...
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1answer
130 views

Field Extensions and their dimensions

There is a powerful theorem with respect to a field $F$ extensions and their dimensions. $F<E<K \ \Rightarrow [K:F] = [K:E][E:F] $ This is analogous to the famous Lagrange's theorem with ...
5
votes
2answers
250 views

Does first isomorphism theorem hold in the category of normed linear spaces?

Consider the category of normed linear spaces over $\mathbb{C}$ with bounded linear maps as morphisms. If $M\subset X$ is a subspace, then the quotient space $X/M$ has a map $\|x+M\|: = \inf_{y\in ...
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81 views

Constructing Dual notions via Duality

First of all, I do not have much mathematical background and I have minimal category theory knowledge. I am just trying to understand one or two things about category theory because the concept sounds ...
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1answer
113 views

Isomorphism through adjunction

An adjunction $F \dashv G$ gives a morphism $\phi(f) : A \to G B$ to each morphism $f : F A \to B$. Does $\phi(f)$ have any special property if I know that $f : F A \to B$ is an isomorphism?
2
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1answer
327 views

Free product of groups as coproduct

Wikipedia says "The coproduct in the category of sets is simply the disjoint union with the maps ij being the inclusion maps. Unlike direct products, coproducts in other categories are not all ...
4
votes
1answer
98 views

Triangulated Categories question from Neeman's book

Lemma 1.2.4 on page 39 of Neeman's book Triangulated Categories states: Suppose we are given a candidate triangle $$X\rightarrow A\oplus Y\stackrel{\left(\begin{array}{cc}1&\alpha\\ ...
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0answers
55 views

Characterization of certain maps in $Hom(A \otimes A^{*}, A \otimes A^{*})$

Let $(M, \otimes, I)$ be a symmetric monoidal category and let $(A, B, \eta, \epsilon)$ be a dual pair in $M$. Consider maps $i_{A}: Hom(A, A) \rightarrow Hom(A \otimes B, A \otimes B)$, $i_{B}: ...
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2answers
407 views

There are two types of functors; covariant and contravariant. Is it right?

In the category theory, there are lots of functors between the categories. I thought that however each functor must be either covariant or contravariant, for instance, the identity functor is ...
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0answers
34 views

When can we complete bundle morphisms when given just a morphism of total spaces?

Consider a bundle $E \rightarrow B$ in some category. The morphisms of it to some other bundle $E' \rightarrow B'$ is simply two morphism $E \rightarrow E'$ and $B \rightarrow B'$ which make the ...
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1answer
185 views

An example of covariant functor.

Let $F$ be the following covariant functor from the category of sets to the category of left module over a ring $R$ with identity. For each set $X$, $F\left(X\right)$ is the free $R$-module on $X$. If ...
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0answers
34 views

EXAMPLE on p.58, in Hungerford(GTM)

In line 6, there is an explanation, '$h$ is an equivalence in $E$ if and only if $h$ is an equivalence in $C$. ' Suppose that $h : B \rightarrow D$ is an equivalence in $C$. Then $h$ is an ...
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1answer
90 views

Some questions in the Category theory.

I have two simple questions about the category theory. In any category, is $Hom(A, B)$ always nonempty? In some typical categories, it seems right but the definition of morphism does not give any ...
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2answers
193 views

Which of these constructions are left adjoints?

A groupoid can be regarded as a small category in which every arrow is an isomorphism A monoid can be regarded as a small category with only a single object A preorder can be regarded as a small ...
2
votes
1answer
118 views

Why does the pushout preserve monic in an abelian category?

In this question the poster says that, if one of the two maps with the same domain is monic, then the corresponding induced map in the pushout diagram is also monic, in an abelian category. ...
4
votes
2answers
78 views

How to identify the object $K$ in $K-\operatorname{Vect}$ categorically?

How does one identify the field $K$ in the category of vector spaces $K-\operatorname{Vect}$ over $K$? The obvious objects are to try are the initial & terminal objects, but these aren't right, as ...
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3answers
869 views

What does “Arrows are more important than objects” really mean?

I am a final year undergraduate student and I am trying to learn category theory. I am familiar with the basic notions. I am reading Pareigis's notes, ...
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1answer
94 views

At a closed monoidal category, how can I derive a morphism $C^A\times C^B\to C^{A+B}$?

Let $A$, $B$ and $C$ be objects of a closed monoidal category which is also bicartesian closed. How can I derive a morphism $C^A\times C^B\to C^{A+B}$? $(-)\times (-)$ denotes the product, $(-)+(-)$ ...
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2answers
171 views

Is this thing really a category?

i am having difficulty understanding why the diagram on page 8 of this presentation is a category. the author claims that it is on page 43. it looks like the two smaller arrows on the left must be ...
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0answers
54 views

How much does a theory tell about categorical properties of its models.

Let take a first order theory $T$ in a language $\mathcal L$. One can form the category $\mathrm{Mod}(T)$ whose objects are the models of $T$ (those $\mathcal L$-structures $\mathfrak M$ such that ...
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1answer
30 views

Can a category have heterogeneous arrows?

For example, Poset has objects all posets and morphisms that preserve structure. Can the arrows be different operations? An example?
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1answer
158 views

Endomorphisms of forgetful functor $\mathbf{Grp}\to \mathbf{Set}$

It is well known endomorphisms of faithful functor form a monoid. I was trying to determine monoid of endomorphisms of forgetful functor $\mathbf{Grp} \to \mathbf{Set}$, and found it to be ...