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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4
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1answer
52 views

Showing that localization is an exact functor

I'm again in this awfully familiar situation where I'm struggling to prove simple statements mostly because I have no idea how a template of a proof should look like in this specified context. I'm ...
2
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1answer
52 views

Braided Hopf algebra - properties

If $(H,\Delta,\nabla)$ is a Hopf algebra in the prebraided monoidal category $(\mathcal{C},\Psi)$ then $\Psi_{H,H}=\left(\nabla\otimes \nabla\right)\left(S\otimes\Delta \nabla\otimes ...
5
votes
2answers
82 views

Left adjoint of the forgetful functor $\mathsf{Grpd} \to \mathsf{Cat}$?

I've heard that there is a left adjoint to the forgetful functor $\mathsf{Grp} \to \mathsf{Mon}$. I wonder if there is also a left adjoint $F : \mathsf{Cat} \to \mathsf{Grpd}$ to the forgetful functor ...
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1answer
39 views

Is the reflective localization $L_WC$ of a category $C$ equivalent to $C$? What am I missing?

This is probably a dumb question but this is going over my head at the moment, I came here from nlab's entry on localization (http://ncatlab.org/nlab/show/localization). Let $C$ be a category, let $W ...
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1answer
38 views

Induced morphisms into pullback

The induced morphism by the universal property of pullback, when is it an epimophism ( I'm looking at it in a regular category, when it's induced from a coproduct )
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0answers
23 views

Replacing a covering in a site with a single arrow

When we work in a site with a pretopology, what is the phenomenon called that one can replace a covering $\lbrace A_i\to B\rbrace$ with a single covering $V\to B$ (is $V$ always the pullback ...
2
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0answers
82 views

Which sequential colimits commute with pullbacks in the category of topological spaces?

Given diagrams of topological spaces $$X_0\rightarrow X_1\rightarrow\ldots$$ $$Y_0\rightarrow Y_1\rightarrow\ldots$$ $$Z_0\rightarrow Z_1\rightarrow\ldots$$ and maps $X_i\rightarrow Z_i$, ...
2
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0answers
110 views

Kock Frobenius algebras and 2D TQFTs

My mathematical career consists of selfstudy of the first 7 chapters of Lee topological manifolds. I want to read the book Kock Frobienius algebras and 2d TQFTs. Can you suggest which books and what ...
1
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1answer
43 views

Is this assignment of the topos of sheaves functorial?

Let $\mathcal{C}$ be a site and for any object $X$ of $\mathcal{C}$ denote by $\text{Sh}(X)$ the category of sheaves on the site $\mathcal{C}/X$. My question is, what can we say about this assignment? ...
3
votes
1answer
56 views

Choosing projective replacement to be functorial

A basic result of homological algebra says that if $\mathsf A$ is an abelian category with enough projectives, then the mapping $P:\mathsf{Obj}(\mathsf A)\rightarrow \mathsf{Obj}(\mathsf{K} ^+(\mathsf ...
3
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1answer
77 views

Universal property and colimit

The free group $F(S)$ is the group given by a set $S$ with the universal property: For every group $G$ and map $f: S \to G$ there is a unique homomorphism $\phi: F(S) \to G$, such that $\phi \circ i = ...
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0answers
34 views

Sifted colimits of models of a Lawvere theory.

While trying to prove that the monad associated to a Lawvere theory is finitary, I came across the following, which troubles me. Let $\mathcal A$ be small category and $\mathcal C$ be the full ...
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0answers
52 views

Standard notation for the transform that turns a function $A \rightarrow (B \rightarrow C)$ into a function $B \rightarrow (A \rightarrow C).$

Suppose we're given sets $A,B$ and $C$. Then to each function $f : A \rightarrow (B \rightarrow C)$, we can assign another function $F : B \rightarrow (A \rightarrow C)$ by defining: $$F(b)(a) = ...
3
votes
1answer
45 views

Natural transformations preserve exactness.

Is there a quick way to see that if F,G are both functors (say between abelian categories or $R-Mod$ to Ab) and $F \cong G$, then F exact implies G exact?
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0answers
22 views

Approximation to FRel in Cat

FRel is the category of Finite sets and relations.  The category of Small categories is locally finitely presentable, meaning it has all Small colimits.  The compact categories are the finite graphs.  ...
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0answers
46 views

Examples of preadditive categories

The obvious examples for preadditive categories / $\textsf{Ab}$-enriched categories are of course: $R$-$\textsf{Mod}$, the category of $R$-modules for any ring $R$ $\textsf{Ab}$, the category of ...
3
votes
1answer
58 views

Definition of exact sequence of functors.

What is meant by an "exact sequence of functors" in an abelian category? My guess is the following: a sequence of the form: $0 \rightarrow A \stackrel{f} \rightarrow B \stackrel{g} \rightarrow C ...
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0answers
53 views

Notation about commutative diagrams and their vertices

Usually vertices of a commutative diagram are labeled with objects like $A\overset{f}{\leftrightarrow} B$. But now I want to distinguish between vertices of the diagram even if they happen to ...
3
votes
1answer
50 views

Construction of a ring from a category

Here is a straightforward construction I haven't seen before: Let $\mathcal{C}$ be a small category and consider the set $\mathbb{Z}\mathcal{C}$ of $\mathbb{Z}$-formal sums of elements from ...
7
votes
1answer
72 views

This is just the Eilenberg-Moore category, right?

Let $F$ denote a monad on $\mathbf{Set}$. Write $\mathbf{Set}_F$ for the corresponding Kleisli category and $\mathbf{Set}^F$ for the Eilenberg-Moore category. On the train home today, it occured to me ...
0
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0answers
20 views

Definition for a generalization of a natural transformation

Let $\mathcal{C}$ be a small category, and suppose I want to label the morphisms $a,b,c,...$ of $\mathcal{C}$ by the elements of a group $G$. I can simply consider a functor $F_1 \colon \mathcal{C} ...
2
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0answers
76 views

Help to write a proof (category theory diagram)

It is known that $f$, $g$, $h$ are isomorphisms. It is known that $g\circ f = h^{-1}$. I need to write down the proof of the following theorem. I am an amateur mathematician and am not an expert in ...
0
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0answers
20 views

Transitive closure of a relation in categorical logic

How is it possible (if at all) to specify the transitive closure of a relation say $r:A \rightarrow A$ in Rel (i.e., catgeory of sets and relations)? Is it possible to build it using a sequence of ...
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0answers
67 views

How do we prove commutativity of a diagram?

How do we prove commutativity of a diagram? There may be an infinite number of paths. We can't enumerate all paths.
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0answers
77 views

A question regarding Grothendieck , topos and (adelic??) points

I am having a look at this conference by Bertrand Toen about Grothendieck's work. At 1:14:30 and after, Toen presents the new objects emerging from topos theory in algebraic geometry. He takes the ...
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0answers
34 views

Can I define a site as a category endowed with a pretopology instead of a topology?

If $K$ is a pretopology on a category $\mathcal{C}$ and $J$ the topology it induces, are the Grothendieck toposes $\text{Sh}(\mathcal{C},K)$ and $\text{Sh}(\mathcal{C},J)$ the same in general? As I ...
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0answers
29 views

Direct limite commute with Direct sum

Let $\{I_i\}_{i\in \Gamma}$ and $\{J_i\}_{i\in \Gamma}$ be tow direct sets of ideal in a commutative ring with identity such that $\Gamma$ is a chain, dose the following ideal isomorphism is true? ...
3
votes
0answers
68 views

If two monoids have equivalent action categories, are they isomorphic?

If $G,H$ are groups such that $G\mathsf{-Set} \simeq H\mathsf{-Set}$, then $G \cong H$; see math.SE/1375309 for a proof by Zhen Lin. Question. Does this also hold when $G,H$ are monoids? Since there ...
4
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0answers
64 views

Free lattice in three generators

By general results for every set $X$ there is a free bounded lattice $L(X)$ on $X$. I would like to understand the element structure of this lattice. The cases $X=\emptyset$, $X=\{x\}$ and $X=\{x,y\}$ ...
3
votes
1answer
90 views

Are there any free or fascist boolean algebras?

A boolean algebra is an algebra with the binary operations $\wedge$, $\vee$, an unary operation $\neg$, and constants $0$, and $1$, satisfying axioms. A heyting algebra is an algebra with the binary ...
2
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0answers
54 views

How to compute (co)limits of enriched categories?

Let $\mathscr{V}$ be a monoidal category. Let $\mathbf{Cat}_{\mathscr{V}}$ be the category of (small) categories. I would like to know how to compute (co)limits in $\mathbf{Cat}_{\mathscr{V}}$. This ...
6
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0answers
61 views

What are the universally effective epimorphisms of topological spaces?

An effective epimorphism in a category is a morphism that is the coequaliser of its kernel pair, and a universally effective epimorphism is a morphism $f : X \to Y$ such that, for every pullback ...
1
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1answer
35 views

sequence of colimit diagrams

Suppose we have a finite diagram with a colimit. Is it possible to then take the colimit diagram as a new base diagram and then have a new colimit of this new diagram? We could build up diagrams ...
5
votes
1answer
69 views

Functorial properties of the compact open topology.

Let $X,Y,Y'$ be topological spaces and $A\subseteq Y$ a subspace. Every set of continuous maps is equipped with the compact-open topology. Is the canonical map ...
1
vote
1answer
58 views

Prove that a morphism $\alpha$ of $Fun(\mathcal{A},\mathcal{B})$ is an isomorphism iff each component $\alpha_A$, is an isomorphism in $\mathcal{B}$

I'm a computational engineer starting with a course of Introduction to Category Theory, and perhaps is extremely basic what I'm asking but I'm trying to learn how to make proofs in category theory ...
1
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1answer
47 views

A question about combinatorial commutative diagrams.

In the comments of this question the directed graphs that can appear as commutative diagrams are axiomatized: A graph is a combinatorial commutative diagram iff it's nonempty and such that any ...
3
votes
1answer
86 views

Which directed graphs correspond to “algebraic” diagrams?

Any diagram for which the question of commutativity make sence is a directed graph, but not any directed graph make the question meaningful. $\require{AMScd}$ \begin{CD} A @>>> B @. A ...
6
votes
1answer
129 views

Is “Categories and Sheaves” a good followup to Aluffi's “Algebra: Chapter 0”?

I'm about to finish Aluffi's "algebra: chapter 0" and am a bit confused as to what should be my next move. I've been planning to read Tom Dieck's Algebraic Topology for some time now. I glimpsed at it ...
0
votes
1answer
57 views

Topology on compactly supported smooth functions

I'm confused by a set of lecture notes I'm reading and would like help in understanding what's going on. First, there is the following nice theorem. Theorem. The topology of a locally convex space is ...
3
votes
1answer
99 views

How similar are pullbacks to products?

Please excuse me if this is a trivial question. Let $f:A\to B$ and $g:C\to B$ be morphisms in a category and consider their pullback. I have seen books that say $projections$ for the morphisms ...
5
votes
0answers
45 views

Why don't we use the constant sheaf as the dualizing sheaf in Verdier Duality?

The Verdier dual of a complex of sheaves $F$ on a complex manifold $X$ is defined in the derived category $D_b(X)$ as $\mathcal{RHom}(F,\mathbb{D}_X)$, where $\mathbb{D}_X$ is the dualizing "sheaf" ...
4
votes
1answer
71 views

The algebra of natural transformations of the n-th power tensor functor

Let $k$ be a $0$ characteristic field, $n$ an positive integer and $S_n$ the $n$-th symmetric group. Let's work in the symmetric monoidal category of $k$-vector spaces and linear maps that we denote ...
1
vote
1answer
41 views

The inverse limit of C$^*$-algebras and whether it commutes with taking the minimal tensor product

Suppose we are given a C$^*$-algebra $A$ and a family of C$^*$-ideals $\mathfrak{I}$ that is upwards directed when ordered by reverse inclusion (i.e. for any $I_1,I_2\in\mathfrak{I}$ there exists a ...
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1answer
23 views

Does the inverse image sheaf have a left adjoint for $\mathsf{Set}$-valued sheaves?

It's known that for sheaves with values in modules, the inverse image sheaf functor $j^\ast$ for $j:U\subset X$ an inclusion of an open set has a left adjoint which is extension by zero. Is there any ...
7
votes
1answer
81 views

Does an equivalence of $G$-sets and $H$-sets imply an isomorphism of $G$ and $H$?

Here $G$-sets denote the category of sets which have a left $G$-action. So the question is whether a functor $F \colon \text{$G$-sets} \to \text{$H$-sets}$ implies that we have an isomorphism of ...
2
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0answers
47 views

Pullbacks in filtered categories?

A sufficient condition for the inclusion of a full subcategory $\mathsf C\hookrightarrow \mathsf D$ to be cofinal is that: Every object of $\mathsf D$ has an arrow into some object of $\mathsf C$. ...
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43 views

topos have colimits

Define an (elementary) topos to be a cartesian closed category with all finite limits and subobject classifiers. I'm looking for a proof of the fact that a topos also has all finite colimits. I know ...
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1answer
44 views

The Relationship between Separable Functors and Faithful Functors

Consider the adjunction $\mathcal{C} \mathrel{\substack{\mathcal{F}\\\rightleftarrows\\ \mathcal{G}}} \mathcal{D} $ together with unit $\eta: I_{\!_{\mathcal{C}}} \rightarrow \mathcal{G} \mathcal{F} $ ...
2
votes
3answers
71 views

Defining a coproduct in $\mathsf{Grp}$ using group presentations

I've encountered this exercise in Aluffi's Algebra: Chapter 0. It might be helpful to say that the book doesn't introduce functors at this stage,, and that the book defines a presentation of a group ...
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1answer
28 views

Copower functor

Computing copowers and "tensoring with sets" often means the same thing. If a locally small category $\mathcal{C}$ has coproducts and if $S$ is a set then for any object $C\in\mathcal{C}$ the copower ...