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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6
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1answer
115 views

map of hom-sets as morphism

I am an undergraduate math student, and while studying cartesian closed categories i encountered a problem i could not solve: Suppose we have a (locally small) cartesian closed category $C$, and let ...
4
votes
1answer
57 views

If $D$ is a triangulated category, and $E_i$ is a set of generators, then $D$ is equivalent to $D(End(\oplus E_i))$?

I am looking for a result along the lines of the following statement: If $D$ is a triangulated category, and $E_i$ is a set of generators (every object can be obtained up to isomorphism by shifts and ...
4
votes
0answers
113 views

Visualizing co-Yoneda lemma

I am studying the Yoneda and co-Yoneda lemmas, and in order to understand them well I am trying to develop particular cases. This one is getting me in trouble (it must be easy, but I cannot get it): ...
1
vote
0answers
50 views

Localization and the universal property

I've been trying to get to grips with the universal property and was looking at the localization of a commutative ring $A$ at some multiplicative set $S$, denoted $S^{-1}A$, to get an idea of what the ...
1
vote
1answer
29 views

Abelian subcategory which is not a Serre subcategory

Is there an example of an abelian subcategory $\mathcal{B}$ of an abelian category $\mathcal{A}$ such that : The inclusion functor $\mathcal{B}\to \mathcal{A}$ is exact. $\mathcal{B}$ is not closed ...
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0answers
32 views

Smallness and Yoneda extension of categories

I have read in Sheaves in Geometry and Logic by Maclane the following theorem: Given a small category and a functor $F: \mathcal{C} \to \mathcal{A}$ with $\mathcal{A}$ cocomplete we can construct ...
4
votes
3answers
368 views

Covariance, contravariance and all that jazz

For the love of God, can someone explain to me the difference between functors of the form $\mathcal{C}^{\text{op}}\to \mathcal{D}$, $\mathcal{C}\to \mathcal{D}^{\text{op}}$ and $\mathcal{C}^{\text{op}...
4
votes
1answer
35 views

Fiber bundles with varying fibers via pullbacks along étale surjections

Suppose $\mathsf C$ is a complete extensive category. I managed to prove that a bundle $\alpha$ is a fiber bundle with fiber $F$ if there exists an associated cover $p:\coprod_iU_i\rightarrow B$ such ...
7
votes
2answers
157 views

In category theory: What is a fibration and why should I care about them?

I stumbled upon the "fibration of points" in the definition of a protomodular category and apparently this is indeed an instance of a fibration. What are fibrations intuition-wise and how are they ...
3
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1answer
69 views

Dual statements involving functors

I know how to construct the dual of a statement concerning objects and morphisms of a category, and understand the duality principle associated, but I am having trouble when various categories and ...
0
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2answers
92 views

Reference for Algebraic Topology

I know undergraduate algebra (groups, rings, fields, Galois etc), undergraduate differential geometry, undergraduate real/complex analysis and now I feel as though to get to the next level, i should ...
0
votes
2answers
70 views

What happens with the empty set in the categorical definition of a presheaf?

I've found in Hartshorne's book on Algebraic Geometry a categorical definition of a presheaf (Ch. 2.1, just after the first definition). He there defines for a topological space $X$ the category $\...
3
votes
1answer
59 views

Amalgam of trees

Definition A tree is a partially ordered set $(T, <)$ such that for each $t \in T$, the set $\{s \in T : s < t\}$ is well-ordered by the relation $<$. For trees $(T,<_T)$, $(S,<_S)$, $...
4
votes
0answers
71 views

What is so special about natural isomorphism?

We know that all finite dimensional vector spaces of the same dimension are isomorphic. In particular, if $V$ is of finite dimension, then $(\star)$$V^{**} \cong V \cong V^{*}$. However, we know that ...
4
votes
1answer
64 views

Natural isomorphism are base independent isomorphisms?

In Linear Algebra courses, often one hears the term "natural isomorphism" to designate isomorphism which behave like $V \cong V^{**}$ (in the finite dimensional case). Usually, one comes across the ...
1
vote
1answer
58 views

Show that if $A$ is free on a set $S$ in Ab is a coproduct of $|S|$ copies of $\mathbb{Z}$

I am working on the following problem: Show that if $A$ is free on a set $S$ in Ab, with map $\mathrm{\Phi}:S \rightarrow A$, then $A$ is also a coproduct of $|S|$ copies of $\mathbb{Z}$, where each $...
6
votes
0answers
62 views

Proving étale maps have discrete fibers by abstract nonsense?

I'm trying to prove that an étale map of spaces has discrete fibers. The first diagram I drew is: $$\require{AMScd} \begin{CD} f^\ast\coprod_i \left\{ x \right\} @>>> f^\ast \coprod_i U_i @&...
6
votes
2answers
787 views

Is TOP a small category?

A quick question... is the category of topological spaces and continuous maps a small category? If so how do we know and if not how do we know not?
2
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1answer
36 views

Isomorphism of Filtered Objects?

Let $\mathscr{C}$ be a category and let $Filt(\mathscr{C})$ denote the category of filtered objects in $\mathscr{C}$. So an object $X$ in $Filt(\mathscr{C})$ look something like $X_0\subset X_1\subset ...
2
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0answers
35 views

Fixed point set as an inverse limit

We can regard a group action on a set as a functor $$ F: BG \to Set\;, $$ where $BG$ is the category with one object and a morphism for each element of $G$, and $Set$ the category of sets. Now, is ...
2
votes
0answers
60 views

Can you prove that modules over a field are vector spaces in a categorical way?

Let $\mathcal{C}$ be the category of abelian groups endowed with tensor product. It is a monoidal category with unit object the integers. Given a monoid $(R, \mu_R , \iota_R)= R$ in this category (i.e....
1
vote
1answer
46 views

Is $Set$ a 2-category?

I've read that $Rel$ (the category of sets and relations) is a 2-category by considering 2-morphisms to be inclusion of relations. Is $Set$ also a 2-category by considering 2-morphisms to be inclusion ...
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0answers
50 views

Locality and base change along effective descent morphisms

Definition. Let $\mathcal M$ be a class of arrows in a site $\mathsf{C}$. An arrow $\pi:E\rightarrow B$ is said to be locally in $\mathcal M$ if there's a covering ${u_i:U_i\rightarrow B}$ such that $...
4
votes
1answer
77 views

What is a composition in category theory?

I'm just beginning to learn category theory. So far, the basic examples (like Set) are making sense. But I'm having a little trouble getting my head around the fundamentals. Suppose I try to define a ...
3
votes
1answer
58 views

When the empty family of arrows to an object is epimorphic, that object must be initial?

Is it true that when the empty family of arrows to an object $E$ in some category is epimorphic, that object $E$ must be the initial object $0$? This is a claim on page 433 (eq. 22) of Mac Lane and ...
0
votes
1answer
40 views

Projective Module equivalence [closed]

I'm trying to prove that given an $A-module$: $P$ The functor $Hom(P,-)$ is exact $\implies$ $\exists Q (A-module): P\oplus Q$ is free Can anyone guide me with some hints? It's the first time I'...
5
votes
0answers
162 views

Can we somehow use the functor $\mathbf{Set}(\mathbb{N},-)$ to define $\mathbb{N}$?

Hom functors can be used define coproducts in terms of products. In particular: $$\mathbf{Set}(A \sqcup B,X) \cong \mathbf{Set}(A,X) \times \mathbf{Set}(B,X)$$ To oversimplify a little: "a function ...
2
votes
0answers
37 views

Characterization of split mono- and split epimorphisms of algebras

Given a concrete category $(\mathcal{A}, U : \mathcal{A} \to \mathsf{Set})$ of universal algebras, obviously if a morphism $f$ in $\mathcal{A}$ is split mono / split epi, then so is $U(f)$. What is ...
0
votes
1answer
36 views

Pullback of a function along identity?

If $f:A\rightarrow B$ is a function, then $A\times _BB= \left\{ (a,b)\mid f(a)=b\right\}$. Isn't this the preimage of the image of $f$? I.e is $A\times_BB\cong f^\ast f_\ast(A)$? If working with ...
5
votes
1answer
81 views

In an algebraic category a morphism is a regular epi iff it is surjective

According to the nLab (see the 4th point under "Examples") in an "algebraic category" a morphism is a regular epi if and only, if it is surjective. Here a morphism $e$ is said to be surjective, if its ...
2
votes
3answers
64 views

Examples for the fact that a pullback of an epimorphism is not necessarily an epimorphism.

I'm reading in Borceux' book Basic Category Theory about pullbacks. It turns out that the pullback of an epimorphism is not necessarily an epimorphism. On the linked page, Borceux gives a ...
2
votes
1answer
33 views

Enough projectives and $F$ preserves limits implies $G$ preserves epi's.

Exercise: Let $\mathcal{C}, \mathcal{D}$ be categories, $G : \mathcal{C} \to \mathcal{D}$ and $F : \mathcal{D} \to \mathcal{C}$ an adjunction $F \dashv G$. Suppose $\mathcal{D}$ has enough projectives ...
3
votes
3answers
113 views

Strange question about free abelian group

I was given the following question for homework, but it makes no sense to me. Let $F$ be a free abelian group over a set $S$ with respect to the function $\varphi \colon S \to F$. Identify the set ...
0
votes
1answer
55 views

post composition as a functor?

For every arrow $f$, we have a post composition functor $f_\ast$. However, one often uses the equality $(gf)_\ast=g_\ast\circ f_\ast$. I was wondering, what's the actual definition of this functor $(-)...
2
votes
0answers
24 views

Modal extensions (operators) for monoidal (categorical) logics

There is nice generalization of first order logic to monoidal (categorical) logics http://www.springer.com/us/book/9783642128202 which has recently been applied extensively as replacement for deontic ...
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vote
0answers
52 views

Existence of pullback of fiber bundles from abstract nonsense?

Let $\mathsf C$ be a superextensive site with products. A trivial fiber bundle is a bundle $\pi:E\rightarrow B$ which is isomorphic to $\pi_1:B\times F\rightarrow B$ in $\mathsf{C}/B$. Let $\mathcal ...
2
votes
1answer
58 views

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products?

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products? I know it preserves products, but it seems to actually create them.
4
votes
0answers
47 views

How to recover a $K$-algebra from endomorphism algebra of forgetful functor?

I'm trying to work out how a finite-dimensional $K$-algebra $B$ can be recovered from the algebra of endomorphisms of the forgetful functor $\omega$ from the category of finitely generated left $B$-...
4
votes
0answers
41 views

Explicit unit/counit of inverse image/direct image adjunction.

Is there a nice explicit description for the unit and counit of the inverse image/direct image adjunction $f^{-1} \dashv f_*$ between sheaves of rings (and in the version $f^* \dashv f_*$ for $\...
9
votes
1answer
410 views

Why is the homotopy category actually important?

Since the homotopy category (of whatever) generally has very few limits and colimits, and these don't coincide with homotopy limits and colimits in the lifted model structure, why do we care about the ...
3
votes
1answer
91 views

“Representability” of $\pi_1:\mathsf{hTop}_\ast \longrightarrow \mathsf{Grp}$?

In this MO question, Qiaochu Yuan asks about limit preservation of "representable" functors which are not $\mathsf{Set}$-valued. The answer gives a simple sufficient condition, possessed by monadic ...
0
votes
1answer
23 views

G adjoint iff initial object in $(D\downarrow G)$

$\def\D{\mathcal{D}}\def\C{\mathcal{C}}$ Let $G: \C \rightarrow \D$ be a functor. For each $D \in \D$ define the category $(D \downarrow G)$ which has as objects pairs $(C,g)$ with $C \in \C$ and $g: ...
1
vote
1answer
32 views

Functor is of the form Set(-,A)

Let $F: Set^{op} \rightarrow Set$ be a functor such that for corresponding functor $\overline{F}: Set \rightarrow Set^{op}$ we have $\overline{F} \dashv F$. With corresponding functor I mean that $F$ ...
4
votes
1answer
79 views

What is this structure involving a monad and a comonad?

Let $F$ be a monad on some category $\mathsf{C}$ and $G$ be a comonad on the same category. Assume further that they "commute" (see below): $FG \cong GF$. Then, for lack of a better name, one can ...
4
votes
1answer
92 views

Functorial modifications of a topology

Let $S$ be a set. Then there are only two ways to attach functorialy a topology $\mathcal{T(S)}$ to it: The discrete and the trivial topology. Functorial means in this case that all maps $f \colon S \...
0
votes
1answer
85 views

The left adjoint to the forgetful functor $G\colon\mathsf{Vect}_\mathbb{C}\to\mathsf{Vect}_\mathbb{R}$ and Barr-Beck

Let $G\colon\mathsf{Vect}_\mathbb{C}\to\mathsf{Vect}_\mathbb{R}$ be the forgetful functor from $\mathbb{C}$-vector spaces to $\mathbb{R}$-vector spaces. I am trying to explicitly construct the left ...
2
votes
0answers
57 views

adjoint functor of inverse image functor

$f: U\hookrightarrow X$ an open immersion of two complex manifolds. $f^{-1}$ is inverse image functor, in usual sense, from category of sheaves of abelian groups $\mathcal{Ab}(X)$ over $X$ to category ...
0
votes
1answer
80 views

Weighted limits in the $Cat$-category of categories

What is a weighted limit in the $Cat$-category of categories, functors and natural transformations? I can find the general definition of a weighted limit for enriched categories in Kelly's book or ...
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0answers
69 views

The free cocompletion of a complete locally small category is complete

$\DeclareMathOperator{\colim}{colim}\newcommand{\cat}{\mathbf}\DeclareMathOperator{\Nat}{Nat}$I'm looking for a reference that talks about the free cocompletion $\hat{\cat C}$ of a (large) locally ...
1
vote
1answer
26 views

principal bundle morphism preserves fundamental group

Two related questions. What is the morphism for principal bundles? Does it "preserve" fundamental groups? Fibre bundle morphisms usually preserve "the structure on the fibre". I am not sure how to ...