2
votes
0answers
44 views

Push-out of product of push-out diagrams

Let $\pi(U\cup V)$ be the push-out of the diagram $\pi(U)\leftarrow \pi(U\cap V) \rightarrow \pi(V)$ that appears when we apply Vam-Kampen Theorem to the open sets $U,V$ in a topological space $X$. ...
3
votes
1answer
48 views

Connected index category and limit of constant functor

This is question 8 from chapter 4 section 2 of MacLane's Category for the Working Mathematician: If the category J is connected, prove for any $c \in C$ that $Lim \triangle c \cong c$ and $Colim ...
1
vote
2answers
25 views

Monic group morphism is an injection.

I am trying to prove that a monic group morphism is an injection. Does this follow from the existence of a free group in one generator? Thank you for any thoughts on this.
0
votes
1answer
46 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 ...
0
votes
2answers
58 views

how to understand that products and coproducts are dual

I am reading some basic category notes, how can one relate the products to coproducts? If given a product, can one build its dual product? for example, the coordinate product $(x,y)$ : $ R \times R$, ...
3
votes
1answer
118 views

Right adjoint unique up to isomorphism

i want to prove the following without the Yoneda Lemma (because it is the exercise): Suppose $F\dashv G$ (with unit $\eta$ and counit $\epsilon$) and $F\dashv G'$ (with unit \eta' and conunit ...
1
vote
1answer
33 views

If a category has pullbacks and a terminal object, then it has (binary) products

$\require{AMScd}$ The questions is in the title. Here is what I have attempted. I want to know if this is enough. Let $1$ be the terminal object in a category $K$. Given objects $X, Y$ consider the ...
1
vote
1answer
39 views

find the left and right adjoints of the inclusion of integers into real numbers

Any ordered set $S$ can be considered a category as follows: The objects are the elements of $S$ and there is a unique morphism $ s \rightarrow s'$ IFF $s \leq s'$ for the inclusion functor $ i : Z ...
2
votes
1answer
72 views

questions about extremal epimorphisms in category theory

let $K$ be a category with equalizers, show that every extremal epimorphism is epic. for composable morphisms $f: A \rightarrow B $ and $g: B \rightarrow C$ in $K$, show that if $gf$ is an extremal ...
1
vote
1answer
86 views

show the following conditions are equivalent for a category $ C $

I need to show the following conditions are equivalent for a category $ C $ (a) $ C $ has binary products, equalizers, and a terminal object; (b) $ C $ has pullbacks and a terminal object; (c) $ C ...
2
votes
1answer
89 views

problem on products in category theory

Let $C$ be the category of torsion abelian groups. (1) How do you prove that products are representable in $C$? (2) Could you also please give me an example where the product in $C$ is not isomorphic ...
1
vote
0answers
58 views

Easy characterization of Cohomology in an Abelian Category

It should be quite an easy question and probably there's also a certain degree of intrinsic silliness in it, but still... Let $\mathcal{C}$ be an abelian category and let $C(\mathcal{C})$ be the ...
1
vote
1answer
68 views

Coproducts in a poset considered as a category.

This is a homework question. We have defined products in a category over an index set $I$ as follows: "A (possibly infinite) product in a category $\mathbb C$ is a limit of a diagram whose shape is a ...
1
vote
1answer
37 views

Monoidal Category - Equalizer

We have a category $\mathbb C$ with finite products and terminal object $1$. Further $\mathsf{Mon}(\mathbb C)$ is the category of monoids in $\mathbb C$ where a monoid is a triple $(M,m:M\times M ...
0
votes
1answer
40 views

Equiv. of Cats. Preserves Product

Show that an equivalence of categories sends products to products and coproducts to coproducts. That is, if $X_i$ are a family of objects in $\mathcal{C}$ with coproduct $X$ then $F(X)$ is the ...
1
vote
1answer
51 views

Isomorphisms in different categories

I know what it means to be a isomorphism in a given category. But now i want to prove the following statements: Isomorphisms are exactly the bijections in the catgeory of Sets (Sets) Isomorphisms ...
3
votes
2answers
59 views

isomorphic coequalizers

Suppose $e: B \rightarrow C$ is the coequalizer of two parallel morphisms $f,g; A \rightarrow B$. Show that if $e$ is monic then it is an isomorphism. I know that $e$ is epic. If $e$ is monic ...
-1
votes
1answer
34 views

Coequalizers of parallel morphisms

Show if $e : B \rightarrow C$ is the coequalizer of two parallel morphisms $f,g: A \rightarrow B$ and $w: W \rightarrow A$ is epic, then $e$ is the coequalizer of parallel morphisms $fw,gw: W ...
1
vote
2answers
76 views

morphisms on topological spaces

In the category of topological spaces: 1.) Show that a morphism is monic IFF it is injective 2.) Show that a morphism is epic IFF it is surjective 3.) Are there any morphisms that are monic and ...
1
vote
0answers
60 views

Functors that have a natural Isomorphism

Find different functors $T, S: Rng \rightarrow Rng$ both identity on objects IE: for each ring $R, T(R)=S(R)=R$, such that there is a natural isomorphism between T and S. I know that a natural ...
0
votes
0answers
44 views

What is an effective descent morphism?

In the book I'm reading(Galois Theories, Francis Borceux & George Janelidze) defines as follows. Definition 4.4.1 Let $C$ be a category with pullbacks. A morphism $f:X\rightarrow Y$ is an ...
10
votes
1answer
164 views

Functoriality of the Fundamental group

The fundamental group is a functor from the category of pointed topological spaces to the category of groups. Therefore every base-point preserving continuous function $f$ between pointed ...
4
votes
1answer
166 views

Left adjoint in a functor category

Edit: Originally put "right adjoint" instead of "left adjoint"; now changed. If I have small categories $\mathcal{C},\mathcal{D}$ with $F : \mathcal{C} \to \mathcal{D}$ a functor then I want to show ...
5
votes
1answer
159 views

Elephant: how do I prove Lemma 2.1.7, section C2.1?

I'm referring (also for notations and terminology) to P. Johnstone, Sketches of an Elephant. A Topos Theory Compendium. Volume I. Clarendon Press. Oxford, 2002. The Lemma can be found at page 540. I ...
11
votes
2answers
297 views

coequalizers+pullbacks implies equalizers?

The question is on the title, I would like a hint on this exercise. This is what I've tried so far: Suppose we're given $f,g:A\rightarrow B$, let $h=\operatorname{Coeq}(f,g)$, then we have parallel ...
3
votes
1answer
135 views

Forgetful Functors Create Limits

I'm working on the following problem but I can't seem to make any headway. A widget is a set $A$ with elements $0,1 \in A$, a ternary operation $[-,-,-]: A^3 \to A$, and for each rational number ...
1
vote
2answers
53 views

Groupoids isomorphism

Let $G, G'$ be two groups and $X=\{x,y\}$ be a set of two elements. Consider a groupoid $\mathcal{G}$ with objects from $X$ such that Hom$(x,x)=G$ and Hom$(y,y)=G'$. Suppose Hom$(x,y) \neq ...
2
votes
1answer
57 views

An example of a map that has no section but each of its fibers are not empty

"Conceptual mathematics" by Lawvere and Schanuel, 2nd ed. on page 82 says: ... If one fiber is empty, the map has no sections. Furthermore, for maps between finite sets the converse is also ...
5
votes
1answer
84 views

Exact sequence from $G=G_0\supset G_1\supset G_2\supset\cdots\supset G_r=\{e\}$

This is Exercise 5.3 from Algebra: Chapter 0. Given a normal series of subgroups \begin{equation}G=G_0\supset G_1\supset G_2\supset\cdots\supset G_r=\{e\}, \end{equation} construct an exact ...
1
vote
1answer
51 views

Eigenvalues and equalizers

I am asked to relate eigen{value,vectors} with equalizers in the category of matrices. However, in the category of matrices $g\circ f$ means the (matrix) product $fg$. And $\lambda$ is an eigenvalue ...
1
vote
1answer
44 views

Product in a preorder

Is the product of two objectos $a,b$ in a preorder category their infimum $a\wedge b$? I can't assume that their infimum exists, can I?
1
vote
2answers
158 views

An epimorphism in $\text{Grp}$ without right inverse?

Exercise 8.24 in Aluffi's Algebra: Chapter 0 asks us to find an epimorphism in $\text{Grp}$ without right inverses. I happen to know that epimorphisms in $\text{Grp}$ are surjective, so we need a ...
6
votes
1answer
106 views

Pushout of open map is open

I have been struggling with the following problem. Consider the pushout for topological spaces (or adjunction space) $B \cup_A C$ obtained by gluing together $B$ and $C$ along $A$ by means of ...
1
vote
2answers
93 views

Full subcategory and inclusion functor

I have the suspicion that if $A$ is a subcategory of $B$, then the inclusion functor $A \rightarrow B$ is full. Is this right?
2
votes
1answer
65 views

Composition of functors

I'm having trouble presenting a proof for the intuitive fact that the composition of two full functors is again full. Mostly, I'm having trouble doing the symbolic transformations that get me to the ...
1
vote
1answer
123 views

Natural transformation monomorphism condition

If I have functors from C to Set for a small category C and a natural transformation between them, how can I show that this natural transformation is monomorphic iff each of its components, indexed by ...
2
votes
1answer
103 views

Questions about adjointness of quantifiers in first-order logic

I have a category theory homework problem which asks: "In first-order logic, why does $\forall$ not have a right adjoint?" The typical argument is that: If for some operator $\cdot$ it could be ...
3
votes
1answer
79 views

Isomorphisms in the category of real vector spaces

The problem is to show that in the category of real vector spaces the direct product of countably infinitely many $\mathbb{R}$ is isomorphic to $\mathbb{R}[[t]]:= \sum\limits_{j=1}^\infty a_j t^j, ...
1
vote
1answer
171 views

Balanced Categories, Full/Faithful Functors and Monomorphic Units/Counits

I wish to show that for an adjunction $F: \mathcal{C} \to \mathcal{D} \dashv G: \mathcal{D} \to \mathcal{C}$, if both the unit $\eta$ and the counit $\epsilon$ are monomorphisms and $\mathcal{C}$ is ...
4
votes
1answer
229 views

Koszul Complex Homology

I'm attempting to understand Eisenbud's proof that: If $x_1,x_2,\ldots,x_i$ is an $M$-sequence, then $H^i(M\otimes K(x_1,...,x_n))=((x_1,\ldots,x_i)M:(x_1,\ldots,x_n))/(x_1,\ldots,x_i)M$. Here ...
1
vote
1answer
66 views

Finding the source and target categories of a certain Functor

From "Algebra of Programming" (Bird, de Moor), the exercise 2.11 reads: For any category $\mathcal{C}$, define: H(A,B) = { f | f : A $\leftarrow$ B in $\mathcal{C}$} H(f,h) g = f $\circ$ ...
2
votes
2answers
170 views

$\operatorname{Func}(J,Ab)$ has enough injectives.

I am trying to show that the functor category $\operatorname{Func}(J,Ab)$ has enough injectives (meaning that for each $F\in \operatorname{Func}(J,Ab)$ there is an injective object $I\in ...
0
votes
1answer
57 views

Prove the composition of these map objects are consistent.

I have been working my way through Lawvere and Schanuel (1997) without too much trouble, but now that I am up to Article V, I am stumped. So, without further ado: Exercise 6: In a category with ...
3
votes
1answer
156 views

Image factorisation in pointed categories

Let $\mathbf{C}$ be a pointed category, that is to say, a category with a zero object $0$, and suppose that $\mathbf{C}$ has all kernels and cokernels, and suppose also that every monomorphism in ...
0
votes
1answer
78 views

monics in Sets with unary operation

I want to find a monic in category OSet defined as "sets with unary operation, $(A,x)$, where $x:A\rightarrow A$, and morphism preserving that operation, that is a morphism from $(A,x)$ to $(B,y)$ ...
0
votes
1answer
281 views

Meaning of 'pullback of a pullback square'

I'm trying to solve a problem which asks me to deduce that "the pullback of a pullback square is a pullback", using the result of http://ncatlab.org/nlab/show/pullback (under 'Pasting of pullbacks') ...
3
votes
1answer
143 views

Do all functors preserve split coequalisers?

I have a homework problem asking me "which kind of functors preserve split coequalisers?" - I have seen multiple online sources (such as the comments in ...
3
votes
0answers
128 views

Simple Category Theory/Grps Problem

I'm doing a little homework but I think my brain has ceased to function late at night, was hoping you could help me out with what I am certain is a very simple group/cat theory problem. Similarly to ...
0
votes
1answer
149 views

Factorization of functors into full+faithful and object-bijective factors

I'm doing a problem to show that any functor $F: \mathcal{A} \to \mathcal{C}$ between categories can be factorized as $F_L: \mathcal{A} \to \mathcal{B},\,F_R: \mathcal{B} \to \mathcal{C}$, where ...
5
votes
0answers
227 views

Abelian categories, axioms AB5 and AB5* and incompatability

This is a homework exercise, so please don't post full solutions to the question below. Grothendieck (I believe) introduced several axioms an abelian category A voluntarily could satisfy. In ...