# Tagged Questions

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### 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 $X$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $$G=G_0\supset G_1\supset G_2\supset\cdots\supset G_r=\{e\},$$ construct an exact ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 $(B,y)$ ...
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### 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') ...
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### 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 ...
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### 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 ...
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 ...