# Questions tagged [coproduct]

For questions related to coproduct. In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces.

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### Coproduct of free objects [duplicate]

In the category of groups, the coproduct is free product, and for any $A,B$ sets: the coproduct of $F(A),F(B)$ is $F(A\sqcup B)$. Does this property hold in any other categories? What is the ...
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### does anyone know of a book where it shows a sketch of the proof of the statement?

I hope everyone is well. A few days ago I was studying a bit about homological algebra and in an article they showed the following: ''Given the finitely generated $R-$modules $L_i,$ $i \in I,$ and the ...
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### Two definitions of Direct sum

In a linear algebra course, a direct sum of two vector spaces $V, W$ is defined as a vector space $V\oplus W$ where $V \cap W = \{ 0 \}$. And in Group Theory, a direct sum of abelian groups $A_i$ is ...
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### When is coproduct called disjoint union?

Disjiont union forms coproduct in the category of sets. The coproduct in the category of topological spaces is given by endowing a natural topology on the disjiont union of the underlying sets of ...
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### How to do the following exercise about showing the sum of two identical two elements group is infinite.

The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes Definition for $\textit{Coproducts in}$ $\textbf{Mon}$ $\textit{and}$ $\textbf{Grp:}$ ...
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### Getting a coproduct from a product

In $\operatorname{Set}$, one can construct the coproduct (here the disjoint union) of some sets $(S_λ)_{λ \in Λ}$ as follows: We take $$\prod_{λ \in Λ} \prod_{x \in S_λ} \{0_x, 1_x\}$$ and then we ...
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### Regarding a morphism going to a coproduct in a category

I've been working with coproducts, and, as expected, dealing with a morphism to a coproduct is quite difficult. One thing that would help me is to work with morphisms that factor through their ...
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### Is true that $f:∐_{σ∈Σ}X_σ⟶X$ is continuous iff $f∘ι_σ$ is continuous when $ι_σ$ is the standard embedding of $X_σ$ into $∐_{σ∈Σ}X_σ$?
Given a collection of topological spaces $$\mathfrak X:=\{X_\sigma:\sigma\in\Sigma\}$$ we let consider the natural inclusion  \iota_\sigma: X_\sigma\ni x_\sigma\longrightarrow(x_\sigma,\sigma)\in\...