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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Why is the coproduct in the category of sets not the cartesian product? [duplicate]
I'm confused as to why the coproduct of sets is the disjoint union, and not just the cartesian product (which, I know, is the categorical product here). Having poured over the definition for a while ...
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Infinite coproduct in a category of groups [duplicate]
Let $\mathbf{Grp}$ be a category of groups.
Then, we know that there exists a coproduct $\bigsqcup_{I\in I} G_i$ for a family of groups $\{G_i\}_{I\in I}$ in $\mathbf{Grp}$ when $I$ is a finite set.
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Kronecker product arising from a coproduct
I'm reading a paper that involves some background on linear algebra, and I came across a sentence that I'm trying to make sense of:
"The tensor/Kronecker product $\otimes$ of representations ...
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Isomorphisms from the external sum of modules to the internal sum
Is this true: Let $R$ be a ring with a $1$. Let $(N_{i})_{i \in I}$ be a collection of $R$-submodules of the $R$-module $M$. If
$$\sum_{i \in I}N_{i} \cong \bigoplus_{i \in I}N_{i}$$
then the map $f: \...
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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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Shuffle product formula for coproduct
I'm studying the coproduct $\Delta$ defined on a tensor algebra $T(V)$ and its action on tensor products of elements from a vector space $V$. The coproduct is given by $\Delta(v) = v \boxtimes 1 + 1 \...
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the free product of two presentations is isomorphic to a third presentation using UP of free product.
Here is the question that I want an answer to it using commutative diagrams (as small number of them as possible):
Prove that the free product of $ \langle g_1, \dots ,g_m | r_1, \dots ,r_n \rangle$ ...
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Distributive property proof.
I was reading this question here: Distributivity of categorical product and sum but I could not understand the statement of the OP that said "If $\textbf{C}$ is $\textbf{Set}$ or $\textbf{Top}$ I'...
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"Asymmetry" in the construction of products and coproducts in concrete categories
Mathematical objects like groups, vector spaces, topological spaces, etc. can be regarded as sets endowed with an additonal structure. The formal setting is that of a concrete category:
A concrete ...
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What exactly is the natural transformation that arises in a coproduct diagram?
Source: Categories for the Working Mathematician, second edition by Saunders Mac Lane.
Given a category $C$, a coproduct diagram is a universal arrow from an object $<a, b>$ of $C\times C$ to ...
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What is, in general, a commutative coproduct, and where can I learn more about it?
This is a follow-up question to this one. I asked about the proof of $U(\mathfrak{g}\oplus \mathfrak{h})\cong U(\mathfrak{g})\otimes U(\mathfrak{h})$ using universal properties, where $U(\mathfrak{g})$...
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Prove that the category of Rings is cocomplete
I am trying to prove that the category of Rings is cocomplete. Of course, it suffices to show that it has coproducts and coequalizers. I looked at this page and I think I get the coequalizer, but I am ...
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Non unital Hopf relation
The following problem is an exercise in Loday-Vallette's Algebraic Operads. I hope I am understanding this correctly. Any suggestions or hints would be appreciated.
Show that the restriction of the
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Awodey's Category Theory Example 3.6. co-product of Top
I'm having problems with Example 3.6 of Awodey's Category Theory Book: In Top, the coproduct of two spaces $X+Y$ is their disjoint union with the topology $ O(X+Y) \cong O(X) × O(Y) $.
But I don't ...
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Definition and construction of disjoint union (topology) using a toy example
Similar to my previous question about quotient spaces, I need to understand the definition and construction of the disjoint union topology. As before, definitions in my textbooks and online has felt ...
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