Why C O K E R ( ⨁ Φ : I → J A I → ⨁ I ∈ I A I ) \mathrm{coker}(\bigoplus_{\varphi:i\to J}A_i\to \bigoplus_{i\in I}A_i) Coker ( ⨁ Φ : I → J ​ A I ​ → ⨁ I ∈ I ​ A I ​ ) Defines The Colimit C O L I M I ∈ I A I \mathrm{colim}_{i\in I}A_i Colim I ∈ I ​ A I ​

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Introduction to Colimits and Cokernels

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In the realm of Homological Algebra, the concepts of colimits and cokernels play a crucial role in understanding the structure of abelian categories. A colimit is a way to construct a new object from a family of objects, while a cokernel is a way to describe the quotient of an object by a subobject. In this article, we will explore why the cokernel of a specific map defines the colimit of a family of objects.

The Definition of Colimits

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A colimit of a family of objects $\{A_i\}_{i\in I}$ in an abelian category $\mathcal A$ is an object $A$ together with a family of morphisms $\mu_i: A_i \to A$ such that for any object $B$ and a family of morphisms $f_i: A_i \to B$, there exists a unique morphism $f: A \to B$ such that $f \circ \mu_i = f_i$ for all $i \in I$. The object $A$ is called the colimit of the family $\{A_i\}_{i\in I}$, and the morphisms $\mu_i$ are called the colimiting morphisms.

The Definition of Cokernels

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A cokernel of a morphism $f: A \to B$ in an abelian category $\mathcal A$ is an object $C$ together with a morphism $\epsilon: B \to C$ such that $\epsilon \circ f = 0$ and for any object $D$ and a morphism $g: B \to D$ such that $g \circ f = 0$, there exists a unique morphism $h: C \to D$ such that $h \circ \epsilon = g$.

The Cokernel of a Specific Map

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In the proof of Small direct sums exists if and only if the abelian category $\mathcal A$ is cocomplete, it is shown that the cokernel of a specific map defines the colimit of a family of objects. The map in question is the map $\bigoplus_{\varphi:i\to j}A_i\to \bigoplus_{i\in I}A_i$, where $\varphi:i\to j$ is a family of morphisms between the objects $A_i$.

The Construction of the Cokernel

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To construct the cokernel of the map $\bigoplus_{\varphi:i\to j}A_i\to \bigoplus_{i\in I}A_i$, we need to consider the quotient of the object $\bigoplus_{i\in I}A_i$ by the subobject generated by the image of the map $\bigoplus_{\varphi:i\to j}A_i\to \bigoplus_{i\in I}A_i$. This quotient object is denoted by $\mathrm{coker}(\bigoplus_{\varphi:i\to j}A_i\to \bigoplus_{i\in I}A_i)$.

The Colimiting Morphisms

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The colimiting morphisms $\mu_i: A_i \to \mathrm{coker}(\bigoplus_{\varphi:i\to j}A_i\to \bigoplus_{i\in I}A_i)$ are defined as follows: for each $i \in I$, the morphism $\mu_i$ is the composition of the inclusion morphism $A_i \to \bigoplus_{i\in I}A_i$ and the quotient morphism $\bigoplus_{i\in I}A_i \to \mathrm{coker}(\bigoplus_{\varphi:i\to j}A_i\to \bigoplus_{i\in I}A_i)$.

The Universality of the Colimit

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The colimit $\mathrm{colim}_{i\in I}A_i$ is universal in the sense that for any object $B$ and a family of morphisms $f_i: A_i \to B$, there exists a unique morphism $f: \mathrm{colim}_{i\in I}A_i \to B$ such that $f \circ \mu_i = f_i$ for all $i \in I$. This universality property is a key feature of colimits, and it is what makes them useful in a wide range of applications.

The Connection to Cokernels

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The cokernel of the map $\bigoplus_{\varphi:i\to j}A_i\to \bigoplus_{i\in I}A_i$ defines the colimit $\mathrm{colim}_{i\in I}A_i$ because the quotient object $\mathrm{coker}(\bigoplus_{\varphi:i\to j}A_i\to \bigoplus_{i\in I}A_i)$ satisfies the universality property of colimits. In other words, the cokernel of the map $\bigoplus_{\varphi:i\to j}A_i\to \bigoplus_{i\in I}A_i$ is the object that is "left behind" when we quotient out the image of the map, and this object is precisely the colimit of the family $\{A_i\}_{i\in I}$.

Conclusion

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In conclusion, the cokernel of a specific map defines the colimit of a family of objects because the quotient object satisfies the universality property of colimits. This connection between cokernels and colimits is a key feature of Homological Algebra, and it has far-reaching implications for the study of abelian categories.

References

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• Weibel, Charles A. (1994). An Introduction to Homological Algebra. Cambridge University Press.
• Mac Lane, Saunders (1998). Categories for the Working Mathematician. Springer-Verlag.
• Freyd, Peter (1966). Abelian Categories. Harper & Row.
  

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Q: What is the significance of the cokernel of a specific map in Homological Algebra?

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A: The cokernel of a specific map is significant because it defines the colimit of a family of objects. In other words, the cokernel of the map $\bigoplus_{\varphi:i\to j}A_i\to \bigoplus_{i\in I}A_i$ is the object that is "left behind" when we quotient out the image of the map, and this object is precisely the colimit of the family $\{A_i\}_{i\in I}$.

Q: What is the universality property of colimits?

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A: The universality property of colimits states that for any object $B$ and a family of morphisms $f_i: A_i \to B$, there exists a unique morphism $f: \mathrm{colim}_{i\in I}A_i \to B$ such that $f \circ \mu_i = f_i$ for all $i \in I$. This property is a key feature of colimits, and it is what makes them useful in a wide range of applications.

Q: How does the cokernel of the map $\bigoplus_{\varphi:i\to j}A_i\to \bigoplus_{i\in I}A_i$ satisfy the universality property of colimits?

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A: The cokernel of the map $\bigoplus_{\varphi:i\to j}A_i\to \bigoplus_{i\in I}A_i$ satisfies the universality property of colimits because the quotient object $\mathrm{coker}(\bigoplus_{\varphi:i\to j}A_i\to \bigoplus_{i\in I}A_i)$ is the object that is "left behind" when we quotient out the image of the map. This object is precisely the colimit of the family $\{A_i\}_{i\in I}$.

Q: What is the connection between cokernels and colimits?

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A: The connection between cokernels and colimits is that the cokernel of a specific map defines the colimit of a family of objects. In other words, the cokernel of the map $\bigoplus_{\varphi:i\to j}A_i\to \bigoplus_{i\in I}A_i$ is the object that is "left behind" when we quotient out the image of the map, and this object is precisely the colimit of the family $\{A_i\}_{i\in I}$.

Q: Why is the connection between cokernels and colimits important in Homological Algebra?

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A: The connection between cokernels and colimits is important in Homological Algebra because it provides a way to construct colimits from cokernels. This is useful in a wide range of applications, including the study of abelian categories and the construction of derived functors.

Q: What are some common applications of the connection between cokernels and colimits?

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A: Some common applications of the connection between cokernels and colimits include:

• The study of abelian categories: The connection between cokernels and colimits is used to construct colimits in abelian categories, which is an important tool in the study of these categories.
• The construction of derived functors: The connection between cokernels and colimits is used to construct derived functors, which are an important tool in Homological Algebra.
• The study of homological algebra: The connection between cokernels and colimits is used to study the homological properties of abelian categories, which is an important area of research in Homological Algebra.

Q: What are some common misconceptions about the connection between cokernels and colimits?

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A: Some common misconceptions about the connection between cokernels and colimits include:

• The idea that cokernels and colimits are equivalent concepts: While cokernels and colimits are related, they are not equivalent concepts. Cokernels are a way to describe the quotient of an object by a subobject, while colimits are a way to construct a new object from a family of objects.
• The idea that the connection between cokernels and colimits is only useful in specific cases: The connection between cokernels and colimits is a general result that applies to a wide range of situations, including the study of abelian categories and the construction of derived functors.

Q: What are some common pitfalls to avoid when working with the connection between cokernels and colimits?

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A: Some common pitfalls to avoid when working with the connection between cokernels and colimits include:

• Failing to distinguish between cokernels and colimits: Cokernels and colimits are related but distinct concepts, and it is easy to get them confused.
• Failing to check the universality property of colimits: The universality property of colimits is a key feature of colimits, and it is essential to check that it holds in any given situation.
• Failing to consider the context in which the connection between cokernels and colimits is being used: The connection between cokernels and colimits is a general result that applies to a wide range of situations, but it is essential to consider the context in which it is being used to avoid misunderstandings.