PlanetPhysics/Grothendieck Category

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Generator, Generator Family and Cogenerator

Let ${\mathcal {C}}$ be a category. Moreover, let $\left\{U\right\}=\left\{U_{i}\right\}_{i\in I}$ be a family of objects of ${\mathcal {C}}$. The family $\left\{U\right\}$ is said to be a family of generators of the category ${\mathcal {C}}$ if for any object $A$ of ${\mathcal {C}}$ and any subobject $B$ of $A$, distinct from $A$, there is at least an index $i\in I$, and a morphism, $u:U_{i}\to A$, that cannot be factorized through the canonical injection $i:B\to A$. Then, an object $U$ of ${\mathcal {C}}$ is said to be a generator of the category ${\mathcal {C}}$ provided that $U$ belongs to the family of generators $\left\{U_{i}\right\}_{i\in I}$ of ${\mathcal {C}}$ ().

By duality, that is, by simply reversing all arrows in the above definition one obtains the notion of a family of cogenerators $\left\{U^{*}\right\}$ of the same category ${\mathcal {C}}$, and also the notion of cogenerator $U^{*}$ of ${\mathcal {C}}$, if all of the required, reverse arrows exist. Notably, in a groupoid-- regarded as a small category with all its morphisms invertible-- this is always possible, and thus a groupoid can always be cogenerated via duality. Moreover, any generator in the dual category ${\mathcal {C}}^{op}$ is a cogenerator of ${\mathcal {C}}$.

Ab-conditions: Ab3 and Ab5 conditions

1. (Ab3) . Let us recall that an Abelian category ${\mathcal {A}}b$ is cocomplete

(or an ${\mathcal {A}}b3$-category) if it has arbitrary direct sums.

1. (Ab5). A cocomplete Abelian category ${\mathcal {A}}b$ is said to be an ${\mathcal {A}}b5$-category if for any directed family $\left\{A_{i}\right\}_{i\in I}$ of subobjects of ${\mathcal {A}}$, and for any subobject $B$ of

${\mathcal {A}}$, the following equation holds

$(\sum _{i\in I}A_{i})\bigcap B=\sum _{i\in I}(A_{i}\bigcap B).$

Remarks

• One notes that the condition Ab3 is equivalent to the existence of arbitrary direct limits .
• Furthermore, Ab5 is equivalent to the following proposition: there exist inductive limits and the inductive limits over directed families of indices are exact , that is, if $I$ is a directed set and $0\to A_{i}\to B_{i}\to C_{i}\to 0$ is an exact sequence for any $i\in I$, then Failed to parse (unknown function "\limdir"): {\displaystyle 0 \to \limdir{(A_i)} \to \limdir{(B_i)} \to \limdir{(C_i)} \to 0} is also an exact sequence.
• By duality, one readily obtains conditions Ab3* and Ab5* simply by reversing the arrows in the above conditions defining Ab3 and Ab5 .

Grothendieck and co-Grothendieck Categories

A Grothendieck category  is an Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b5}
 category

with a generator.

As an example consider the category Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b} of Abelian groups such that if $\left\{X_{i}\right\}_{i\in I}$ is a family of abelian groups, then a direct product $\Pi$ is defined by the Cartesian product $\Pi _{i}(X_{i})$ with addition defined by the rule: $(x_{i})+(y_{i})=(x_{i}+y_{i})$. One then defines a projection $\rho :\Pi _{i}(X_{i})\rightarrow X_{i}$ given by $p_{i}((x_{i}))=x_{i}$. A direct sum is obtained by taking the appropriate subgroup consisting of all elements $(x_{i})$ such that $x_{i}=0$ for all but a finite number of indices $i$. Then one also defines a structural injection , and it is straightforward to prove that Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b} is an Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b6} and Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b4^*} category. (viz . p 61 in ref. ).

A co-Grothendieck category  is an ${\mathcal {A}}b5^{*}$ category that has a set of cogenerators,

i.e., a category whose dual is a Grothendieck category.

Remarks

1. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \mathcal{\A}} be an abelian category and ${\mathcal {C}}$ a small category.

One defines then a functor Failed to parse (unknown function "\A"): {\displaystyle k_c: \mathcal{\A} \rightarrow [\mathcal{C},\mathcal{\A}]} as follows: for any Failed to parse (unknown function "\A"): {\displaystyle X \in Ob \mathcal{\A}} , Failed to parse (unknown function "\A"): {\displaystyle k_{\mathcal{C}}(X) : \mathcal{C} \rightarrow \mathcal{\A}} is the constant functor which is associated to $X$. Then Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}} is an Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A'' b5} category} (respectively, Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b5^*} ), if and only if for any directed set $I$, as above, the functor $k_{I}$ has an exact left (or respectively, right) adjoint.

1. With Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \mathcal{\A}b4} , Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b5} , Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b4^*} , and Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b6}

one can construct categories of (pre) additive functors.

1. A preabelian category is an \htmladdnormallink{additive category {http://planetphysics.us/encyclopedia/DenseSubcategory.html} with the additional (Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b1} ) condition} that for any morphism $f$ in the category there exist also both $kerf$ and $cokerf$;
2. An Abelian category can be then also defined as a \em{preabelian category} in which for any morphism $f:X\to Y$, the morphism ${\overline {f}}:coimf\to imf$ is an isomorphism (the Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b2} condition).