PlanetPhysics/Grothendieck Category

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Generator, Generator Family and Cogenerator

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

By [[../TrivialGroupoid/|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 $𝒞$, and also the notion of cogenerator $U^{*}$ of $𝒞$, if all of the required, reverse arrows exist. Notably, in a groupoid-- regarded as a [[../Cod/|small category]] with all its morphisms invertible-- this is always possible, and thus a [[../QuantumOperatorAlgebra5/|groupoid]] can always be cogenerated via duality. Moreover, any generator in the dual category ${𝒞}^{op}$ is a cogenerator of $𝒞$.

Ab-conditions: Ab3 and Ab5 conditions

1. (Ab3) . Let us recall that an Abelian category $𝒜b$ is cocomplete

(or an $𝒜b3$-category) if it has arbitrary direct sums.

1. (Ab5). A [[../CocompleteAbelianCategory/|cocomplete Abelian category]] $𝒜b$ is said to be an $𝒜b5$-category if for any directed family ${\left\{A_i\right\}}_{i\in I}$ of subobjects of $𝒜$, and for any subobject $B$ of

$𝒜$, 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 [[../Predicate/|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 [[../TrivialGroupoid/|Abelian groups]] such that if ${\left\{X_i\right\}}_{i\in I}$ is a family of abelian groups, then a direct product $\mathrm{\Pi }$ is defined by the Cartesian product ${\mathrm{\Pi }}_i(X_i)$ with addition defined by the rule: $(x_i)+(y_i)=(x_i+y_i)$. One then defines a projection $\rho :{\mathrm{\Pi }}_i(X_i)\to X_i$ 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 $𝒜b{5}^{*}$ category that has a set of cogenerators,

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

Remarks

1. Let Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}} be an [[../AbelianCategory2/|abelian category]] and $𝒞$ a small category.

One defines then a [[../TrivialGroupoid/|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 (unknown function "\A"): {\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 $\bar{f}:coimf\to imf$ is an [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphism]] (the Failed to parse (unknown function "\A"): {\displaystyle \mathcal{\A}b2} condition).