Overcategory

 ( Category Theory ) 

In mathematics, an overcategory (also called a slice category) is a construction from category theory used in multiple contexts, such as with covering spaces (espace étalé). They were introduced as a mechanism for keeping track of data surrounding a fixed object $X$ in some category $\backslash mathcal\{C\}$. The dual notion is that of an undercategory (also called a coslice category).

---

Definition Let $\backslash mathcal\{C\}$ be a category and $X$ a fixed object of $\backslash mathcal\{C\}$^{pg 59}. The overcategory (also called a slice category) $\backslash mathcal\{C\}/X$ is an associated category whose objects are pairs $(A,\; \backslash pi)$ where $\backslash pi:A\; \backslash to\; X$ is a morphism in $\backslash mathcal\{C\}$. Then, a morphism between objects $f:(A,\; \backslash pi)\; \backslash to\; (A\text{'},\; \backslash pi\text{'})$ is given by a morphism $f:A\; \backslash to\; A\text{'}$ in the category $\backslash mathcal\{C\}$ such that the following diagram commutes

There is a dual notion called the undercategory (also called a coslice category) $X/\backslash mathcal\{C\}$ whose objects are pairs $(B,\; \backslash psi)$ where $\backslash psi:X\backslash to\; B$ is a morphism in $\backslash mathcal\{C\}$. Then, morphisms in $X/\backslash mathcal\{C\}$ are given by morphisms $g:\; B\; \backslash to\; B\text{'}$ in $\backslash mathcal\{C\}$ such that the following diagram commutes

These two notions have generalizations in 2-category theory and higher category theory^{pg 43}, with definitions either analogous or essentially the same.

---

Properties Many categorical properties of $\backslash mathcal\{C\}$ are inherited by the associated over and undercategories for an object $X$. For example, if $\backslash mathcal\{C\}$ has finite products and , it is immediate the categories $\backslash mathcal\{C\}/X$ and $X/\backslash mathcal\{C\}$ have these properties since the product and coproduct can be constructed in $\backslash mathcal\{C\}$, and through universal properties, there exists a unique morphism either to $X$ or from $X$. In addition, this applies to limits and colimits as well.

---

Examples

---

Overcategories on a site Recall that a site $\backslash mathcal\{C\}$ is a categorical generalization of a topological space first introduced by Grothendieck. One of the canonical examples comes directly from topology, where the category $\backslash text\{Open\}(X)$ whose objects are open subsets $U$ of some topological space $X$, and the morphisms are given by inclusion maps. Then, for a fixed open subset $U$, the overcategory $\backslash text\{Open\}(X)/U$ is canonically equivalent to the category $\backslash text\{Open\}(U)$ for the induced topology on $U\; \backslash subseteq\; X$. This is because every object in $\backslash text\{Open\}(X)/U$ is an open subset $V$ contained in $U$.

---

Category of algebras as an undercategory The category of commutative $A$-algebras is equivalent to the undercategory $A/\backslash text\{CRing\}$ for the category of commutative rings. This is because the structure of an $A$-algebra on a commutative ring $B$ is directly encoded by a ring morphism $A\; \backslash to\; B$. If we consider the opposite category, it is an overcategory of , $\backslash text\{Aff\}/\backslash text\{Spec\}(A)$, or just $\backslash text\{Aff\}\_A$.

---

Overcategories of spaces Another common overcategory considered in the literature are overcategories of spaces, such as schemes, , or topological spaces. These categories encode objects relative to a fixed object, such as the category of schemes over $S$, $\backslash text\{Sch\}/S$. Fiber products in these categories can be considered intersections (e.g. the scheme-theoretic intersection), given the objects are of the fixed object.

---

See also

• Comma category

Post Comment