Subcategories and Products of Categories

Czeslaw Bylinski

Warsaw University, Bialystok

Supported by RPBP.III-24.C1.

Summary.

The {\it subcategory} of a category and product of categories is defined. The {\it inclusion functor} is the injection (inclusion) map $E \atop \hookrightarrow$ which sends each object and each arrow of a Subcategory $E$ of a category $C$ to itself (in $C$). The inclusion functor is faithful. {\it Full subcategories} of $C$, that is, those subcategories $E$ of $C$ such that $\hbox{Hom}_E(a,b) = \hbox{Hom}_C(b,b)$ for any objects $a,b$ of $E$, are defined. A subcategory $E$ of $C$ is full when the inclusion functor $E \atop \hookrightarrow$ is full. The proposition that a full subcategory is determined by giving the set of objects of a category is proved. The product of two categories $B$ and $C$ is constructed in the usual way. Moreover, some simple facts on $bifunctors$ (functors from a product category) are proved. The final notions in this article are that of projection functors and product of two functors ({\it complex} functors and {\it product} functors).

MML Identifier: CAT_2

Contents (PDF format)

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