Cartesian Categories

Journal of Formalized Mathematics
Volume 4, 1992
University of Bialystok
Copyright (c) 1992 Association of Mizar Users

Cartesian Categories

Czeslaw Bylinski: Warsaw University, Bialystok

Summary.

We define and prove some simple facts on Cartesian categories and its duals co-Cartesian categories. The Cartesian category is defined as a category with the fixed terminal object, the fixed projections, and the binary products. Category $C$ has finite products if and only if $C$ has a terminal object and for every pair $a, b$ of objects of $C$ the product $a\times b$ exists. We say that a category $C$ has a finite product if every finite family of objects of $C$ has a product. Our work is based on ideas of [10], where the algebraic properties of the proof theory are investigated. The terminal object of a Cartesian category $C$ is denoted by $\hbox{\bf 1}_C$. The binary product of $a$ and $b$ is written as $a\times b$. The projections of the product are written as $pr_1(a,b)$ and as $pr_2(a,b)$. We define the products $f\times g$ of arrows $f: a\rightarrow a'$ and $g: b\rightarrow b'$ as $:a\times b\rightarrow a'\times b'$.\par Co-Cartesian category is defined dually to the Cartesian category. Dual to a terminal object is an initial object, and to products are coproducts. The initial object of a Cartesian category $C$ is written as $\hbox{\bf 0}_C$. Binary coproduct of $a$ and $b$ is written as $a+b$. Injections of the coproduct are written as $in_1(a,b)$ and as $in_2(a,b)$.

MML Identifier: CAT_4

The terminology and notation used in this paper have been introduced in the following articles [12] [5] [13] [8] [11] [14] [2] [3] [9] [1] [6] [4] [7]