# Direct products in a category of funcoids

ADDED: I’ve proved that subatomic product is the categorical product.

There are defined (\hrefhttp://www.mathematics21.org/algebraic-general-topology.htmlsee my book) several kinds of product of (any possibly infinite number) funcoids:

1. 1.

cross-composition product

2. 2.

subatomic product

3. 3.

displaced product

There is one more kind of product, for which it is not proved that the product of funcoids are (pointfree) funcoids:

 $\left\langle f_{1}\times f_{2}\right\rangle x=\bigsqcup\left\{\left\langle f_{% 1}\right\rangle X\times^{\mathsf{\operatorname{FCD}}}\left\langle f_{2}\right% \rangle X\hskip 10.0pt|\hskip 10.0ptX\in\operatorname{atoms}x\right\}.$

It is considered natural by analogy with the category Top of topological spaces to consider this category:

• Objects are endofuncoids on small sets.

• Morphisms between a endofuncoids $\mu$ and $\nu$ are continuous (that is corresponding to a continuous funcoid) functions from the object of $\mu$ to the object of $\nu$.

• Composition is induced by composition of functions.

It is trivial to show that the above is really a category.

The product of functions is the same as in Set.

Title Direct products in a category of funcoids DirectProductsInACategoryOfFuncoids 2013-09-12 19:40:35 2013-09-12 19:40:35 porton (9363) porton (9363) 11 porton (9363) Definition msc 54J05 msc 54A05 msc 54D99 msc 54E05 msc 54E17 msc 54E99