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# Direct products in a category of funcoids

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

There are defined (see my book) several kinds of product of (any possibly infinite number) funcoids:

- 1.
cross-composition product

- 2.
subatomic product

- 3.
displaced product

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

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

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.

## Mathematics Subject Classification

54J05*no label found*54A05

*no label found*54D99

*no label found*54E05

*no label found*54E17

*no label found*54E99

*no label found*

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