direct product

A construction of direct product in the category of continuous maps between endo-funcoids โ˜…โ˜…โ˜…

Author(s): Porton

Consider the category of (proximally) continuous maps (entirely defined monovalued functions) between endo-funcoids.

Remind from my book that morphisms $ f: A\rightarrow B $ of this category are defined by the formula $ f\circ A\sqsubseteq B\circ f $ (here and below by abuse of notation I equate functions with corresponding principal funcoids).

Let $ F_0, F_1 $ are endofuncoids,

We define $ F_0\times F_1 = \bigsqcup \left\{ \Phi \in \mathsf{FCD} \,|\, \pi_0 \circ \Phi \sqsubseteq F_0 \circ \pi_0 \wedge \pi_1 \circ \Phi \sqsubseteq F \circ \pi_1 \right\} $

(here $ \pi_0 $ and $ \pi_1 $ are cartesian projections).

Conjecture   The above defines categorical direct product (in the above mentioned category, with products of morphisms the same as in Set).

Keywords: categorical product; direct product

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