# categorical product

## A construction of direct product in the category of continuous maps between endo-funcoids ★★★

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).

\begin{conjecture} The above defines categorical direct product (in the above mentioned category, with products of morphisms the same as in **Set**).\end{conjecture}

Keywords: categorical product; direct product

## Hedetniemi's Conjecture ★★★

Author(s): Hedetniemi

\begin{conjecture} If $G,H$ are simple finite graphs, then $\chi(G \times H) = \min \{ \chi(G), \chi(H) \}$. \end{conjecture}

Here $G \times H$ is the \Def[tensor product]{tensor product of graphs} (also called the direct or categorical product) of $G$ and $H$.

Keywords: categorical product; coloring; homomorphism; tensor product