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November 23, 2013 / porton

Changes to my article about products in certain categories

There are two changes in Products in dagger categories with complete ordered Mor-sets draft article:

1. I’ve removed the section on relation of subatomic product with categorical product saying that for funcoids they are the same. No, they are not the same. My claim that they are the same was false.

2. Added section “Special case of funcoids” with a theorem and two new open problems:

Proposition \prod^{\mathsf{FCD}} a \mathrel{\left[ \prod f \right]} \prod^{\mathsf{FCD}} b \Leftrightarrow \forall i \in \mathrm{dom}\, f : a_i \mathrel{[ f_i]} b_i for an indexed family f of funcoids and indexed families a and b of filters where a_i \in \mathfrak{F} ( \mathrm{Src}\, f_i), b_i \in \mathfrak{F} ( \mathrm{Dst}\, f_i) for every i \in \mathrm{dom}\, f.

Conjecture \left\langle \prod f \right\rangle x = \prod^{\mathsf{FCD}}_{i \in \mathrm{dom}\, f} \langle f_i \rangle \Pr^{\mathsf{FCD}}_i x for an indexed family f of funcoids and x \in \mathrm{atoms}^{\mathsf{FCD} ( \lambda i \in \mathrm{dom}\, f : \mathrm{Src}\, f_i)} for every n \in  \mathrm{dom}\, f.

A weaker conjecture:

Conjecture \langle f \times g \rangle x = \langle f \rangle \mathrm{dom}\, x \times^{\mathsf{FCD}} \langle g \rangle \mathrm{im}\, x for funcoids f and g and x \in \mathrm{atoms}^{\mathsf{FCD} ( \mathrm{Src}\, f ; \mathrm{Src}\, g)}.

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One Comment

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  1. porton / Nov 23 2013 21:51

    I hastened. The prior statement that subatomic product of funcoids coincides with categorical product was indeed true. Consequently the above conjectures are also true.

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