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The product of two categories $\A$ and $\B$ is the category $\A \times\B$ with objects being pairs $(A,B)$, having $A \in \A$ and $B \in \B$, and morphism similarly defined by pairs of appropriate morphisms. This definition simply extends to $n$-arry products.
It also seems that there is a simple extension to a family of categories $\C_i$ indexed by the set $I$. Category $ \prod_{i \in I}\C $ should have dependend functions $X : I \to \C_i, X : i \mapsto X_i$ as objects with evident morphisms. What concerns me here is consistence of certain $\infty$-variable Galois connections.
For example, for every topological space it is possible $X$ to define small order category $\T X$ with open sets as objects, and morphisms corresponding to the relationship $(\subseteq)$. Evidently, if we use above definition of the product of categories for the infinite set $I$, and product topology defined by natural property for the product of topological spaces, then:
$$ \prod_{i \in I} \T X_i \not\subseteq \T\prod_{i \in I} X_i $$
in general, as topology on $\prod_{i \in I} X_i$ will be generated by products of open sets $\prod_{i\in I}U_i$ with only finite number of $U_i \neq X_i$. This is bad as results in topology and measure theory involving infinite products may not translate well to the categorical language.
To overcome this issue I propose definition of special product of categories, $\prod^\wedge_{i \in I} \C_i$ with dependent functions as above for objects, but with only finite number of values not being terminal, or dually $\prod^\vee_{i \in I} \C_i$ with only finite number of values not being initial. Probably, this type of products need to be define only for categories with these universal objects. Finite products will be equal to normal products. Furthermore, any motphisms will be defined by a finite number of arrows from original categories.
In my example, then
$$
{\prod_{i \in I}}^\wedge \T X_i \subseteq \T \prod_{i \in I} X_i,
$$
and in fact $\prod_{i \in I}^\wedge \T X_i$ is the base for the topology of $\prod_{i \in I} X_i$.
Probably such definitions of products only may make sense in more narrow order theory of even lattice theory. I haven"t seen anything like this defined before.
Are non-standard products of categories like these used somewhere (Not necessarily exactly like these)? Сan you provide a reference if that is a case?
Thank you for reading this long question.
Best Answer
It seems to me that your construction should be something similar (i.e. equivalent if not isomorphic to the following).
Let $(\mathbf C_i)_{i \in I}$ be a family of categories with a terminal object. Then we have a natural diagram $D$ parametrized by the posetetal category of finite subsets of $I$, this diagram associates
to each finite subset $J \subseteq I$ the category $\prod_{i \in J}\mathbf C_i$
to each inclusion $J_1 \subseteq J_2$ the natural embedding $\prod_{i \in J_1} \mathbf C_i \to \prod_{i \in J_2} \mathbf C_i$.
The colimit of this diagram should provides your special product.
Note that this is basically the direct-sum construction for rings and modules (over a fixed ring), in the last case this construction is also the coproduct, and it should specialize in the case of $\mathbf {Lex}$ to the coproduct described by ne-.
Hope this helps.