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$\newcommand{A}{\mathcal{A}} \newcommand{B}{\mathcal{B}} \newcommand{C}{\mathcal{C}} \newcommand{T}{\mathsf{T}}$

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.

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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.

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  • $\begingroup$ @NikPronko yep, sorry for the typo. $\endgroup$ – Giorgio Mossa Jul 22 '18 at 18:31
  • $\begingroup$ I accepted your answer because of vox populi $\endgroup$ – Nik Pronko Jul 23 '18 at 19:03
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I don't know a reference, but your construction appears to come up when thinking about infinite coproducts in the category $\mathrm{Lex}$ with

  • Objects: small categories with finite limits,

  • Morphisms: functors that preserve finite limits.

(By Gabriel-Ulmer duality, it would also come up when thinking about infinite products in the category of locally finitely presentable categories; you might be able to find references along these lines.)

I claim that if $(\mathcal{C}_i)_{i\in I}$ is a collection of objects in $\mathrm{Lex}$, then their categorical coproduct $\coprod_{i\in I}^{\mathrm{Lex}} \mathcal{C}_i$ is exactly the full subcategory $\prod_{i\in I}^{\wedge}\mathcal{C}_i$ of the product category consisting of those sequence $(c_i)_{i\in I}$ where $c_i$ is terminal for all but finitely many $i\in I$. Observe first that $\prod_{i\in I}^{\wedge}\mathcal{C}_i$ lives in $\mathrm{Lex}$ (whereas the coproduct $\coprod_{i\in I}^{\mathrm{Cat}}\mathcal{C}_i$ does not). Moreover, given maps $f_i\colon \mathcal{C}_i\rightarrow \mathcal{D}$ in $\mathrm{Lex}$, we obtain a unique map $f\colon \prod_{i\in I}^{\wedge}\mathcal{C}_i\rightarrow\mathcal{D}$ in $\mathrm{Lex}$ given by $f((c_i)_{i\in I}) = \prod_{i\in I}f_i(c_i)$; this product exists as most $f_i(c_i)$ are terminal.

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  • $\begingroup$ Thanks, This is interesting. I need to think it over a bit more. $\endgroup$ – Nik Pronko Jul 21 '18 at 17:25

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