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Let $C$ be a category and let $C'$ be the wide subcategory whose maps are projections, that is maps in $C$ which belong to some limiting cone (over a discrete base). Since limiting cones compose, $C'$ is indeed a subcategory of $C$ (containing all isomorphisms of $C$).

For instance, if $C$ is the category of finite sets, a mapping $f:A\to B$ is in $C'$ if and only if all fibers $f^{-1}y$ have the same cardinality. Thus the posetal reflection of $C'$ is equivalent to the poset of natural numbers ordered by divisibility. Note that this $C'$ has not finite products (although a terminal set is terminal also in $C'$). Or (dually) one can consider topological spaces obtaining the category of embeddings which are summands.

My question:

has this notion been already considered?

are there interesting instances?

Improve this question
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    $\begingroup$ We've found some use for the finite sets example that you mention in categorical probability, see Definition 3.1.4 in A Probability Monad as the Colimit of Spaces of Finite Samples. $\endgroup$
    – Tobias Fritz
    Commented Jan 9, 2023 at 18:05
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    $\begingroup$ I would not consider it to be a subcategory but rather a class of morphisms. In a category with finitary products, this would be the smallest class of morphisms containing the isomorphisms and the projections to the terminal object and closed under pullback. In particular, in a category with pullbacks, this is the smallest class of fibrations making it a category of fibrant objects. $\endgroup$
    – Zhen Lin
    Commented Jan 9, 2023 at 22:59
  • $\begingroup$ @ZhenLin, re, what is the difference between a wide subcategory and a class of morphisms closed under composition? $\endgroup$
    – LSpice
    Commented Jan 10, 2023 at 0:44
  • $\begingroup$ You might say it’s a difference in perspective. Personally I don’t find it useful to think of a class of fibrations as a category in its own right, so why should I think of it as a subcategory at all? $\endgroup$
    – Zhen Lin
    Commented Jan 10, 2023 at 2:12
  • $\begingroup$ Because "wide subcategory" is shorter than "class of morphisms containing all identities and closed under composition"? (-: $\endgroup$
    – Mike Shulman
    Commented Jan 11, 2023 at 4:39

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