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I would like to know if there is a special name for the following concept, papers that feature something similar or a general reference. Let $\mathcal{C}$ be a category and $\mathcal{D}$ a subcategory (or more generally a subclass of objects and morphisms). For a diagram $F:J\to\mathcal{C}$, say that the $\mathcal{D}$-limit of $F$ is a cone in $\mathcal{C}$ that satisfy a universal property respect to all cones whose vertex and morphisms to the vertex are in $\mathcal{D}$. $\mathcal{D}$-colimit would be the same concept but with cocones. It is clear that a $\mathcal{C}$-limit is precisely a usual limit, while any cone is a $\emptyset$-limit.

I am particularly interested in understanding in greater generality the behaviour of the examples in the following questions:

-Do colimits of manifolds coincide with underlying colimits as topological spaces?

-Are GIT's good categorical quotients just locally ringed space coequalizers?

-Colimits of manifolds

M-complete category is certainly related but too specific.

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This is such a natural notion, that I feel it must be written about somewhere, but I don't think it's in any of my usual sources like Borceux's books, Mac Lane's Categories for the working mathematician, or Emily Riehl's book Category Theory in Context.

It seems to me that such a notion, if previously invented, might be called a "relative limit" or "relative colimit." It feels like the sort of thing Grothendieck could have written about. Googling led me to writings of Lurie definining the analogous notion for $\infty$-categories, e.g., in Higher Topos Theory (4.3.1) and at this link in Kerodon.

The theory of Mahavier limits is also related, as this paper by Ittay Weiss spells out.

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We can view a category $C_0 \hookrightarrow C$ equipped with a specified subcategory as an $\mathscr M$-category, which is a category enriched in the category of injections and commutative squares. As with any enriched category, we have the notion of weighted limit, which is an enriched notion of limit. Roughly speaking, a weighted limit for an $\mathscr M$-category comprises a limit in $C$ whose projections are in $C_0$, and for which the mediating morphism induced by the limit is in $C_0$ if each of its composite with each of the projections is in $C_0$. This seems to fit the description you give.

Unfortunately, I'm not aware of any explicit references for weighted limits in this setting: the closest reference I can find gives a description of weighted limits for $\mathscr F$-categories, which are a 2-categorical generalisation of $\mathscr M$-categories: see Proposition 3.6 of Lack–Shulman's Enhanced 2-categories and limits for lax morphisms. Alternatively, we can see these limits as kinds of limits in the double category where one class of morphisms is given by $C_0$ and the other is given by $C$: see Grandis–Paré's Limits in double categories, for instance, although this specific example is not mentioned.

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    $\begingroup$ I am not very familiarised with the concept of $\mathscr M$-category, but I think this would be only a special case. In particular, I understand that $C_0$ would have the same objects as $C$, that is, would be a wide subcategory. Some "nice" properties as the unicity of the weighted limit would hold, while this doesn't happen in general. Also, the unique morphism that always from the limit to other cones doesn't have to exist either, in the general case. Still, it is useful to consider a "well-behaved" case, thank you. $\endgroup$
    – Arturo
    Commented Dec 12, 2022 at 11:18
  • $\begingroup$ Yes, this is true. $\endgroup$
    – varkor
    Commented Dec 12, 2022 at 11:33

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