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I am looking for a reference on the different possible definitions of subobjects. According to a particular friend of mine, subobjects should be at least monomorphisms (up to slice isomorphism) and at most regular monomorphisms. However, I wish to fact-check this information. When I asked my friend for clarification on this at a later time, he said that it was primarily from his own intuition and not from any particular reference. Here is a quote from my friend to give more context on what he could have meant:

"e.g., finite posets and finite directed acyclic graphs are roughly the same objects, but because of how we use graphs vs posets they have different classes of subobjects ... for posets we're only interested in regular monomorphisms ... for DAGs we're interested in any monomorphism"

I also added the order theory tag because he is primarily someone who does order theory and that might be further important context.

Post edited based on moderator feedback.

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    $\begingroup$ I like the general thrust of the question, since it's asking for intuitions that often do not see the formal light of day. But I feel some inner resistance towards the formulation. The use of "at most" and "at least" looks backwards to me, because to me "at least" would refer to a minimal set of conditions, not a more stringent set. "It is generally agreed" is not something you can assert if you're getting your information "purely from a friend". As a category theorist, I'm not sure I agree that there's always a "right" notion of subobject for a given category. Depends on what you want to do. $\endgroup$ Commented Jul 25 at 2:10
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    $\begingroup$ @ToddTrimble: “at most”/“at least” can easily switch based on phrasing, since there’s always a duality between the natural order on conditions (weaker ≤ stronger) and the classes of things satisfying them (smaller ≤ larger). So “widgets should be assumed at least foos and at most bars” is equivalent to “widgets should include at least the bars and at most the foos”. // In the bigger view, though, while I like questions that probe issues of intuition, I think this one is slightly too open-ended to admit any reasonable answer. $\endgroup$ Commented Oct 27 at 10:37
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    $\begingroup$ Even in the case of posets and graphs, we often want to consider only induced subposets or induced subgraphs (I guess this is analogous to full subcategories mentioned in Martin Brandenburg's answer). $\endgroup$ Commented Nov 3 at 2:41

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Basically there is only one definition of the general term "subobject" of $X$. It is an isomorphism class of monomorphisms into $X$. (Some sources remove the isomorphism class part and just take monomorphisms as the definition. Ok that makes two definitions then.)

It just turns out that in many categories this condition is much too weak. For example, in the category of schemes, the monomorphisms are the radicial formally unramified morphisms, but the term subscheme refers to something much more restrictive and adapted to schemes appearing in practice. Same for submanifolds. To remedy this, one may fix a class $\mathcal{M}$ of monomorphisms and define a $\mathcal{M}$-subobject to be an isomorphism class of morphisms into $X$ that belong to $\mathcal{M}$. See Definition 7.77 in Joy of Cats. One should assume that $\mathcal{M}$ contains all isomorphisms and is closed under composition, but this isn't always the case.

There are all sorts of general types of monomorphisms (see Remark 7.76 in op.cit. for a summary: iso => section => regular => strict => swell => strong => extremal => mono), but in concrete categories we can choose whatever fits our purpose, even though sometimes it just so happens that one of the general types already fits (subposets = regular subobjects). For example, a subscheme is a $\mathcal{I}$-subobject in the category of schemes for the class $\mathcal{I}$ of locally closed immersions. To answer the question in the title: Of course, this example does not change the general meaning of "subobject" for this category.

Many definitions can now be made relative to $\mathcal{M}$, for example the notions of $\mathcal{M}$-wellpowered category (Definition 7.82 in op.cit.), $\mathcal{M}$-injective object and $\mathcal{M}$-essential morphisms (Definition 9.22 in op.cit.), $\mathcal{M}$-partial morphism (Definition 28.1 in op.cit.) and $\mathcal{M}$-topos (Definition 28.7 in op.cit.), where a $\mathrm{Mono}$-topos is the same as a topos in the usual sense.

Having said this, it should be clear that even for a fixed category there is not a correct choice of $\mathcal{M}$-subobjects to look at. As Todd already commented, it depends on what you want to do with them. Also in the example of the category of all $\mathcal{U}$-categories, in some situations we need subcategories (which indeed are just subobjects in the categorical sense), but in others we need full subcategories, wide subcategories, or replete subcategories.

The story also dualizes, for example we get the notion of a regular-quotient, which captures what we usually understand as quotients of algebraic structures, and of a regular-projective object, which by definition is able to lift morphisms along regular epimorphisms. For any algebraic theory, free algebraic structures are regular-projective, but not necessarily projective since there might be too many epimorphisms.

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