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I added links to the horizontal categorifications in group object and created groupoid object.
In groupoid object in an (infinity,1)-category I read the conspicious statement: ”an internal ∞-group or internal ∞-groupoid may be defined as a group(oid) object internal to an (∞,1)-category C with pullbacks” - but this terminology seems to hinder distinguishing between them and ∞-groupoid objects in (∞,1)-categories.
Yes, there’s a bit of the implicit infinity-category theory convention going on in the definition of “groupoid object in an (infinity,1)-category”, which took me a while to realize. An internal 1-groupoid object in an $(\infty,1)$-category requires some additional “1-truncation” property. I’m not sure what I think we should do about that terminology-wise on the nLab. Perhaps it depends somewhat on how interesting 1-groupoid objects in $(\infty,1)$-categories are.
Stephan,
to amplify on what Mike said: the definition of “groupoid in an $(\infty,1)$-category” really is already the notion of $\infty$-groupoid. They need not be distinguished. Instead one has to do extra work, as Mike says, not to automatically get the $\infty$-groupoids.
I find it helpful to think about it this way: one of the equivalent definitions of groupoid object in an $(\infty,1)$-category says that it is a complete Segal space object which is groupoidal.
But a Segal space object is category object in an $(\infty,1)$-category, which however externally comes out as being actually an $(\infty,1)$-category. The reason is that its definition really says that the ordinary associativity axioms on a category are to be formulated internally to the $(\infty,1)$-category, where we cannot but formulate them up to homotopy, so they automatically come out in the $\infty$-version.
Not sure if I said this well. Probably not.
I shall ask a naive question:
Why should not every object in an $(\infty, 1)$-category be, in itself, construed as an internal $\infty$-groupoid? (Insofar as the presheaf it represents is a presheaf of $\infty$-groupoids; the analogy I am thinking is to the fact that, for essentially algebraic structures, an internal $X$ in a $1$-category amounts to the same thing as a presheaf of $X$s whose underlying sets are representable functors)
Why should not every object in an (∞,1)-category be, in itself, construed as an internal ∞-groupoid?
Yes, they are! They are the homotopy colimits of those “groupoid objects”.
Let’s discuss this first for the case of group objects, where it is maybe a bit easier to see.
A “group object in an $(\infty,1)$-category” is defined to be a simplicial object satisfying the Segal property that starts out
$\cdots \stackrel{\to}{\stackrel{\to}{\stackrel{\to}{\to}}} G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *$Here the middle arrow in the triple of arrows encodes a product on $G$, the quadruple of arrows encodes an associator and so on. So this defines in total an object $G$ in the $\infty$-category that is equipped with the structure of a groupal $A_\infty$-algebra.
Taken the $\infty$-colimit over this diagram in the ambient $\infty$-category produces the object $\mathbf{B}G$, which is the delooping of $G$. This is now just a bare object in the ambient $\infty$-category (not equipped with extra structure). From it we reobtain the group object by forming the loop space object.
Now for groupoid objects it is “the same but -oided” story.
So conversely, take any object $A$ in the $\infty$-topos and pick any effective epimorphism. $U \to A$.
If you have been brought up with stacks, think of $U \to A$ as an atlas of the $\infty$-stack $A$. If you rather like logic, think of this as an equivalence relation on $U$ (things in the same fiber are regarded to be equivalent).
The corresponding “internal groupoid object” is the Cech nerve of this morphism $U \to A$.
$\cdots U \times_A U \times_A U \stackrel{\to}{\stackrel{\to}{\to}} U \times_A U \stackrel{\to}{\to} U \,.$Here we see explicitly how the $\infty$-groupoid that $A$ already is, is taken apart into its pieces.
For consider the case that $A$ is a stack presented by a groupoid $A_1 \stackrel{\to}{\to} A_0$ and take $U = A_0$ and $U \to A$ the canonical inclusion. Then $U \times_A U \simeq A_1$ and hence the groupoid object corresponding to the object $A$ of the $\infty$-topos is indeed
$\cdots \stackrel{\to}{\stackrel{\to}{\stackrel{\to}{\to}}} A_1 \times_{t,s} A_1 \times\stackrel{\to}{\stackrel{\to}{\to}} A_1 \stackrel{\to}{\to} A_0 \,.$So this extracts the morphism-components of the $\infty$-groupoid $A$.
I see (well, not entirely, because I still don’t really have infinity-categories under my belt, but somewhat, I think). Would it be fair to say, then, that a “groupoid object” as defined as one of these diagrams is just a particular presentation of an ostensible object (which may not actually exist, if the appropriate colimits are not around), with any particular object having potentially many such presentations? In the same way that an internal equivalence relation in a 1-category is just a particular presentation of an ostensible object (the corresponding quotient object, which may not actually exist if a suitable coequalizer is not around), with any particular object having potentially many such presentations?
So, for example, my question was a bit like asking “Why should not ’internal setoid’ in a 1-category just refer to any old object, rather than to a diagram specifying an equivalence relation imposed on an object?”?
Would it be fair to say, then, that a “groupoid object” as defined as one of these diagrams is just a particular presentation of an ostensible object (which may not actually exist, if the appropriate colimits are not around), with any particular object having potentially many such presentations? In the same way that an internal equivalence relation in a 1-category is just a particular presentation of an ostensible object (the corresponding quotient object, which may not actually exist if a suitable coequalizer is not around), with any particular object having potentially many such presentations?
Yes, I think that is exactly what is going on. Indeed, the “groupoid objects in an $(\infty,1)$-category” are precisely the equivalence relations in the $(\infty,1)$-category.
So in particular any object $A$ is presented by the groupoid object / $\infty$-equivalence relation which is “trivial on $A$”:
$\cdots A \times A \times A \stackrel{\to}{\stackrel{\to}{\to}} A \times A \stackrel{\to}{\to} A \,.$And also this here is precisely the right 1-categorical analogy to keep in mind:
So, for example, my question was a bit like asking “Why should not ’internal setoid’ in a 1-category just refer to any old object, rather than to a diagram specifying an equivalence relation imposed on an object?”?
Since you say “was” I guess this is clear to you now, but let’s just say it anyway for the record and for the sake of other reader:
because there may be many different equivalence relations that define the same quotient.
An internal setoid in a category should be an object $S$ and a “relation object” $R$, namely an object equipped with two morphisms
$R \stackrel{\to}{\to} S$satisfying some conditions. From any such you get an actual object in the category, defined to be the colimit of this diagram, and every object arises this way (in more than one way). This colimit happens to be a coequalizer here, but with an eye towards generlization to higher categories it helps to keep in mind that this is really a colimit over a truncated simplicial object.
I am glad that we had (or are having) this discussion, since it shows which kind of information was missing on the relevant $n$Lab pages.
I have started drawing some first consequences and made the following edits:
I have made all three entries equivalence relation, congruence, groupoid object in an (infinity,1)-category point out the relation to the respective other two.
Then I have added at the third of these after the sentence
A groupoid object is then accordingly the many-object version of a group object.
The paragraph
But notice the following. Since this is defined internal to an (∞,1)-category, externally these look like genuine ∞-groupoid and ∞-group objects. For instance a group object in a (2,1)-category such as Grpd is, externally, a 2-group.
Also notice that if the ambient $(\infty,1)$-category is in fact an (∞,1)-topos, then every object in there may already be thought of as an “∞-groupoid with geometric structure” (see for instance the discussion at cohesive (∞,1)-topos, but this is true more generally). The relation between the internal groupoid objects then and the objects themselves is (an oid-ification) of that of looping and delooping. Notably for $G$ any internal group object (externally an ∞-group) the corresponging ordinary object is its delooping object $\mathbf{B}G$, and every pointed connected object in the $(\infty,1)$-topos arises this way from an internal group object.
This is what I have time for right now, but much more could be added. For instance Examples-sections with more details. Feel free to do so, or to improve on the above paragraphs.
Thanks! In particular to Urs for the fruitful comments in succession of my question (I was indisposed, so I just now come to read them; I doubted the terminology because of stict omega-groupoids with I imagined to be definable in a globular way in a category). Maybe we could create a page colimits over infinity groupoids with contents of this discussion - but I didn’t yet look at the latest changes at equivalence relation, congruence, groupoid object in an (infinity,1)-category to see if this is desirable or better to have it just here- which seems to be along the lines of the motivation of ∞-groupoids in ”Higher Topos Theory” by Jacob Lurie. He emphasizes there the role of effective ∞-groupoids being effective congruences in our nomenclature - a property which is not prominent in the above.
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