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One axiomatisation of set theory, the Elementary Theory of the Category of Sets, or ETCS for short, comes from category theory and states that sets and functions form a locally cartesian-closed, finitely complete and co-complete Heyting pretopos with a subobject classifier and a natural numbers object, whose generating object is the terminal object and whose epimorphisms are split.

Is there a corresponding axiomatic charactersation of the category of groupoids, in which one could do groupoid theory in, that does not involve first defining the concept of $\infty$-groupoid or homotopy types or other infinity categorical structures, and if so, what are the axioms?

And just as ETCS could be considered as a foundation of mathematics that only require sets and propositions, could this Elementary Theory of the Category of Groupoids serve as a more general foundation of mathematics that includes everything that could be done in ETCS as well as providing proper foundations for $1$-category theory? The collection of objects $Ob(C)$ in a category $C$ tend to be a groupoid; when $Ob(C)$ is a set, the category is called strict, so category theory as defined in ETCS or another set theory like ZFC could only speak of strict categories rather than general categories.

Edit: In a finitely complete category, finite limits are saturated under the terminal object and pullbacks. Does this still remain true when one moves to (2,1)-terminal objects and (2,1)-pullbacks and (2,1)-limits in finitely (co)complete (2,1)-categories? For ($\infty$,1)-categories, it doesn't seem to be the case that finite ($\infty$,1)-limits are saturated under ($\infty$,1)-terminal objects and ($\infty$,1)-pullbacks, if I am reading the nLab article on Lurie's ($\infty$,1)-pretopos correctly. In a (2,2)-category, (2,2)-terminal objects and (2,2)-pullbacks are known not to be enough for all finite (2,2)-limits; (2,2)-powers with the interval category are also needed.

Edit 2: I commented somewhere below that this theory as a foundational theory should be expressed in first order logic with isomorphisms, rather than first order logic with equality. I don't think this is the case anymore; ETCS should be the theory expressed in first order logic with isomorphism, as sets in ETCS are only isomorphic rather than equal. Rather, ETCG should be expressed in first order logic with equivalence of groupoids. It is only models of ETCG internal to ETCG that are expressed in first order logic with isomorphism, in the same way that models of ETCS internal to ETCS are expeessed in first order logic with equality.

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    $\begingroup$ I'm nervous to try to write up the answer myself because I'll mess up something about universes, but this is the sort of thing that should be pretty straightforward if you understand the HoTT book and ETCS. You want to do the same kind of translation that relates Martin-Löf type theory with Axiom K (i.e. all types are 0-truncated) to ETCS but instead applied to type theory with a 1-truncation axiom. $\endgroup$ – Noah Snyder Feb 3 at 20:23
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    $\begingroup$ And I'm reasonably certain that there are papers out there studying type theory with a 1-truncaction axiom. The analog of the issue in the last paragraph of my answer in this setting is that you have to figure out what form of univalence you'll have. $\endgroup$ – Tim Campion Feb 3 at 20:48
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    $\begingroup$ @MadeleineBirchfield That's a good point. For that the canonical place to point you would be Makkai's FOLDS. $\endgroup$ – Tim Campion Feb 4 at 1:49
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    $\begingroup$ In the $(2,1)$-case the situation is terminologically a little awkward. There is a strict notion of limit which still exists if you're working with strict (2,1)-categories (which it is often convenient to do) which is defined up to isomorphism, and there is also a weak notion of (2,1)-limit, which should really be defined only up to equivalence, but can also be given an up-to-isomorphism definition as a pseudolimit. The analog of the statement in question is that a (2,1)-category with a pseudoterminal object and pseudopullbacks has all finite pseudolimits up to equivalence. $\endgroup$ – Tim Campion Feb 4 at 3:18
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    $\begingroup$ But an often more convenient way to work with (2,1) limits is to construct them from products, iso-inserters, and equifiers -- so called PIE limits. The precise relationship to pseudolimits is not something I have at my fingertips, but they should be essentially equivalent. $\endgroup$ – Tim Campion Feb 4 at 3:20
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Robert Harper and Dan Licata studied the topic under the name 2-dimensional type theory, see their Canonicity for 2-Dimensional Type Theory and possibly these slides. As they are computer scientists there is a lot of talk about the syntactic properties of such theories, but underneath it really is just groupoids (and they make the connection explicit). If you stare long enough at their rules, they can all be understood in terms of groupiod structures.

In the context of homotopy type theory one can formulate an axiom stating that all types are groupoids: $$\Pi(A : \mathsf{Type}) (x, y : A) (p, q : x = y) (\alpha, \beta : p = q)\,.\, \alpha = \beta.$$ It states that for any type $A$, points $x, y : A$, parallel paths $p, q$ from $x$ to $y$, any 2-cells between $p$ and $q$ are equal. This is a straightforward modification of Streicher's axiom $K$ which states that all types are sets. This may look simpler than Harper's and Licata's 2-dimensional type theory, but is actually not capturing groupoids directly – it's more like 2-truncated $\infty$-groupoids, where all the higher structure is still around, but is declared to be contractible.

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  • $\begingroup$ An extremely lazy search of the slides did not reveal use of the word "univalence". Was I off-base in asserting above that formulating a correct (weak) version of univalence is essentially equivalent to understanding the analog of a (sub)object classifier and hence to getting a good theory? $\endgroup$ – Tim Campion Feb 4 at 16:37
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    $\begingroup$ The paper explains that their type theory is suitably univalent. $\endgroup$ – Andrej Bauer Feb 4 at 16:56
  • $\begingroup$ But yes, understanding the univalent universe is always the hard and illuminating. $\endgroup$ – Andrej Bauer Feb 4 at 17:04
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My rambling comments are starting to cohere into a quasi-answer, though not a definitive one:

  1. In addition to ETCS, Lawvere also formulated ETCC, the elementary theory of the category of categories.

One could very likely arrive at a similar axiomatization of groupiods, either by adapting Lawvere's ETCC axioms, or by directly using the fact that groupoids form a full subcategory of categories.

Of course, the way you formulate the axioms of ETCS reflects a great deal of conceptual development which has happened since Lawvere originally wrote down the theory. I don't know of a similar modernized, sleek way to package Lawvere's axioms for the category of categories, and I think that's partly because it would be harder to do / the necessary concepts may not have been studied (yet?). So if one were to groupoid-ify ETCC, there would still be more work to make the theory more "palatable".

  1. You might also be interested in groupoid models of type theory, where groupoids are thought of in a similar way to what you suggest, lying somewhere between sets and homotopy types and undelrying a theory which can be used foundationally.

  2. I'll also point out that your idea of thinking of the underlying groupoid of a category as a sort of "improved version" of the set of objects of the category does come up. For example, this way of thinking leads to the idea of a complete Segal space as opposed to a Segal category.

Because of the difficulties I alluded to above in groupoid-ifying ETCC, it may be better to return to the comparison to ETCS. The formulation you give, though it doesn't directly invoke the notion of an elementary topos, is closely related to it. A groupoid version of this axiomatization, then, would be talking about something close in spirit to a higher theory of elementary topoi. There has been a great deal of interest in formulating such a theory for a long time.

Going beyond ordinary topoi, the most famous development is Lurie's theory of (Grothendieck) $\infty$-topoi. Nima Rasekh has given a theory of elementary $\infty$-topoi, but the theory is very much in its infancy.

You want something in the middle -- a (2,1)-topos. (The "2" means that you want your topos to be a 2-category; the "1" means that only 1-morphisms and not 2-morphisms will be non-invertible in your category.) I believe that people like Mike Shulman have thought about notions in this area, but I don't know if anything is published.

I haven't thought carefully about this, but the main thing I'd think needs to be straightened out is the following. In ordinary topoi, the subobject classifier does a great deal of work, but in in $\infty$-topoi the concept needs to be strengthened to an object classifier, and size issues cause technical annoyances. The other way of thinking about what the object classifier encodes is in terms of descent (a scary category-theoretic word which means different things to different people, but which in each individual case turns out to be not scary). Thinking in these terms, the theory of $\infty$-topoi actually turns out to be in some sense cleaner than the theory of ordinary topoi. In fact the following is true in $\infty$-topoi but (its corresponding version) is false in ordinary topoi: if $\mathcal E$ is an $\infty$-topos, the "slice" functor $\mathcal E^{op} \to CAT_\infty$, $X \mapsto \mathcal E/X$ preserves limits. I'm not sure whether a (2,1)-topos is expected to be more like $\infty$-topoi or 1-topoi in this regard.

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  • $\begingroup$ Since a subobject classifier is simply an object that classifies $-1$-morphisms; i.e. internal trith values; I would think that the corresponding classifier would be something that could classify $0$-morphisms, or the discrete objects of the groupoid. This might be some internal notion of a groupoid of sets. $\endgroup$ – Madeleine Birchfield Feb 3 at 20:11
  • $\begingroup$ Yes. I should also mention that there has been work on 2-topoi (i.e. (2,2)-topoi in the terminology from above. In this terminology, what is normally called an $\infty$-topos should be called an $(\infty,1)$-topos, and what is normally called a topos or 1-topos should be called a (1,1)-topos).Just as the canonical 1-topos is $Set$, and the canonical $(2,1)$-topos is $Gpd$, the canonical $(2,2)$-topos is $Cat$.The "subobject classifier" of the 2-topos$Cat$ is $Set$. The "subobject classifier" of the (2,1)-topos of groupoids, would be the groupoid of sets $\endgroup$ – Tim Campion Feb 3 at 20:17
  • $\begingroup$ However, the notion of the groupoid of sets in a category of groupoids has size issues, similar to the case in infinity topos theory, so it probably would make more sense to talk about internal groupoid of $\kappa$-small sets. $\endgroup$ – Madeleine Birchfield Feb 3 at 20:26
  • $\begingroup$ Yes. I believe in Weber's theory of 2-topoi, two universes are used. Size issues are important to think about here, and potentially very delicate. This is especially so if one is trying to work foundationally, because a lot of the ways we're used to thinking about size are potentially quite tied up with the details of the metatheory, which is usually ZFC (but if we're trying to build a foundational theory, we're probably trying to avoid using something like ZFC in any kind of metatheory). As Noah Snyder indicates, thinking about this from a type-theoretic perspective would at least mean no ZFC $\endgroup$ – Tim Campion Feb 3 at 20:29
  • $\begingroup$ In the (Grothendieck) $\infty$-topos case, Lurie formally uses universes in the metatheory. That's not really relevant to the issue of thinking about size inside an $\infty$-topos, where he ends up with $\kappa$-object classifiers for every regular cardinal $\kappa$ or something. (I'm trying to say that your $\kappa$-small sets need not technically form a "universe" but just some sort of "weak universe"...) Anyway, the existing stuff outside of type theory is foundationally complicated. But type theorists do think carefully about these sorts of things. $\endgroup$ – Tim Campion Feb 3 at 20:34
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Since groupoids are categories then Maclane's axiomatisation of categories, which he covers in his book, Categories for the Working Mathematician might be a possibility. I don't recall whether he stated this was a first-order axiomatisation but it shows that the sense that category theory could itself be a foundation for mathematics came early in it's development. And I think importantly, given the hegemony of ZFC in foundational thinking.

He expresses the axioms in terms of what he calls meta-categories, and in fact, of a simpler notion, meta-graphs. He does without objects, noting that these can be recovered from the arrows. I've checked, and he doesn't mention whether or not it's first order.

There's also Makkai's FOLDS, which is first order logic with dependent types. It has a weakened law of equality where it's impossible to say two objects are equal to each other and according to NLab it can even be used to formulate higher category theory.

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    $\begingroup$ ETCS is an elementary axiomatisation of the category of SETS, not categories, Lawvere did also try to write an ETCC, but this has some foundational shortcomings. In any case, it is not what Birchfield is asking for. $\endgroup$ – Steven Gubkin Feb 3 at 17:59
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    $\begingroup$ He almost gives (but does not formally give) an axiomatization of a category, but not the category of all categories. Birchfield is asking for a direct axiomatization of the category of groupoids, not a single groupoid (which is easy). $\endgroup$ – Steven Gubkin Feb 3 at 19:02
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    $\begingroup$ A model of MacLane's axioms would be a single category. For instance, his axioms are satisfied by the empty category. Birchfield is looking for a collection of axioms which "capture" the category of groupoids. Whatever those axioms are, the empty category will not be a model. I would guess that any model of these (as yet to be determined) axioms should at least contain all finite groupoids. I hope that clarifies the issue. $\endgroup$ – Steven Gubkin Feb 4 at 11:51
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    $\begingroup$ Mac Lane did his thesis in logic, so he was conscious of the foundational issues around formalising large categories, and this care is present in the 1945 paper introducing the definition of a category. But you have completely misunderstood what the OP wanted, so please don't keep beating the dead horse of trying to justify your non-answer. What the OP wanted is something with a technical definition that is known and understood within the category theory community, and it is not what you think Mac Lane is providing. $\endgroup$ – theHigherGeometer Feb 14 at 5:07
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    $\begingroup$ @MoziburUllah Please stop. I am a category theorist and I am telling you that you have misunderstood the question, and Mac Lane has absolutely nothing to say about the OP's question. Please take it from me. What the OP wants is a characterisation, up to equivalence, of the category whose objects are groupoids. What you are offering is the definition of a (meta)category. This is as useless as offering the group axioms when asked by a group theorist for a characterisation of the Monster group up to isomorphism. I will not add any more and please do not @ me again. $\endgroup$ – theHigherGeometer Feb 14 at 9:07

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