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Above thoughts, albeit superficial and naive without neither strict logic nor axiomatized semantics, propelled me to search for clues and hints in symposia, books, seminars, minicourses, and other accessible ways. As new ingredients from a variety of theories add to them, some questions were solved or partly solved, and new questions emerged, mixing and reacting with those partly unsolved old ones. The process circles over and over to make inspirations explode in my mind, driving me overwhelmed and lost in these endless puzzles. Grothendieck’s theories which construct a marvelous empire with solid foundation of mathematical logic reorganized and enriched my unfettered thoughts well but don’t clear my confusions on some key is- sues. Thanks to the series of mini courses on Derived (abbreviated as DAG) which was given by Bertrand To¨en et cetera in Octo- ber, 2019, presenting to me concrete illustration to a number of the riddles. Consequently, I proceeded to learn DAG, starting with preliminaries like higher category theory, higher topos theory, and higher algebra, which con- stitute the logical basis of the renewed systematic theoretical frameworks. Before diving deeper into DAG, I am going to set out to explore some topics on higher orbifolds in next section. Since it is not finished yet despite that a plenty of work has been done, only certain parts of axiomatized seman- tics will be interpreted. Meanwhile, definitions, lemmas, propositions and theorems given or proved in bibliographies cited in this article would not be repeated unless necessary. Notwithstanding the fact that DAG theories involved in this essay are based on both ’s, Bertrand To¨en and Gabriele Vezzosi’s, for convenience, the terminology and notations involved will be borrowed from Jacob Lurie’s if there are ambiguities or discrepancies between the two.

2 2 Model Structure on Higher Orbifolds

Orbifolds were known as singular spaces that are locally modelled on quo- tients of open subsets of Rn by finite group actions. The orbifold structure encodes not only the structure of the underlying quotient space, but also that of the isotropy subgroups. Atlases are used to describe the orbifold structure. However, it is complicated and inconvenient to define the mor- phisms, not to mention the composition of morphisms, hindering categorical operations on orbifolds and further steps.

I. Moerdijk and D. A. Pronk has shown in [MP97] that orbifolds are essentially the same as certain “proper” groupoids. E. Lerman mentioned in [Ler10] that a proper ´etale Lie groupoid is locally isomorphic to an finite action of groupoid. From then on, geometers are used to define orbifolds as proper ´etale Lie groupoids [Moe02]. What’s more, as groupoids can also be supposed to be atlases on orbifolds, there is a way of thinking of a groupoid as coordinates on a corresponding stack. In another word, orbifolds can also be seen as stacks.

The concept of stack (i.e. 2-sheaf) is the categorical analogue of sheaf, which is a generalization of principal bundle. Recall that a principal G- bundle can be defined as a collection of transition functions gij : Uij → G on the double intersections Uij of some open covering {Ui}i∈I of base man- ifold M, satisfying the cocycle condition gijgjk = gik compatible with the singular cohomology group H1(M; G). These transition functions construct morphisms of groupoids from the C`ech groupoid Uij ⇒ Ui associated to ` ` the open covering {Ui}i∈I to the Lie groupoid G ⇒ ∗. Meanwhile, a princi- pal G-bundle P over a manifold M canonically determines a homotopy class of maps from M to the classifying space BG of the group G. In effect, the set of isomorphism classes of G-principal bundles over M is in bijection with the set of homotopy classes of maps M → BG ≡ K(G, 1), where K(G, 1) denotes the Eilenberg-MacLane space. Thus, stacks can be viewed as a kind of classifying construction which coincides with C`ech covering.

Considering morphisms between G -torsor and H -torsor which is gen- erally called 2-morphisms, we can define bibundles [Bre10] ,2-groupoids and 2-stacks satisfying certain descent condition. In particular, G-gerbes are equivalent to AUT (G)-principal 2-bundles, for AUT (G) the automorphism 2-group of G [GS15]. Take G-bibundle as a prototype, we can construct a similar classifying stack BBG ≡ K(G, 2) associated with 2-C`ech covering satisfying 2-cocycle condition, which is compatible with H2(M; G). As for Hn(M; G), where n ≥ 2, K(G, n) is a higher n-stack [Sim96] classifying “higher” pincipal G-bundles [Vez11], whose n-C`ech covering (essentially the same with hyper-covering) nerve is n-hypergoupoid [Dus79]. Equivalently,

3 the singular cohomology groups Hn(M; G) of a nice topological space M 1 with coefficients in an abelian group G (sheaf cohomology Hsheaf (M; G) of M with coecients in the constant sheaf G associated to G for a gen- eral space M) is actually a representable functor of M. That is, there exists an Eilenberg-MacLane space K(G, n) and a universal cohomology class η ∈ Hn(K(G, n); G) such that, for any nice topological space X, pull- back of η determines a bijection [X; K(G, n)] → Hn(X; G) [Lur09a]. Here [X; K(G, n)] denotes the set of homotopy classes of maps from X to K(G, n).

Extending n to be ∞, ∞-groupoid can be defined to be an ∞-category [Lur09a] (quasi-category by Joyal and weak Kan complex by Boardman and Vogt) in higher category [Lur20] to be an abstract homotopical model for topological spaces. Therefore, stacks over ∞-groupoid generalize those over groupoid, requiring a new topology corresponding to higher category, which is formally named higher topos [Lur09a], represented by model topos.

Back to orbifolds which have been previously mentioned to be proper ´etale Lie groupoids. As Lie groupoids are essentially differentiable stacks up to Morita equivalence [BX06], it is reasonable to assume higher orbifolds to be ´etale differentiable higher stacks. Start from higher (n-)stack which can also be regarded as (n-)truncated ∞-stack (also denoted by (∞,n)- stack), or n-geometric D-stack in [BT08]. Since ∞-stack (also known as (∞, 1)-sheaf) is endowed with a geometry [Lur09b], forming a simplicial en- riched category whose all internal hom-objects are Kan complexes (which are fibrant-cofibrant objects), there exists a n-truncated geometry. We may literally transplant Jacob Lurie’s frameworks in [Lur09a] to obtain local and global model structures on (∞,n)-stacks, i.e. n-orbifolds, although detailed machinery is rather complicated. David Carchedi gave another description in [Car15b] [Car15a] [Car13].

3 Quantum Cohomology of Higher Orbifolds

W. Chen and Y. Ruan have asserted in [CR00] [CR04] that an important feature of orbifold cohomology groups is degree shifting, i.e. shifting up the degree of cohomology classes of X(g) by 2ι(g), defining the orbifold cohomol- ogy group of degree d to be the direct sum

d d−2ι H (X; Q)= ⊕ ∈ H (g) X ; Q , orb (g) T (g)  for any rational number d ∈ [0, 2n], where X is a closed almost complex orbifold with dimC X = n, X = p, (g)G ∈ X˜ | (g)G ∈ (g) called a (g) n p  p o twisted sector for (g) 6= (1), and ι(g) called degree shifting numbers. More- over, Y. Ruan explored twisted orbifold cohomology and its relation to dis-

4 crete torsion in [Rua00].

Now that the geometry and topology behavior of higher orbifolds can be detected in section 2, it should be possible to compute their quantum coho- mology but will obviously be a significant amount of work. However, when it comes to derived orbifolds, things should be radically different. And what will derived twisted orbifolds look like? How would the twisted sector and degree shifting numbers change? Will them get simpler compared with the general case? Techniques are supposed to be totally different with existing literatures’.

I apologize to suspend it here because of time.

References

[Bre10] Lawrence Breen. Notes on 1- and 2-Gerbes, pages 193–235. Springer New York, New York, NY, 2010.

[BT08] G.V. Bertrand Ton. Homotopical Algebraic Geometry II: Geomet- ric Stacks and Applications. American Mathematical Soc., 2008.

[BX06] Kai Behrend and Ping Xu. Differentiable stacks and gerbes, 2006.

[Car13] David Carchedi. Higher orbifolds and deligne-mumford stacks as structured infinity topoi, 2013.

[Car15a] David Carchedi. On the homotopy type of higher orbifolds and haefliger classifying spaces, 2015.

[Car15b] David Carchedi. On the tale homotopy type of higher stacks, 2015.

[CR00] Weimin Chen and Yongbin Ruan. Orbifold quantum cohomology, 2000.

[CR04] Weimin Chen and Yongbin Ruan. A new cohomology theory of orbifold. Communications in Mathematical Physics, 248(1):131, May 2004.

[Dus79] John Williford Duskin. Higher dimensional torsors and the coho- mology of topoi : The abelian theory. 1979.

[GS15] Gr´egory Ginot and Mathieu Sti´enon. G-gerbes, principal 2-group bundles and characteristic classes. Journal of Symplectic Geome- try, 13(4):1001–1048, January 2015.

[Ler10] Eugene Lerman. Orbifolds as stacks? Enseign. Math. (2), 56(3- 4):315–363, 2010.

5 [Lur09a] J. Lurie. Higher Topos Theory (AM-170). Annals of Mathematics Studies. Princeton University Press, 2009.

[Lur09b] Jacob Lurie. Derived algebraic geometry v: Structured spaces, 2009.

[Lur20] Jacob Lurie. Kerodon. https://kerodon.net, 2020.

[Moe02] Ieke Moerdijk. Orbifolds as groupoids: an introduction, 2002.

[MP97] I. Moerdijk and Dorette Pronk. Orbifolds, sheaves and groupoids. K-Theory, 12:3–21, 07 1997.

[Rua00] Yongbin Ruan. Discrete torsion and twisted orbifold cohomology, 2000.

[Sim96] Carlos Simpson. Algebraic (geometric) n-stacks, 1996.

[Vez11] Gabriele Vezzosi. What is ...a derived stack? Notices of the Amer- ican Mathematical Society, 58, 01 2011.

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