Sometimes for categories having some fixed property and/or structure, one can produce a recipe which gives (up to suitable equivalence) all the examples (and nothing else).
There are several typical classes of reconstruction theorems (which are all to some extent related).
These theorems reconstruct an algebraic symmetry object (group, groupoid, gerbe, Hopf algebra, Hopf algebroid) from the monoidal category of representations of that object (typically rigid and symmetric or braided). The correspondence between the symmetry object and the corresponding category of representations is called Tannaka duality.
Examples include classical Tannaka theorem and Krein theorem, Doplicher-Roberts reconstruction theorem in physics, Deligne's theorem on tensor categories, Woronowicz’s Tannaka duality for compact matrix pseudogroups, Saavedra-Rivano and Deligne reconstruction theorems for neutral and mixed Tannakian categories, Ulrich’s reconstruction theorem, reconstruction theorems of Majid, Nori Tannakian theorem, Grothendieck’s version of Galois group in algebraic geometry and so on. The notion of the fiber functor (due to Grothendieck) is central to these considerations.
These theorems for schemes (or varieties only) reconstruct a scheme (variety) out of the category of quasicoherent or only coherent sheaves (or a derived category version of them). In that class one can find Gabriel–Rosenberg theorem, Bondal-Orlov reconstruction theorem, reconstruction theorems of P. Balmer and of G. Garkusha and so on. There is also a class of reconstructions where for some derived categories a realization as derived categories of representation of quivers can be reconstructed.
There is a large class of abelian reconstruction theorems, for example the Gabriel–Popescu theorem. In topos theory the Giraud theorem is also a reconstruction theorem (of a site out of a topos, though a nonuniqueness of the resulting site is involved, not affecting cohomology, hence, according to Grothendieck, nonessential).
Lawvere's reconstruction theorem reconstructs a Lawvere theory $C$ from its category of finitely generated free models.
Gabriel-Ulmer duality is an equivalence of 2-categories LFP of locally finitely presentable categories and Lex of finitely complete categories. It is related to syntax-semantics adjunction and to Tannaka type reconstruction for coalgebra-like objects, with which has a common generalization (enriched Tannaka duality of Day).
Typically in the proofs of most reconstruction theorems an implicit use of the Yoneda lemma is involved. Various embedding theorems of classes of categories (as well as theorems on realization as quotient categories) are closely related, e.g. Barr embedding theorem and Freyd–Mitchell embedding theorem.
André Joyal, Ross Street, An introduction to Tannaka duality and quantum groups, in Part II of Category Theory, Proceedings, Como 1990, eds. A. Carboni, M. C. Pedicchio and G. Rosolini, Lec. Notes in Mathematics 1488, Springer, Berlin, 1991, pp. 411–492 doi:10.1007/BFb0084207.
P. Deligne, Catégories Tannakiennes, Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) pp.111–195.
Alexander L. Rosenberg, The existence of fiber functors, in ‘The Gelfand Mathematical Seminars 1996–1999’, pp. 145–154. Birkhäuser, Boston, MA, 2000.
A. L. Rosenberg, Reconstruction of groups, Selecta Math. N.S. 9:1 (2003) doi
N. Saavedra Rivano, Catégories Tannakiennes, Springer LNM 265, 1972
Bodo Pareigis, Quantum groups and noncommutative geometry, WS 1999, chapter 3, online notes.
S. Majid, Foundations of quantum group theory, chapter 9, Camb. Univ. Press 1995, 2002.
S. Majid, Tannaka–Krein theorem for quasiHopf algebras and other results, Contemp. Math. 134 (1992) 219–232.
A. L. Rosenberg, Reconstruction of groups, Selecta Math. N.S. 9:1 (2003)doi:10.1007/s00029-003-0322-x (nlab remark: this paper is on a generalization of Tannaka–Krein and not of the Gabriel–Rosenberg kind of reconstruction)
A. Rosenberg, The spectrum of abelian categories and reconstructions of schemes, in Rings, Hopf Algebras, and Brauer groups, Lectures Notes in Pure and Appl. Math. 197, Marcel Dekker, New York, 257–274, 1998; MR99d:18011; and Max Planck Bonn preprint Reconstruction of Schemes, MPIM1996-108 (1996).
A. L. Rosenberg, Spectra of noncommutative spaces, MPIM2003-110 ps dvi, Underlying spaces of noncommutative schemes, MPIM2003-111, dvi, ps
P. Gabriel, Des catégories abéliennes, Bulletin de la Société Mathématique de France 90 (1962), 323–448, numdam
A. I. Bondal, D. O. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compos. Math. 125 (2001), 327–344 doi:10.1023/A:1002470302976
K. Szlachányi, Fiber functors, monoidal sites and Tannaka duality for bialgebroids arxiv:0907.1578
Phùng Hô Hai, Tannaka–Krein duality for Hopf algebroids, arxiv:math.QA/0206113
Hélène Esnault, Phùng Hô Hai, Gauß–Manin connection and Tannaka duality, math.AG/0509111
H. Pfeiffer, Tannaka–Krein reconstruction and a characterization of modular tensor categories, arxiv:math.QA/0711.1402
S. L. Woronowicz, Tannaka–Kreĭn duality for compact matrix pseudogroups. Twisted $SU(N)$ groups, Invent. Math. 93 (1988), no. 1, 35–76
Michael Müger, Abstract duality for symmetric tensor $*$-categories (App. to Hans Halvorson: ‘Algebraic Quantum Field Theory’.) In: J. Butterfield and J. Earman (eds.): “Handbook of the Philosophy of Physics”, p. 865–922. North Holland, 2007; arxiv:math-ph/0602036; cf. The String Coffee Table, Müger on Doplicher–Roberts
S. Doplicher, J. E. Roberts, A new duality theory for compact groups, Inventiones Math., 98(1):157–218, 1989.
P. Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005), pp. 149–168 dvi pdf ps.
M. Prest, G. Garkusha, Reconstructing projective schemes from Serre subcategories, J. Algebra 319 (3) (2008), 1132–1153 (pdf).
P. Deligne, J. S. Milne, Tannakian categories, Lect. notes in math. 900, 101–228, Springer 1982.
A. Bruguières, On a tannakian theorem due to Nori, pdf; Théorie tannakienne non commutative, Comm. Algebra 22, 5817–5860, 1994
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