Topos Theory

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In higher category theory




For CC a small category, and PSh(C)PSh(C) its presheaf topos, we have (by the discussion at Profunctor – In terms of colimit preserving functors on presheaf categories) that a cocontinuous, i.e., colimit-preserving, functor PSh(C)SetPSh(C) \to Set is equivalently itself a copresheaf on CC:

[PSh(C),Set] cocCoPSh(C). [PSh(C), Set]_{coc} \simeq CoPSh(C) \,.

If we replace in this statement presheaves with sheaves, we obtain the notion of cosheaf on CC:

[Sh(C),Set] cocCoSh(C). [Sh(C), Set]_{coc} \simeq CoSh(C) \,.



Let CC be a site. A cosheaf on CC is a copresheaf

F:CSet F : C \to Set

such that it takes covers to colimits: for each covering family {U iU}\{U_i \to U\} in CC we have

F(U)lim ( ijF(U i× UU j) iF(U i)) F(U) \simeq \lim_{\to} \left( \coprod_{i j} F(U_i \times_{U} U_j) \stackrel{\to}{\to} \coprod_i F(U_i) \right)

Write CoSh(C)CoPSh(C)CoSh(C) \subset CoPSh(C) for the full subcategory of cosheaves.



There is a natural equivalence of categories

CoSh(C)Func coc(Sh(C),Set), CoSh(C) \simeq Func_{coc}(Sh(C), Set) \,,

where on the left we have the category of cosheaves from def. 1 and on the right we have the category of colimit-preserving functors on the sheaf topos of CC.

Equivalently: a copresheaf is a cosheaf precisely if its Yoneda extension PSh(C)SetPSh(C) \to Set factors through the sheafification functor PSh(C)Sh(C)PSh(C) \to Sh(C).

This is (Bunge-Funk 06, prop. 1.4.3).


In AQFT and higher AQFT

Cosheaves of algebras, or notions similar to this, appear in AQFT as local nets of observables. Similar structures in higher category theory are factorization algebras, factorization homology, and topological chiral homology. Notably the definition of factorization algebra typically explicitly involves the notion of cosheaf.


  • G. E. Bredon, Sheaf Theory , Springer Heidelberg 1997. (chap. V-VI)

  • Marta Bunge and Jonathon Funk, Singular coverings of toposes , Lecture Notes in Mathematics vol. 1890 Springer Heidelberg (2006). (sec. 1.4)

  • Jonathon Funk, The Display Locale of a Cosheaf , Cah. Top. Géom. Diff. Cat. 36 (1995) pp.53-93.

  • Justin M. Curry, Abstract existence of cosheafification (pdf)

  • Andrei V. Prasolov, Cosheafification (arXiv:1605.01555)

Revised on May 25, 2017 02:59:55 by David Corfield (