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I want to study the theory of sheaves from a categorical point of view with an emphasis on applications in algebraic topology and differential geometry and I'm looking for a good introductory book to the subject with these qualifications in mind.

I know that the theory of sheaves is fundamentally a categorical theory, what I mean by "a categorical point of view" is that I'm searching for a book that doesn't shy away from getting into (somewhat) advanced category theory in general to explain concepts related to sheaves.

However, I haven't studied yet algebraic topology (I'm taking a course in the subject this semester) so the material shouldn't be too advanced. Additionally, I have taken only a very short introductory course in differential geometry (classical formulation) so this should also be taken into account. I've taken an introductory course in category theory which covered the basics of the field (up to adjoint functors), so I do have some background there.

Any recommendations for books that hit those marks would be more than welcome.

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    $\begingroup$ I think this is exactly not what you're asking, but some standard references are Iversen's Cohomology of sheaves and Dimca's Sheaves in topology (the latter omits many 'standard' proofs). $\endgroup$ Mar 14, 2023 at 13:21
  • $\begingroup$ Of course, one of the main applications of sheaf theory in algebraic topology and differential geometry concerns cohomology theories. If this is suitable, I can suggest some references (other than general ones). $\endgroup$
    – ARA
    Mar 14, 2023 at 13:27
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    $\begingroup$ Maybe helpful if you tell us about your motivation to study sheaves, too. I mean, are you interested in sheaf theory generally, or do you want to learn it for some special topic? (I didn't get it from the question). For example, if you want to study sheaves in general you must have advanced categorical backgrounds (in particular, a familiarity with limits is crucial). But, e.g., in differential geometry, there is no need to be involved with advanced categorical backgrounds, instead, you must have more homological backgrounds. $\endgroup$
    – ARA
    Mar 14, 2023 at 14:28
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    $\begingroup$ I think Iversen is a great introduction to sheaves in topology (Dimca is a bit more advanced), but it's maybe not specifically categorical (but I suppose any text will use some basic category theory). But have a look, maybe this is exactly what you're looking for! $\endgroup$ Mar 14, 2023 at 14:42
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    $\begingroup$ I'd also suggest Bott and Tu's book Differential Forms in Algebraic Topology, even though sheaves are not discussed much in the book. $\endgroup$
    – Deane Yang
    Mar 14, 2023 at 16:08

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I have no idea if this is the sort of thing you want, but the first thing I thought of when reading your question was the work of my colleague David Carchedi.

As written on his website, his work is generally about "applications of higher category theory to topology and differential and algebraic geometry."

Here is his webpage: https://math.gmu.edu/~dcarched/

In particular, you might start with his doctoral thesis which has a nice introduction:

Categorical Properties of Topological and Differentiable Stacks

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From a categorical point of view, sheaves are considered on a Grothendieck site. If you want to start with the general setting, I recommend the following comprehensive reference in which mainly the properties of the category of sheaves are investigated. Moreover, all of the needed categorical materials are discussed.

  • S. Mac Lane, I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer-Verlag, 1992.

As for the application, you have to choose the desired site, e.g., that of open sets and that of Euclidean spaces. Of course, one of the main applications of sheaf theory in algebraic topology and differential geometry concerns cohomology theories. See, e.g.,

  • G.E. Bredon, Sheaf Theory, Graduate Texts in Mathematics, Vol. 170, Springer-Verlag, 1997.

  • Frank W. Warner, Foundations of differentiable manifolds and Lie groups. Vol. 94. Springer Science & Business Media, 1983. (Chapter 5)

  • L.I. Nicolaescu, Lectures on the Geometry of Manifolds, World Scientific, 2008. (Chapter 7)

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  • $\begingroup$ What does "on a Grothendieck site" mean? $\endgroup$ Mar 15, 2023 at 18:09
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    $\begingroup$ A Grothendieck site is an underlying category together with a Grothendieck topology (See the mentioned reference or jmilne.org/math/CourseNotes/LEC.pdf). From a categorical point of view, a sheaf is defined on a Grothendieck site. For example, the category of open subsets of a topological space (take open subsets as objects and inclusions as morphisms) with a well-known Grothendieck topology is an example of a Grothendieck site. $\endgroup$
    – ARA
    Mar 15, 2023 at 18:55
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Additionally to Bredon's and Warner's excellent books and Nicolaescu's lecture notes mentioned in ARA's answer and specifically for applications of sheaves in Differential Geometry I would suggest the following notes and books, in no particular order:

With focus on distributions (in the sense of Differential Geometry):

  • A. Lewis - Generalized Subbundles and Distributions, Comprehensive Review (2014) (link: https://mast.queensu.ca/~andrew/notes/abstracts/2011a.html) (Even though this topic is perhaps slightly more specific, I suggest it because there you can see more interactions with sheaf theory, and arguably distributions are among the central objects in Differential Geometry.)

With focus on analysis on manifolds:

  • S. Ramanan - Global Calculus (2005) (Chapter 4 is dedicated to the comparison of the various cohomologies, i.e Cech cohomology, Sheaf cohomology, Singular cohomology, de Rham cohomology. However, sheaves pop up in many of the other chapters as well, including the last one on the consequences of the analysis of elliptic operators done in the previous chapter. The last chapter includes elliptic complexes, some basic Hodge Theory, Kodaira's Vanishing Theorem etc. Of course, the latter topics are more comprehensively treated in books on complex geometry.)

A comprehensive introductory treatment specifically of sheaves on manifolds:

  • T. Wedhorn - Manifolds, Sheaves, and Cohomology (2016) (As far as sheaf theory in differential geometry is concerned, this book is much more comprehensive than Warner's and contains nice examples. This would be my go-to recommendation for anyone who wants to learn about the sheaf-theoretic perspective of differentiable manifolds.)

Finally, there are the two volumes of

  • Mallios - Geometry of Vector Sheaves, An Axiomatic Approach to Differential Geometry (1998),

which might be an interesting second read after getting acquinted with the basics.

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