Book HoTT

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## Idea

Book HoTT is the [[dependent type theory]] which appears in the [[HoTT book]]. It is notable in that, unlike most other dependent type theories which have been formally written in [[natural deduction]], it does not have a separate [[type]] [[judgment]]. Instead, it only has term judgments, and an infinite sequence of [[Russell universes]] indexed by a [[natural numbers]] primitive. Types are represented by terms of a Russell universe.

However, when formally defining the natural numbers primitive, in order to ensure that the dependent type theory does not have a separate type judgment, one has to define the natural numbers primitive in a separate [[layer]], meaning that formal book HoTT is a [[layered type theory]]. There are many different ways to define the layer containing the natural numbers primitive:

* One could define the [[first-order theory]] of a [[category]] with [[finite products]] and a parameterized [[natural numbers object]], and use the [[hom-set]] $\mathrm{Hom}(1, \mathbb{N})$ to index the [[Russell universes]].
* One could define the simply typed first order theory of [[ZFC]], and use the von Neumann natural numbers or the Zermelo natural numbers to index the Russell universes.
* One could define the simply typed first order theory of [[Peano arithmetic]], and use the natural numbers in Peano arithmetic to index the Russell universes.
* One could define a separate [[dependent type theory]] with a [[natural numbers type]], such as the [[extensional type theory]] in [[two-level type theory]], and use the natural numbers type $\mathbb{N}$ to index the Russell universes.

## Formal presentation

The presentation of formal book HoTT we have chosen is a  type theory with [[judgments]] for contexts

* $\Gamma \; \mathrm{ctx}$, that $\Gamma$ is a [[context]]

### Universe levels and Peano arithmetic

Then we have judgments for universe levels and predicate logic:

* $i \; \mathrm{level}$, that $i$ is a universe level,

* $\phi \; \mathrm{prop}$, that $\phi$ is a [[proposition]],

* $\phi \; \mathrm{true}$, that $\phi$ is a [[true]] proposition,

and the formal [[signature (in logic)|signature]] and [[inference rules]] of [[first-order theory|first-order]] [[Heyting arithmetic]] or [[Peano arithmetic]].

These rules ensure that there are an [[infinite]] number of indices, which are strictly ordered with [[strict total order]] $\lt$ and upwardly unbounded, where $i \lt s(i)$ is true for all indices $i$.

### Typing judgments and Russell universes

Now, we introduce the typing judgment $a:A$, which says that $a$ is a term of the type $A$. Instead of type judgments, we introduce a special kind of type called a [[Russell universe]], whose terms are the types themselves. Russell universes are formalized with the following rules:

$$\frac{\Gamma \vdash i \; \mathrm{level}}{\Gamma \vdash U_i:U_{s(i)}} \qquad \frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i}{\Gamma \vdash A:U_{s(i)}}$$

In addition, we have rules for contexts which state that one could add typing judgments to the list of contexts:

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i}{(\Gamma, a:A) \; \mathrm{ctx}}$$

### Structural rules

There are three structural rules in dependent type theory, the [[variable rule]], the [[weakening rule]], and the [[substitution rule]].

The variable rule states that we may derive a typing judgment if the typing judgment is in the context already:

$$\frac{\Gamma, a:A, \Delta \; \mathrm{ctx}}{\Gamma, a:A, \Delta \vdash a:A}$$

Let $\mathcal{J}$ be any arbitrary judgment. Then we have the following rules:

The weakening rule:

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i \quad \Gamma, \Delta \vdash \mathcal{J}}{\Gamma, a:A, \Delta \vdash \mathcal{J}}$$

The substitution rule:

$$\frac{\Gamma \vdash a:A \quad \Gamma, b:A, \Delta \vdash \mathcal{J}}{\Gamma, \Delta[a/b] \vdash \mathcal{J}[a/b]}$$

The weakening and substitution rules are admissible rules: they do not need to be explicitly included in the type theory as they could be proven by induction on the structure of all possible derivations.

### Structural rules for judgmental equality

Judgmental equality has its own structural rules: introduction rules for judgmentally equal terms, reflexivity, symmetry, transitivity, the principle of substitution, and the variable conversion rule.

* Introduction rules for judgmentally equal terms
$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i \quad \Gamma \vdash a \equiv b:A}{\Gamma \vdash a:A}$$
$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i \quad \Gamma \vdash a \equiv b:A}{\Gamma \vdash b:A}$$

* Reflexivity of judgmental equality
$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i \quad \Gamma \vdash a:A}{\Gamma \vdash a \equiv a:A}$$

* Symmetry of judgmental equality
$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i \quad \Gamma \vdash a \equiv b:A}{\Gamma \vdash b \equiv a:A}$$

* Transitivity of judgmental equality
$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i \quad \Gamma \vdash a \equiv b:A \quad \Gamma \vdash b \equiv c:A}{\Gamma \vdash a \equiv c:A}$$

* Principle of substitution:
$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i \quad \Gamma \vdash a \equiv b : A \quad \Gamma, x:A, \Delta \vdash c:B}{\Gamma, \Delta[b/x] \vdash c[a/x] \equiv c[b/x]: B[b/x]}$$

* Variable conversion rule:
$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A \equiv B:U_i \quad \Gamma, x:A, \Delta \vdash \mathcal{J}}{\Gamma, x:B, \Delta \vdash \mathcal{J}}$$

We also have a rules saying that equality is preserved across universe levels:

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash j \; \mathrm{level} \quad \Gamma \vdash i = j \; \mathrm{true}}{\Gamma \vdash U_i \equiv U_j:U_{s(i)}}$$

### Dependent product types

* Formation rules for dependent product types:

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i \quad \Gamma, x:A \vdash B:U_i}{\Gamma \vdash \prod_{x:A} B(x):U_i}$$

* Introduction rules for dependent product types:

$$\frac{\Gamma, x:A \vdash b:B}{\Gamma \vdash \lambda(x:A).b(x):\prod_{x:A} B(x)}$$

* Elimination rules for dependent product types:

$$\frac{\Gamma \vdash f:\prod_{x:A} B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash f[a/x]:B[a/x]}$$

* Computation rules for dependent product types:

$$\frac{\Gamma, x:A \vdash b:B \quad \Gamma \vdash a:A}{\Gamma \vdash \lambda(x:A).b(x)[a/x] \equiv b[a/x]:B[a/x]}$$

* Uniqueness rules for dependent product types:

$$\frac{\Gamma \vdash f:\prod_{x:A} B(x)}{\Gamma \vdash f \equiv \lambda(x).f(x):\prod_{x:A} B(x)}$$

### Dependent sum types

* Formation rules for dependent sum types:

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i \quad \Gamma, x:A \vdash B:U_i}{\Gamma \vdash \sum_{x:A} B(x):U_i}$$

* Introduction rules for dependent sum types:

$$\frac{\Gamma, x:A \vdash b:B \quad \Gamma \vdash a:A \quad \Gamma \vdash b:B[a/x]}{\Gamma \vdash (a, b):\sum_{x:A} B(x)}$$

* Elimination rules for dependent sum types:

$$\frac{\Gamma \vdash z:\sum_{x:A} B(x)}{\Gamma \vdash \pi_1(z):A} \qquad \frac{\Gamma \vdash z:\sum_{x:A} B(x)}{\Gamma \vdash \pi_2(z):B[\pi_1(z)/x]}$$

* Computation rules for dependent sum types:

$$\frac{\Gamma, x:A \vdash b:B \quad \Gamma \vdash a:A}{\Gamma \vdash \pi_1(a, b) \equiv a:A} \qquad \frac{\Gamma, x:A \vdash b:B \quad \Gamma \vdash a:A}{\Gamma \vdash \pi_2(a, b) \equiv b:B[\pi_1(a, b)/x]}$$

* Uniqueness rules for dependent sum types:

$$\frac{\Gamma \vdash z:\sum_{x:A} B(x)}{\Gamma \vdash z \equiv (\pi_1(z), \pi_2(z)):\sum_{x:A} B(x)}$$

### Sum types

* Formation rules for sum types:

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i \quad \Gamma \vdash B:U_i}{\Gamma \vdash A + B:U_i}$$

* Introduction rules for sum types:

$$\frac{\Gamma \vdash a:A}{\Gamma \vdash \mathrm{inl}(a):A + B} \qquad \frac{\Gamma \vdash b:B}{\Gamma \vdash \mathrm{inr}(b):A + B}$$

* Elimination rules for sum types:

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma, z:A + B \vdash C:U_i \quad \Gamma, x:A \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash e:A + B}{\Gamma \vdash \mathrm{ind}_{A + B}(z.C, x.c, y.d, e):C[e/z]}$$

* Computation rules for sum types:

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma, z:A + B \vdash C:U_i \quad \Gamma, x:A \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash a:A}{\Gamma \vdash \mathrm{ind}_{A + B}(z.C, x.c, y.d, \mathrm{inl}(a)) \equiv c[a/x]:C[\mathrm{inl}(a)/z]}$$
$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma, z:A + B \vdash C:U_i \quad \Gamma, x:A \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash b:A}{\Gamma \vdash \mathrm{ind}_{A + B}(z.C, x.c, y.d, \mathrm{inr}(b)) \equiv d[b/x]:C[\mathrm{inr}(b)/z]}$$

### Empty type

* Formation rules for the empty type:

$$\frac{\Gamma \vdash i \; \mathrm{level}}{\Gamma \vdash \mathbb{0}:U_i}$$

* Elimination rules for the empty type:

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma, x:\mathbb{0} \vdash C:U_i \quad \Gamma \vdash p:\mathbb{0}}{\Gamma \vdash \mathrm{ind}_\mathbb{0}(x.C, p):C[p/x]}$$

### Unit type

* Formation rules for the unit type:

$$\frac{\Gamma \vdash i \; \mathrm{level}}{\Gamma \vdash \mathbb{1}:U_i}$$

* Introduction rules for the unit type:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash *:\mathbb{1}}$$

* Elimination rules for the unit type:

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma, x:\mathbb{1} \vdash C:U_i \quad \Gamma, y:\mathbb{1} \vdash c[y/x]:C[y/x] \quad \Gamma \vdash p:\mathbb{1}}{\Gamma \vdash \mathrm{ind}_\mathbb{1}(x.C, y.c, p):C[p/x]}$$

* Computation rules for the unit type:

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma, x:\mathbb{1} \vdash C:U_i \quad \Gamma, y:\mathbb{1} \vdash c[y/x]:C[y/x]}{\Gamma \vdash \mathrm{ind}_\mathbb{1}(x.C, y.c, *) \equiv c[*/x]:C[*/x]}$$

### Natural numbers

* Formation rules for the natural numbers:
$$\frac{\Gamma \vdash i \; \mathrm{level}}{\Gamma \vdash \mathbb{N}:U_i}$$

* Introduction rules for the natural numbers:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 0:\mathbb{N}} \qquad \frac{\Gamma \vdash n:\mathbb{N}}{\Gamma \vdash s(n):\mathbb{N}}$$

* Elimination rules for the natural numbers:
$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma, x:\mathbb{N} \vdash C:U_i \quad \Gamma \vdash c_0:C[0/x] \quad \Gamma, x:\mathbb{N}, y:C \vdash c_s:C[s(x)/x] \quad \Gamma \vdash n:\mathbb{N}}{\Gamma \vdash \mathrm{ind}_\mathbb{N}(x.C, c_0, x.y.c_s, n):C[n/x]}$$

* Computation rules for the natural numbers:
$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma, x:\mathbb{N} \vdash C:U_i \quad \Gamma \vdash c_0:C[0/x] \quad \Gamma, x:\mathbb{N}, y:C \vdash c_s:C[s(x)/x]}{\Gamma \vdash \mathrm{ind}_\mathbb{N}(x.C, c_0, x.y.c_s, 0) \equiv c_0:C[0/x]}$$

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma, x:\mathbb{N} \vdash C:U_i \quad \Gamma \vdash c_0:C[0/x] \quad \Gamma, x:\mathbb{N}, y:C \vdash c_s:C[s(x)/x]}{\Gamma \vdash \mathrm{ind}_\mathbb{N}(x.C, c_0, x.y.c_s, s(n) \equiv c_s(n, \mathrm{ind}_\mathbb{N}(x.C, c_0, x.y.c_s, n)):C[s(n)/x]}$$

### Identity types

There is another version of equality in book HoTT, called [[typal equality]]. Typal equality is represented by the [[identity type]].

* Formation rule for identity types:
$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A}{\Gamma \vdash a =_A b:U_i}$$

* Introduction rule for identity types:
$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i \quad \Gamma \vdash a:A}{\Gamma \vdash \mathrm{refl}_A(a) : a =_A a}$$

* Elimination rule for identity types:
$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma, x:A, y:A, p:a =_A b \vdash C:U_i \quad \Gamma, z:A \vdash t:C[z/a, z/b, \mathrm{refl}_A(z)/p] \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash q:a =_A b}{\Gamma \vdash J(x,y,p.C, z.t, a, b, q):C[a, b, q/x, y, p]}$$

* Computation rules for identity types:
$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma, x:A, y:A, p:a =_A b \vdash C:U_i \quad \Gamma, z:A \vdash t:C[z/a, z/b, \mathrm{refl}_A(z)/p] \quad \Gamma \vdash a:A}{\Gamma \vdash J(x.y.p.C, z.t, a, a, \mathrm{refl}(a)) \equiv t:C[a, a, \mathrm{refl}_A(a)/x, y, p]}$$

### Univalence

Given types $A:U_i$ and $B:U_i$, a function $f:A \to B$ is an [[equivalence of types]] if the fiber of $f$ at each element of $B$ has exactly one element. The property of the type $A$ having exactly one element is represented by the [[isContr]] modality which states that the type $A$ is contractible. The fiber of $f$ at an element $b:B$ is given by the type
$$\sum_{b:B} f(a) =_B b$$

We formally define the property of $R$ being a one-to-one correspondence or an equivalence of types as the type:

$$\mathrm{isEquiv}_{A, B}(f) \coloneqq \prod_{b:B} \mathrm{isContr}\left(\sum_{a:A} f(a) =_B b\right)$$

We define the type of equivalences from $A$ to $B$ as

$$A \simeq B \coloneqq \sum_{f:A \to B} \mathrm{isEquiv}_{A, B}(f)$$

There is a function

$$\mathrm{idtoequiv}_{A, B}:(A =_{U_i} B) \to (A \simeq B)$$

which is inductively defined on reflexivity to be the identity function on $A$

$$\mathrm{idtoequiv}_{A, A}(\mathrm{refl}_{U_i}(A)) \equiv id_A$$

The [[univalence axiom]] then states that $\mathrm{idtoequiv}_{A, B}$ is an equivalence for all external natural numbers $i \in \mathcal{N}$ and types $A:U_i$ and $B:U_i$:

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i \quad \Gamma \vdash B:U_i}{\Gamma \vdash \mathrm{ua}_{U_i}(A, B):\mathrm{isEquiv}_{A =_{U_i} B, A \simeq B}(\mathrm{idtoequiv}_{A, B})}$$

## Related pages

Some of the kinds of inductive definitions mentioned in the HoTT Book are:

* [[W-types]]
* [[inductive families]]
* [[mutually inductive types]]
* [[inductive-inductive types]]
* [[inductive-recursive types]]
* [[higher inductive types]]
* [[higher inductive-inductive types]]
* [[higher inductive-recursive types]]

## See also

* [[dependent type theory]]

* [[homotopy type theory]]

* [[univalent type theory]]

## References

* *Homotopy Type Theory: Univalent Foundations of Mathematics*, The [[Univalent Foundations Project]], Institute for Advanced Study, 2013. ([web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf))

[[!redirects book HoTT]]
[[!redirects book homotopy type theory]]
[[!redirects Book homotopy type theory]]
[[!redirects Book Homotopy Type Theory]]


books about string theory

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This page lists literature on _[[string theory]]_.

(See also at _[[string theory FAQ]]_.)

# Contents
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{: toc}

## Mathematically inclined monographs about string theory

There is to date no textbook on string theory genuinely digestible by the standard pure mathematician. Even those that claim to be are not, as experience shows. But here are some books that make a strong effort to go beyond the vagueness of the "mainstream" books, which are listed further below.

* {#Cheng13} Miranda Cheng, _Mathematical tools for string theorists_, lecture notes 2013 ([pdf](http://www.math.jussieu.fr/~chengm/lecture_notes/Thalys_School.pdf))

  This is an elementary set of lecture notes introducing the required basics of [[differential geometry]] ending with [[2d CFT]], the [[Witten genus]] and [[orbifolds]].

* [[Pierre Deligne]], [[Pavel Etingof]], [[Dan Freed]], L. Jeffrey, [[David Kazhdan]], [[John Morgan]], D.R. Morrison and [[Edward Witten]] (eds).  _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

  This is a long collection of (in parts) long lectures by many top string theorists and also by some genuine top mathematicians. Correspondingly it covers a lot of ground, while still being introductory. Especially towards the beginning there is a strong effort towards trying to formalize or at least systematize much of the standard lore. But one can see that eventually the task of doing that throughout had been overwhelming. Nevertheless, this is probably the best source that there is out there. If you only ever touch a single book on string theory, touch this one.

* [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fre]], _[[Supergravity and Superstrings - A Geometric Perspective]]_

  This focuses on the discussion of
  [[supergravity]]-aspects of string theory from the point of view of the [[D'Auria-Fre formulation of supergravity]]. Therefore, while far, far from being written in the style of a mathematical treatise, this book stands out as making a consistent proposal for what the central ingredients of a mathematical formalization might be: as explained at the above link, secretly this book is all about describing supergravity in terms of [[connection on an infinity-bundle|infinity-connections]] with values in [[super L-infinity algebra]]s such as the [[supergravity Lie 3-algebra]].

* [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS (2011)

  This volume tries to give an impression of the rather recent massive progress that has happened in the mathematical understanding of fundamental ingredients of perturbative string theory, revolving around the proof of the [[cobordism hypothesis]] and related topics of [[higher category theory and physics]]. This is not an introductory textbook, even though some contributions do contain introductory material. Rather, this is meant to be read by people who already understand the basic idea of string theory and would like to see what the mathematical picture behind it all is going to be.

* [[Igor Dolgachev]], *Introduction to string theory for mathematicians*, lecture at *[1999 Summer on Algebraic Geometry and Physics](https://people.sissa.it/~bruzzo/sagp99/sagp.html)* (1999) [[pdf](https://dept.math.lsa.umich.edu/~idolga/stringy.pdf), [[Dolgachev-IntroductionStringTheory.pdf:file]]]

* [[Paul Aspinwall]], [[Tom Bridgeland]], Alastair Craw, [[Michael Douglas]], Mark Gross, _Dirichlet branes and mirror symmetry_, Amer. Math. Soc. Clay Math. Institute 2009.

See also

* [[Gregory Moore]], _[[The Impact of D-Branes on Mathematics]]_, talk at [PolchinskiFest 2014](http://online.kitp.ucsb.edu/online/joefest-c14/) ([pdf](http://www.physics.rutgers.edu/~gmoore/JOEFEST-THOUGHTS.pdf))

* {#Moore14} [[Gregory Moore]], _[[Physical Mathematics and the Future]]_, talk at [Strings 2014](http://physics.princeton.edu/strings2014/) ([talk slides](http://physics.princeton.edu/strings2014/slides/Moore.pdf), [companion text pdf](http://www.physics.rutgers.edu/~gmoore/PhysicalMathematicsAndFuture.pdf), [[MooreVisionTalk2014.pdf:file]])

## Mainstream physics monographs
 {#MainstreamPhysicsMonographs}

### More elementary

* [[Jan Troost]], _[Beyond String Theory](http://www.phys.ens.fr/~troost/beyondstringtheory/index2.html)_

  (rough survey for laymen)

* [[Barton Zwiebach]], _A first course in string theory_, Cambridge University Press (2009)

  (meant to be a course for undergraduates)

* [[Katrin Becker]], [[Melanie Becker]], [[John Schwarz]], _String theory and M-theory: a modern introduction_,  Cambridge University Press (2006) ([spire:744404](http://inspirehep.net/record/744404))

* [[Elias Kiritsis]], _String theory in a nutshell_, Princeton UP 2007, 608 pp. ([description](http://press.princeton.edu/titles/8456.html)); 1998 early version _Introduction to superstring theory_ (244 pp) is available as [hep-th/9709062](http://xxx.lanl.gov/abs/hep-th/9709062)

* [[Michael Dine]], _Supersymmetry and string theory: beyond the standard model_, Cambridge University Press (2006, 2007) [[ISBN:9781009290920](https://www.cambridge.org/ae/academic/subjects/physics/particle-physics-and-nuclear-physics/supersymmetry-and-string-theory-beyond-standard-model-2nd-edition?format=HB), [pdf](http://www.nucleares.unam.mx/~alberto/apuntes/dine.pdf)]

* [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], *Basic Concepts of String Theory*, Theoretical and Mathematical Physics, Springer (2013) [[doi:10.1007/978-3-642-29497-6](https://doi.org/10.1007/978-3-642-29497-6)]

* {#Kane17} [[Gordon Kane]], _String theory and the real world_, Morgan & Claypool, 2017 (doi:0.1088/978-1-6817-4489-6)

### More advanced

* {#GreenSchwarzWitten88} [[Michael Green]], [[John Schwarz]], [[Edward Witten]], _Superstring theory_, 3 vols. Cambridge Monographs on Mathematical Physics 1988 (Vol 1: [spire:250488](http://inspirehep.net/record/250488), [ISBN:9781107029118](https://www.cambridge.org/us/academic/subjects/physics/theoretical-physics-and-mathematical-physics/superstring-theory-25th-anniversary-edition-volume-1?format=HB&isbn=9781107029118); Vol 2: [spire:1384879](http://inspirehep.net/record/1384879), [doi:10.1017/CBO9781139248570](https://doi.org/10.1017/CBO9781139248570))

* {#Polchinski98} [[Joseph Polchinski]], _[[String theory]]_, 2 vols. , Cambridge University Press, 1998 (Vol 1: [doi:10.1017/CBO9780511816079](https://doi.org/10.1017/CBO9780511816079), Vol 2: [ISBN:9780521633048](https://www.cambridge.org/us/academic/subjects/physics/theoretical-physics-and-mathematical-physics/string-theory-volume-2?format=HB&isbn=9780521633048))

* [[Joseph Polchinski]], _Joe's Little Book of String_, class notes, UCSB Phys 230A, String Theory, Winter 2010, [pdf](http://www.kitp.ucsb.edu/sites/default/files/users/joep/JLBS.pdf)

* {#West12} [[Peter West]], _[[Introduction to Strings and Branes]]_, Cambridge University Press 2012

* [[Alexander Polyakov]], _Gauge fields and strings_,

* Brian Hatfield, _Quantum field theory of point particles and strings_,  Frontiers in Physics, 752 pages, Westview Press 1998

* [[Clifford Johnson]], _D-branes_

* [[Richard Szabo]], _An introduction to string theory and D-brane dynamics_

* Кетов С.В. "Введение в квантовую теорию струн и суперструн" [djvu](http://www.stringworld.ru/files/Ketov_S.V._Vvedenie_v_kvantovuju_teoriju_strun_i_superstrun.djvu)

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

## More mathematical

* [[Jürgen Jost]], *Bosonic Strings: A Mathematical Treatment*, AMS/IP Stud. Adv. Math. **21** (2001) [[ISBBN:978-0-8218-4336-9](https://bookstore.ams.org/view?ProductCode=AMSIP/21.S.B), [spire:1388134](https://inspirehep.net/literature/1388134)]

## Physics lecture notes

* E. Alvarez, P. Meessen, _String primer_ ([hep-th/9810240](http://arxiv.org/abs/hep-th/9810240))

* David Tong, _Lectures on string theory_ ([arxiv/0908.0333](http://front.math.ucdavis.edu/0908.0333))

* [[Clifford Johnson]], _D-Brane primer_ ([arXiv:hep-th/0007170](http://arxiv.org/abs/hep-th/0007170))

* [[Michael Douglas]], [[Elias Kiritsis]] et. al. (eds.), _[[String theory and the real world]]_, Les Houches  Session LXXXVII 2007

## Popular level books and string propaganda

* [[Brian Greene]], _The elegant universe: superstrings, hidden dimensions, and the quest for the ultimate theory_

* Michio Kaku, various volumes

* video and slides of Witten's KITP overview [Future of String Theory](http://online.itp.ucsb.edu/online/kitp25/witten)

## Big mathematically inclined surveys

* [[Hisham Sati]], _[[Geometric and topological structures related to M-branes]]_, comprehensive survey

* [[Anton Kapustin]], [[Dmitri Orlov|D. O. Orlov]], _Lectures on mirror symmetry, derived categories, and $D$-branes_, Russian Mathematical Surveys, 2004, 59:5, 907–940 (Russian version: [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=772&volume=59&year=2004&issue=5&fpage=101&what=fullt&option_lang=eng), [arxiv version: [arxiv:math.AG/0308173](http://arxiv.org/abs/math.AG/0308173)).

## History

* [[Mike Duff]], _[[The World in Eleven Dimensions]]: Supergravity, Supermembranes and M-theory_ IoP 1999

* {#TheBirthOfStringTheory} Andrea Cappelli, Elena Castellani, Filippo Colomo, Paolo Di Vecchia (eds.), _The Birth of String Theory_
Cambridge University Press (2012). ([additional material](http://theory.fi.infn.it/colomo/string-book/), [publisher webpage](http://www.cambridge.org/gb/knowledge/isbn/item5979248/?site_locale=en_GB))

## Other lists of bibliography

* [String Theory Wiki](http://www.stringwiki.org/w/index.php?title=String_Theory_Wiki)

* [[books and reviews in mathematical physics]]

* AdS/CFT [list](http://www.personal.uni-jena.de/~p5thul2/notes/adscft.html) in Jena (see also nlab page [[AdS-CFT]]).

category: reference

[[!redirects books about string theory]]
[[!redirects books on string theory]]


books and reviews in mathematical physics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

The intention of this page is to list a wide choice of main books and comprehensive reviews in [[mathematical physics]]. The irrelevant repetitions and minor, too specialized and obsolete books in any major respect should be avoided. We avoid references for quantum groups as they are many and the main ones can be found at the [[quantum group]] entry; similarly we avoided the relevant books on Kac-Moody algebras and groups but included the books on related VOAs and the Pressley-Segal book.

#Contents#
* table of contents
{:toc}

## Classical mathematical physics

Here [[PDEs]], [[differential equation|ODE]]s, and integral equation of mathematical physics, special functions, [[generalized function]]s, analytic functions, basic [[functional analysis]], potential theory:

* R. Courant, [[David Hilbert]], _Methods of mathematical physics_, 2 vols.

* P. M. Morse, H. Feshbach, _Methods of theoretical physics I, II, [publisher](http://www.feshbachpublishing.com)

* Michael Reed, Barry Simon, _Methods of modern mathematical physics_, 4 vols. (emphasis on functional analysis)

* V. Vladimirov, _Equations of mathematical physics_, Moscow, Izdatel'stvo Nauka, (1976. 528 p. Russian;  English edition, Mir 198x); _Generalized functions in mathematical physics_, Moscow

Focus on [[differential geometry]]:

* [[Yvonne Choquet-Bruhat]], [[Cécile DeWitt-Morette]], *Analysis, manifolds and physics*, North Holland (1982, 2001) [[ISBN:9780444860170](https://www.elsevier.com/books/analysis-manifolds-and-physics-revised-edition/choquet-bruhat/978-0-444-86017-0)]

* [[Mikhail O. Katanaev]], *Geometrical methods in mathematical physics* (in Russian) [[arXiv1311.0733](https://arxiv.org/abs/1311.0733)]

## Classical mechanics

* [[Vladimir Arnold]], *[[Mathematical methods of classical mechanics]]*, Springer (1989).

* [[Michael Spivak]], *Physics for Mathematicians, Mechanics I*, xvi + 733 pages, Publish or Perish 2010 [contents](http://olivier.thill.perso.neuf.fr/books/bospphma.htm), [amazon](http://www.amazon.com/Physics-Mathematicians-Mechanics-Michael-Spivak/dp/0914098322); _Elementary mechanics from a mathematician's viewpoint_, [pdf](http://www.math.uga.edu/~shifrin/Spivak_physics.pdf)

* R. Abraham, J. Marsden, _The Foundations of Mechanics_, Benjamin Press, 1967, Addison-Wesley, 1978; large pdf 86 Mb free at [CaltechAuthors](http://resolver.caltech.edu/CaltechBOOK:1987.001)

* [[William Lawvere]], [[Stephen Schanuel]] (eds.), _[[Categories in Continuum Physics]]_, Lectures given at a Workshop held at SUNY, Buffalo 1982, Lecture Notes in Mathematics 1174, 1986

* [[Vicente Cortés]], Alexander S. Haupt, _Lecture Notes on Mathematical Methods of Classical Physics_, ([arXiv:1612.03100](https://arxiv.org/abs/1612.03100))

## Mathematical introduction to quantum mechanics

On [[quantum mechanics]]:

* Anthony Sudbery, _Quantum mechanics and the particles of nature: An outline for mathematicians_

* [[Leon A. Takhtajan]], *[[Quantum mechanics for mathematicians]]*, Graduate Studies in Mathematics __95__, Amer. Math. Soc. 2008.

* Brian C. Hall, _Quantum theory for mathematicians_, Springer GTM __267__ (has also a big chapter on geometric quantization)

## Geometry and symmetries in classical and QM, quantization (but no QFT)

In addition to the geometrically written titles under classical mechanics above,

* Bernard F. Schutz, *Geometrical methods of mathematical physics* (elementary intro) [amazon](http://www.amazon.com/Geometrical-Methods-Mathematical-Physics-Bernard/dp/0521298873), [google](http://books.google.hr/books?id=HAPMB2e643kC&lpg=PP1&dq=schutz%20geometrical%20methods&pg=PP1#v=onepage&q&f=false)

* [[Peter Olver]], *Equivalence, invariants, and symmetry*, Cambridge Univ. Press 1995; *Applications of Lie groups to differential equations*, Springer.

* [[Shlomo Sternberg]], *Group theory and physics*, Cambridge University Press 1994.

* [[Victor Guillemin]], [[Shlomo Sternberg]], *Symplectic techniques in physics*, Cambridge University Press (1990)

* [[Theodore Frankel]], _[[The Geometry of Physics - An Introduction]]_, Cambridge Univ. Press

* [[Chris Isham]], _[[Modern Differential Geometry for Physicists]]_

* [[Klaas Landsman]], _Mathematical topics between classical and quantum mechanics_, Springer Monographs in Mathematics 1998. xx+529 pp.

* Sean Bates, [[Alan Weinstein]], _Lectures on the geometry of quantization_, [pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)

* A. Cannas da Silva, A. Weinstein, _Geometric models for noncommutative algebras_, 1999, [pdf](http://math.berkeley.edu/%7Ealanw/Models.pdf)

* [[Nikolai Reshetikhin]], _Lectures on quantization of gauge systems_, [arxiv/1008.1411](http://arxiv.org/abs/1008.1411)

## Lorentzian geometry and general relativity

Global aspects of the geometry of spacetimes:

* John K. Beem, Paul E. Ehrlich, Kevin L. Easley, _Global Lorentzian geometry_ ([ZMATH](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0846.53001&format=complete)) (global aspects)

* Arthur L. Besse, _Einstein manifolds_, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 10, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987, xii + 510 pp. (differential geometry of solutions to Einstein equations, constant negative curvature, classification results, encyclopaedic; for a review see [Bull. AMS](http://projecteuclid.org/euclid.bams/1183554925)

After the introduction emphasis on asymptotics of spacetimes far from gravitation objects:

* John Stewart, _Advanced general relativity_ ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0752.53048&format=complete))

Despite its title the next monograph does not just present the Kerr spacetime, it illustrates many core features of GR with the Kerr spacetime as the prominent example:

* Barrett O'Neill, _The geometry of Kerr black holes._ ([ZMATH](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0828.53078&format=complete))

Here is an introduction to spinors in GR:

* Peter O'Donnell, _Introduction to 2-spinors in general relativity._  ([ZMATH] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1021.83001&format=complete))

while the classic reference for this is:

* Roger Penrose, Wolfgang Rindler, _Spinors and spacetimes_ (2 vols, [vol 1, ZMATH](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0538.53024&format=complete))

* Eric Poisson, _A relativist's toolkit. The mathematics of black-hole mechanics._ ([ZMATH](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1058.83002&format=complete)) (computationally oriented)

See also the above book by Ward and Wells; and mainstream theoretical physics gravity textbooks by Misner, Thorne and Wheeler; Schutz; Landau-Lifschitz vol. 2; Wald; [[Chandrasekhar]] etc. For the [[supergravity]] see the appropriate chapters in the above listed collection by Deligne et al. or the references listed at [[supergravity]].

## Integrable systems and solitons

On [[integrable system]]s and [[soliton]]s:

* O. Babelon, D. Bernard, M. Talon, *Introduction to classical integrable systems*, Cambridge Univ. Press 2003.

* T. Miwa, M. Jimbo, E. Date, _Solitons: Differential equations, symmetries and infinite dimensional algebras_, Cambridge Tracts in Mathematics __135__, translated from Japanese by Miles Reid

* V.E. Korepin, N. M. Bogoliubov, A. G. Izergin, _Quantum inverse scattering method and correlation functions_, Cambridge Univ. Press 1997.

* Ludwig D. Faddeev, Leon Takhtajan, _Hamiltonian methods in the theory of solitons_, Springer

## Modern mathematical approaches to QFT and strings

On [[quantum field theory]] and [[string theory]]:

* Yvonne Choquet-Bruhat, Cecile Dewitt-Morette, *Analysis, manifolds and physics*, 1982 and 2001

* Albert Schwartz, *Quantum field theory and topology*, Grundlehren der Math. Wissen. __307__, Springer 1993. (translated from Russian original)

* Howard Georgi: _Lie Algebras in Particle Physics. From isospin to unified theories._ ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0505.00036&format=complete))

* [[Eberhard Zeidler]], *Quantum field theory. A bridge between mathematicians and physicists*. I: Basics in mathematics and physics. , II: Quantum electrodynamics

* {#HenneauxTeitelboim} [[Marc Henneaux]], [[Claudio Teitelboim]], _[[Quantization of Gauge Systems]]_, Princeton University Press 1992.

* Charles Nash, *Differential topology and quantum field theory*, Acad. Press 1991.

* P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and E. Witten, eds.  _Quantum fields and strings, A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

* [[Gregory L. Naber]], *Topology, Geometry and Gauge fields -- Foundations*,  Texts in Applied Mathematics **25** (2011) [[doi:10.1007/978-1-4419-7254-5](https://doi.org/10.1007/978-1-4419-7254-5)]

* [[Gregory L. Naber]], *Topology, Geometry and Gauge fields -- Interactions*, Applied Mathematical Sciences **141** (2011) [[doi:10.1007/978-1-4419-7895-0](https://doi.org/10.1007/978-1-4419-7895-0)]

* [[Mikio Nakahara]], *[[Geometry, Topology and Physics]]*

* Marian Fecko, *Differential geometry and Lie groups for physicists*

* V. S. Varadarajan, *Supersymmetry for mathematicians: an introduction*, AMS and Courant Institute, 2004.

* R. S. Ward, R. O. Wells, *Twistor geometry and field theory* (CUP, 1990)

* R. E. Borcherds, A. Barnard, *Lectures on QFT*, [arxiv:math-ph/0204014](http://arxiv.org/abs/math-ph/0204014)

* [[Anton Kapustin]], _Topological field theory, higher categories, and their applications_, survey for ICM 2010, [arxiv/1004.2307](http://arxiv.org/abs/1004.2307)

* Siddhartha Sen, [[zoranskoda:Kumar Gupta|Kumar Sankar Gupta]], [[zoranskoda:Gupta-Sen book|Many-body physics, topology and geometry]]

### Branes (mathematical aspects)

* Paul Aspinwall, Tom Bridgeland, Alastair Craw, Michael R. Douglas, Mark Gross, _Dirichlet branes and mirror symmetry_, Amer. Math. Soc. Clay Math. Institute 2009.

* Hisham Sati, [[Geometric and topological structures related to M-branes]], part I ([arXiv:1001.5020](http://arXiv.org/abs/1001.5020)), part _II: Twisted $String$ and $String^c$ structures_ ([arxiv/1007.5419](http://arxiv.org/abs/1007.5419)); part _III: Twisted higher structures_ ([http://arxiv.org/abs/1008.1755](http://arxiv.org/abs/1008.1755))

* A. N. Kapustin, [[Dmitri Orlov|D. O. Orlov]], _Lectures on mirror symmetry, derived categories, and $D$-branes_, Russian Mathematical Surveys, 2004, 59:5, 907–940 (Russian version: [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=772&volume=59&year=2004&issue=5&fpage=101&what=fullt&option_lang=eng), [arxiv version: [arxiv:math.AG/0308173](http://arxiv.org/abs/math.AG/0308173)).

## Conformal field theory and vertex algebras

On [[conformal field theory]] and its chiral part ([[vertex operator algebras]], [[chiral algebras]]):

* Edward Frenkel, David Ben-Zvi: _Vertex algebras and algebraic curves_, Math. Surveys and Monographs __88__, AMS 2001,
xii+348 pp. (Bull. AMS. [review](http://www.ams.org/journals/bull/2002-39-04/S0273-0979-02-00955-2/S0273-0979-02-00955-2.pdf), [ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1106.17035&format=complete))

* Martin Schottenloher, *A mathematical introduction to conformal field theory* (CFT on the plane)

* Philippe Di Francesco, Pierre Mathieu, David Sénéchal, _Conformal field theory_, Springer 1997 (comprehensive textbook for theoretical physicists)

* Ralph Blumenhagen, Erik Plauschinn, _Introduction to conformal field theory: with applications to string theory_, Springer Lecture Notes in Physics (2011)

* V. Kac, _Vertex algebras for beginners_, Amer. Math. Soc.

* B. Bakalov, A. Kirillov, _Lectures on tensor categories and modular functors_, AMS, University Lecture Series, (2000) ([web](http://www.math.sunysb.edu/~kirillov/tensor/tensor.html))

* [[Alexander Beilinson]], [[Vladimir Drinfeld]], __Chiral algebras__, Colloqium Publications __51__,  Amer. Math. Soc. 2004, [gbooks](http://books.google.hr/books?id=yHZh3p-kFqQC&lpg=PP1&vq=%22Two-dimensional%20conformal%20geometry%20and%20vertex%20operator%20algebras%22&dq=isbn%3A0817638296&pg=PP1#v=onepage&q=%22Two-dimensional%20conformal%20geometry%20and%20vertex%20operator%20algebras%22&f=false)

The related subject of positive energy representations for loop groups is represented in unavoidable reference

* A. Pressley, G. Segal, _Loop groups_, Oxford Math. Monographs, 1986.

## Axiomatic quantum/statistical field theory and rigorous approaches to path integral

* N. N. Bogoliubov, A. A. Logunov, I. T. Todorov, *Introduction to axiomatic quantum field theory*, 1975

* R. Haag, _Local quantum physics: fields, particles, algebras_, Springer 1992 [MR94d:81001](http://www.ams.org/mathscinet-getitem?mr=94d:81001), 1996 [MR98b:81001](http://www.ams.org/mathscinet-getitem?mr=98b:81001)

* Huzihiro Araki: _Mathematical theory of quantum fields._ Oxford University Press 1999 [ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0998.81501&format=complete).

* James Glimm, Arthur Jaffe, *Quantum physics: a functional integral point of view*, Springer

* Sergio Albeverio, Jens Erik Fenstad, Raphael Hoegh-Krohn, _Nonstandard methods in stochastic analysis and mathematical physics_, Acad. Press 1986 (there is also a Dover 2009 edition, and a 1990 Russian translation)

## Other reference lists

* [http://en.wikibooks.org/wiki/Introduction_to_Mathematical_Physics/References](http://en.wikibooks.org/wiki/Introduction_to_Mathematical_Physics/References) (that list has mostly references in the area of theoretical physics and just a minority with rigor of mainstream mathematical physics)

* [www.stringwiki.org](http://www.stringwiki.org/w/index.php?title=String_Theory_Wiki)
-- wiki contains entries on several dozens of string theory related topics, where each entry has several references, mainly papers available online

* [[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]], Proceedings of Symposia in Pure Mathematics __83__, Amer. Math. Soc. (2011)

* MathOverflow discussion: _[Companion to theoretical physics for working mathematicians](http://mathoverflow.net/questions/191609/companion-to-theoretical-physics-for-working-mathematicians)_

* reference list of Movshev's sunysb [course on QM](http://www.math.sunysb.edu/~mmovshev/MAT570Spring2008/syllabusfinal.html)

* geometrical background for [[Anton Kapustin]]'s course at Caltech: [list](http://theory.caltech.edu/~kapustin/Ph229/recbooks.html)

* John Baez: _How to Learn Math and Physics_  ([here] (http://math.ucr.edu/home/baez/books.html))

* [[Frédéric Paugam]], _Towards the mathematics of quantum field theory_,  Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 59, Springer (2014), (a draft formerly available [here](http://people.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics.pdf)).

* [[books about string theory]]
* various lectures notes in mathematical physics, Ulrich Theis' list at [Jena](http://www.personal.uni-jena.de/~p5thul2/notes/math.html)

category: reference
[[!redirects books in mathematical physics]]


books in algebraic geometry

# Books in algebraic geometry

We should limit to books which we can really recommend, either by their special content, approach or pedagogical value. Historically fine but outdated books are in a separate historical section below. The outstanding surveys may be added to the lists if they are not too specialized to minor directions: the subfields may be covered in separate lists. See also MathOverflow discussions [best-algebraic-geometry-text-book-other-than-hartshorne](http://mathoverflow.net/questions/2446/best-algebraic-geometry-text-book-other-than-hartshorne), [life-after-hartshorne-the-book](http://mathoverflow.net/questions/61515/life-after-hartshorne-the-book-not-the-person). To orient in the basic standard terminology, the wikipedia's [Glossary of algebraic geometry](https://en.wikipedia.org/wiki/Glossary_of_algebraic_geometry) is decent.

* table of contents
{: toc}

## Introductory level but excellent textbooks

* [[Miles Reid]], _Undergraduate algebraic geometry_, London Math. Soc. Student Texts __12__
* [[Joe Harris]], _Introductory algebraic geometry_ (varieties)
* [[Igor Shafarevich]], _Basic algebraic geometry_ (varieties and schemes)
* [[Shigeru Mukai]], _An introduction to invariants and moduli_, Cambridge Studies in Adv. Math. __81__
* [[William Fulton]], _Algebraic curves. An introduction to algebraic geometry_, 3rd ed. 2008 (varieties)
* [[James Milne|J. S. Milne]], _Algebraic geometry_, 2017 [pdf](http://www.jmilne.org/math/CourseNotes/AG.pdf)

## Schemes, standard sources

* [[Robin Hartshorne]], _Algebraic geometry_, Springer
* Qing Liu, _Algebraic geometry and arithmetic curves_, 592 pp. Oxford Univ. Press 2002
* D. Eisenbud, J. Harris, _The geometry of schemes_, Springer Grad. Texts in Math.
* [[David Mumford]], _Red book of varieties and schemes_ (cf. also unfinished sequel notes for the later part of Mumford's course, coauthored with Oda, [ch. 1-6 pdf](http://www.math.upenn.edu/~chai/624_08/mumford-oda_chap1-6.pdf), [ch. 7-8 pdf](http://www.math.upenn.edu/~chai/624_08/mumford-oda_chap7-8.pdf))
* Amnon Neeman, _Algebraic and analytic geometry_, London Math. Soc. Lec. Note Series __345__
* M. Demazure, P. Gabriel, _Groupes algebriques_, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970
* Ravi Vakil's Stanford [course notes](http://math.stanford.edu/~vakil/0708-216)
* William Fulton, _Intersection theory_, Springer 1984
* Ulrich Görtz, Torsten Wedhorn, _Algebraic Geometry I: Schemes_ With Examples and Exercises, 2nd Edition,
Springer Studium Mathematik - Master,
Springer Nature, 2020.

## Grothendieck school sources

* [[EGA]]
* [[SGA]]
* [[FGA]], [[FGA explained]]
* R. Hartshorne, _Residues and duality_
* A. Grothendieck et al. _Dix exposes sur la cohomologie des schemas_, North Holland, Amsterdam, 1968.

## Background in commutative algebra

* M. Atiyah, I. G. Macdonald, _Introduction to commutative algebra_, 1969, 1994
* H. Matsumura, _Commutative algebra_, 2 vols.; see also the online summary notes by D. Murfet, [Matsumura.pdf](http://therisingsea.org/notes/Matsumura.pdf), [Matsumura-Part2.pdf](http://therisingsea.org/notes/Matsumura-Part2.pdf)
* D. Eisenbud, _Commutative algebra: with a view toward algebraic geometry_, Grad. Texts in Math. 150, Springer-Verlag 1995.
* James Milne, _A primer of commutative algebra_, (online notes in progress) [webpage](http://www.jmilne.org/math/xnotes/ca.html), [pdf](http://www.jmilne.org/math/xnotes/CA.pdf)

## Algebraic groups/group schemes; Fourier-Mukai transform

* William C. Waterhouse, _Introduction to affine group schemes_, GTM 66, Springer 1979
* [[Armand Borel]], _Linear algebraic groups_, Springer GTM, 2 editions
* Tonny A. Springer, _Linear algebraic groups_, Progress in Mathematics, 9 (2nd ed.), BBirkhäuser Boston 1998, [MR1642713](http://www.ams.org/mathscinet-getitem?mr=1642713)
* [[David Mumford]], _Abelian varieties_, Oxford Univ. Press 1970
* A. Polishchuk, _Abelian varieties, theta functions and the Fourier transform_, Cambridge Univ. Press 2003
* [[M. Demazure]], [[P. Gabriel]], _Groupes algebriques_, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970 -- has functor of points point of view (listed also under scheme theory); for review see
Bull. London Math. Soc. (1980) 12 (6): 476-478, [doi](http://dx.doi.org/10.1112/blms/12.6.476b)
* SGA III
* J. S. Milne, _Abelian varieties_, course notes, [pdf](http://www.jmilne.org/math/CourseNotes/AV.pdf)
* [[Daniel Huybrechts]], _Fourier-Mukai transforms in algebraic geometry_, Oxford Mathematical Monographs. 2006. 307 pages.
* André Weil, _Courbes algébriques et variétés abéliennes_, Paris: Hermann 1971

* C. Bartocci, [[Ugo Bruzzo]], D. Hernandez Ruiperez, _Fourier-Mukai and Nahm transforms in geometry and mathematical physics_, Progress in Mathematics __276__, Birkhauser 2009.

## Complex-analytic approach

* P. Griffiths, J. Harris, _Principles of algebraic geometry_
* Phillip A. Griffiths, _Introduction to algebraic curves_
* [[Daniel Huybrechts]], _Complex geometry - an introduction_, Springer (2004). Universitext. 309 pages.
* [[Donu Arapura]], _Algebraic geometry over the complex numbers_, Springer Universitext 223, 329 pp.
* H. Grauert, R. Remmert, _Coherent Analytic Sheaves
* C. Bănică, O. Stănăşilă, _Algebraic methods in the global theory of complex spaces
* H. Grauert, Th. Peternell, R. Remmert, _Several complex variables VII: sheaf-theoretical methods in complex analysis
* G. Fischer, _Complex Analytic Geometry

## Mori program and birational geometry

* János Kollár, Shigefumi Mori, _Birational geometry of algebraic varieties_,
With the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original. Cambridge Tracts in Math. 134 (1998), viii+254 pp.
* Kenji Matsuki, _Introduction to the Mori program_, Universitext. Springer 2002. xxiv+478 pp. [MR2002m:14011](http://www.ams.org/mathscinet-getitem?mr=1875410)
* Herbert Clemens, János Kollár, Shigefumi Mori, _Higher-dimensional complex geometry_, Astérisque __166__ (1988), 144 pp. (1989).

## Arithmetic aspects

* Goro Shimura, _Abelian varieties with complex multiplication and modular functions_, Princeton Univ. Press 1997
* Dale Husemöller, _Elliptic curves_, Graduate Texts in Mathematics. 111 (2nd ed.). Springer 2004, ISBN 0-387-95490-2.
* Anthony Knapp, _Elliptic curves_, Math Notes. 40. Princeton University Press 1992
* Neal Koblitz, _Introduction to elliptic curves and modular forms_, Graduate Texts in Mathematics. __97__ (2nd ed.). Springer-Verlag 1993
* Joseph H. Silverman, _The Arithmetic of Elliptic Curves_, Graduate Texts in Math. __106__, Springer 1986; _Advanced Topics in the Arithmetic of Elliptic Curves_ Graduate Texts in Mathl. __151__, Springer 1994.
* J. H. Silverman, John Tate, _Rational Points on Elliptic Curves_, Springer 1992.
* [[Gerd Faltings]], _Lectures on arithmetic Riemann-Roch theorem_, Annals of Math. Studies __127__, Princeton Univ. Press 1992
* S. Bosch, W. Lütkebohmert, M. Raynaud, _Néron models_, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer 1990. x+325 pp. [MR91i:14034](http://www.ams.org/mathscinet-getitem?mr=1045822)

## Cohomology theories for schemes

* J. S. Milne, _Etale cohomology_, Princeton Univ. Press 1980, [gbooks](http://books.google.com/books?id=uAT4T4BXC50C&lpg=PP1&dq=milne%20etale%20cohomology)
* P. Berthelot, A. Ogus, _Notes on crystalline cohomology_, Princeton Univ. Press 1978. vi+243, ISBN0-691-08218-9
* ...list basic literature on [[motive]]s
* [[Marc Levine]], _Mixed motives_, Math. Surveys and Monographs __57__, Amer. Math. Soc.  1998, free [pdf](http://www.ams.org/online_bks/surv57/surv57.pdf)
* F. Hirzebruch, _Topological methods in algebraic geometry_
* Ulrich Görtz, Torsten Wedhorn, _Algebraic Geometry II: Cohomology of Schemes_
With Examples and Exercises,
Springer Studium Mathematik - Master,
Springer Nature, 2023.

## Modern extensions of scheme theory

These are advanced books or long foundational expositions.

* D. Knutson, _Algebraic spaces_, Springer 1971
* [[Ofer Gabber]], Lorenzo Ramero, _Almost ring theory_, arxiv and published
* [[Jacob Lurie]], _Derived algebraic geometry_, several issues, arxiv
* [[Bertrand Toen]], [[Gabrielle Vezzosi]], HAG and DAG
* [[Nikolai Durov]], _A new approach to Arakelov geometry_, arxiv
* something basic on log schemes, e.g. from Kato/Ogus/Olsson

## Algorithmic and computational methods

Things like Groebner bases, combinatorical methods with toric varieties etc.

* David A. Cox, John B. Little, Don O'Shea, _Ideals, varieties, and algorithms_

## Historically important but now outdated

While many of these fine books are still pleasure for some readers, they do not capture the modern viewpoint and have maybe too old notation to be regularly used. But one should be aware of them, and of sometimes unique material exposed there.

* W. V. D. Hodge, Daniel Pedoe, _Methods of algebraic geometry_, 3 vols. (see review by Coxeter in Bull. Amer. Math. Soc. __55__, 3, part 1 (1949), 315-316, [euclid](http://projecteuclid.org/euclid.bams/1183513548))
* F. Severi, Vol. I (1942): Serie, sistemi d'equivalenza e correspondenze algebriche sulle varieta algebriche. Vol. I I (1958) and I I I (1959): Geometria dei sistemi algebrici sopra una superficie e sopra una varieta algebrica.

category: reference

[[!redirects book in algebraic geometry]]
[[!redirects books on algebraic geometry]]
[[!redirects book on algebraic geometry]]
[[!redirects algebraic geometry books]]
[[!redirects algebraic geometry book]]


BoolAlg

# The category of Boolean algebras
* automatic table of contents goes here
{:toc}

## Definition

__$Bool Alg$__ is the [[category]] whose [[objects]] are [[boolean algebras]] and whose [[morphisms]] are lattice [[homomorphisms]], that is [[functions]] which preserve finitary [[meets]] and [[joins]] (equivalently, binary meets and joins and the [[top]] and [[bottom]] elements); it follows that the homomorphisms preserve [[negation]].  $Bool Alg$ is a [[subcategory]] of [[Pos]], in fact a [[replete subcategory]] of both [[DistLat]] and [[HeytAlg]].

$Bool Alg$ is given by a finitary [[variety of algebras]], or equivalently by a [[Lawvere theory]], so it has all the usual properties of such categories. As usual, the [[Lawvere theory]] is the category opposite to the category of [[finitely generated object|finitely generated]] [[free object|free]] Boolean algebras.

## Free Boolean algebras

The free Boolean algebra $Bool(X)$ generated by a finite set $X$ is isomorphic to the _double [[power set]]_ $\mathcal{P}\mathcal{P}X$ of $X$; an element $a$ of $X$ is interpreted as the set of all those [[subsets]] of $X$ to which $a$ belongs, and the boolean algebra operations on the [[intersection]] and [[union]] as usual.

The __free Boolean algebra__ on an arbitrary set $X$ is more complicated; it can be described in several ways. By [[abstract nonsense]], it can be described as a [[filtered colimit]] of finitely generated free Boolean algebras

$$Bool(X) = colim_{S \subseteq_{fin} X} Bool(S)$$

(where $\subseteq_{fin}$ indicates a [[finite set|finite subset]]) in the category of Boolean algebras; here the colimit is over the [[poset]] of finite subsets and inclusions between them.

A second, more concrete description is

$$Bool(X) = \mathcal{P}_{fin} \mathcal{P}_{fin} X$$

where $\mathcal{P}_{fin}(X)$ denotes the set of [[Dedekind finite set|Dedekind finite]] [[subset]]s of $X$.

However, the Boolean operations are not what one might naively expect. The simplest way of describing the operations is to consider Boolean algebras as equivalent to Boolean rings where binary addition is given by [[symmetric difference]], as explained further at [[Boolean algebra]].
We can construct free Boolean rings by analogy with the [[polynomial algebra]] construction, where one forms the free [[vector space]] generated by a monoid of monomials.

To do this, we first make $\mathcal{P}_{fin}: Set \to Set$ into the [[monad]] for commutative idempotent monoids (that is, [[semilattices]]).  Here we use the fact that that $\mathcal{P}_{fin} X$ with the monoid operation given by union, is the free semilattice on $X$.  Call $\mathcal{P}_{fin}$ with this monad structure $M$, for "multiplication".

Next, let $S: Set \to Set$ be the monad for vector spaces over $\mathbb{F}_2$.  We can take the functor $S$ to have the same action on objects as $\mathcal{P}_{fin}$, since $\mathcal{P}_{fin}X$ with symmetric difference as addition gives the free $\mathbb{F}_2$-vector space on $X$. But beware: $S$ acts differently on morphisms.

Finally, note that there is a [[distributive law]] $M \circ S \Rightarrow S \circ M$ that maps any product of sums to a sum of products in the usual way.  This makes the composite $S \circ M \colon Set \to Set$ into a monad, which in fact is the monad for Boolean rings.   This monad has the same action on objects as the functor $\mathcal{P}_{fin} \mathcal{P}_{fin}: Set \to Set$, but it behaves differently on morphisms.

Using the equivalence between Boolean rings and Boolean algebras, we thus obtain:

\begin{theorem}
The functor $M \circ S \colon Set \to Set$ has a monad structure making it into the monad for Boolean algebras.  That is, the [[Eilenberg-Moore category]] of this monad is equivalent to $BoolAlg$.
\end{theorem}

(A more naive prescription, where one uses the usual intersection and union on the Boolean algebra $\mathcal{P}_{fin} Y$ for $Y = \mathcal{P}_{fin} X$, is guaranteed to fail because this is an [[atomic Boolean algebra]], whereas the free Boolean algebra on an infinite set $X$ is atomless.)

A third description comes from Stone duality (see below).

## Stone duality
 {#StoneDuality}

Classical [[Stone duality]] comes about as follows. The two-element Boolean algebra can be regarded as a Boolean algebra object $\mathbf{2}$ in the category of [[compact Hausdorff spaces]] $CH$. Thus, for each finitary Boolean algebra operation $\theta\colon \mathbf{2}^n \to \mathbf{2}$, there is a corresponding operation on the representable functor $CH(-, \mathbf{2}): CH^{op} \to Set$ given by

$$CH(-, \mathbf{2})^n \cong CH(-, \mathbf{2}^n) \stackrel{CH(-, \theta)}{\to} CH(-, \mathbf{2})$$

and therefore we obtain a lift

$$CH(-, \mathbf{2}): CH^{op} \to BoolAlg$$

A _[[Stone space]]_ is by definition a totally disconnected compact Hausdorff space. Let $Stone \hookrightarrow CH$ denote the full subcategory of Stone spaces.

\begin{theorem}
The representable functor $CH(-, \mathbf{2}): CH^{op} \to BoolAlg$ restricts to an [[equivalence of categories]] $Stone^{op} \to BoolAlg$.
\end{theorem}

This important theorem can be exploited to give a third description of the free Boolean algebra on a set $X$:

$$Bool(X) \cong CH(2^X, \mathbf{2})$$

where $2$ denotes the 2-element compact Hausdorff space, and $2^X$ the product space $\prod_X 2$. Indeed, the inverse equivalence

$$BoolAlg^{op} \to Stone$$

takes a Boolean algebra $B$ to its [[spectrum]], i.e., the space of Boolean algebra maps $Bool(B, 2)$ (*this* $2$ is the two-element Boolean algebra $\mathbb{Z}_2$!) equipped with the [[Zariski topology]]. Applied to $B = Bool(X)$, we have

$$BoolAlg(B, 2) \cong Set(X, 2) = 2^X$$

where the Zariski topology coincides with the [[product topology]] on $2^X$. By the equivalence, we therefore retrieve $Bool(X)$ as $CH(2^X, \mathbf{2})$. This in turn is identified with the Boolean algebra of [[clopen subset]]s of the generalised [[Cantor space]] $2^X$.

A second description of the inverse equivalence $BoolAlg^{op} \to Stone$ comes about through the yoga of [[ambimorphic object|ambimorphic objects]]. Namely, the Boolean compact Hausdorff space $\mathbf{2}$ can equally well be seen as a [[compact Hausdorff object]] in the category of Boolean algebras. Thus, the representable functor $Bool(-, \mathbf{2}): Bool^{op} \to Set$ lifts canonically to a functor
$$Bool^{op} \to CH$$
and in fact part of the Stone representation theorem is that this factors through the inclusion $Stone \hookrightarrow CH$ as the inverse equivalence $Bool^{op} \to Stone$. In particular this lift determines the topology, providing an description alternative to the description in terms of the Zariski topology (although they are of course the same).

## In weak foundations

Boolean algebras are less interesting in [[constructive mathematics]], since [[power sets]] are not boolean algebras.  However, they are still a perfectly good algebraic construct, and the explicit construction of free algebras in terms of finite subsets is still correct.  Stone duality also works in constructive mathematics, but it must be done using [[locales]] instead of standard [[topological spaces]].

In [[predicative mathematics]], the explicit construction of free algebras works if we have general [[inductive object]]s; the [[natural numbers object]] alone is not enough.

category: category

[[!redirects BoolLat]]
[[!redirects Bool Lat]]
[[!redirects BoolAlg]]
[[!redirects BoolALg]]
[[!redirects Bool Alg]]
[[!redirects BoolRing]]
[[!redirects Bool Ring]]

[[!redirects free boolean algebra]]
[[!redirects free Boolean algebra]]
[[!redirects free boolean algebras]]
[[!redirects free Boolean algebras]]
[[!redirects free boolean lattice]]
[[!redirects free Boolean lattice]]
[[!redirects free boolean lattices]]
[[!redirects free Boolean lattices]]
[[!redirects free boolean ring]]
[[!redirects free Boolean ring]]
[[!redirects free boolean rings]]
[[!redirects free Boolean rings]]

[[!redirects double powerset]]
[[!redirects double power set]]


Boole -- The Laws of Thought

This page compiles links related to the book:

* [[George Boole]]:

  \linebreak

  **An Investigation of the Laws of Thought**

  **on Which are Founded the Mathematical Theories**

  **of Logic and Probabilities**

  (1854)

  \linebreak

  republished by:

  Cambridge University Press (2009)

  [doi:10.1017/CBO9780511693090](https://doi.org/10.1017/CBO9780511693090)

  \linebreak

  [Wikipedia entry](https://en.wikipedia.org/wiki/The_Laws_of_Thought)

  [Project Gutenberg eBook](https://gutenberg.org/ebooks/15114)

on [[formal logic]] (for which [[George Boole|Boole]] became widely famous, cf. *[[Boolean algebra]]*) but also on [[probability theory]], where  (as observed by [Pitowsky 1989a](Bell's+inequalities#Pitowsky89a), reviewed, e.g., in [Khrennikov 2008](Bell's+inequalities#Khrennikov08)) Boole's book derives the inequalities that later became known as *[[Bell's inequalities]]* in [[quantum probability theory]].

## References

Review includes:

* I. Grattan-Guinness, *George Boole, An investigation of the laws of thought on which are founded the mathematical theory of logic and probabilities (1854)*, Chapter 36 (p. 470-479) in: *Landmark Writings in Western Mathematics 1640-1940* (2005) [[doi:10.1016/B978-044450871-3/50117-0](https://doi.org/10.1016/B978-044450871-3/50117-0)]

category: reference


Boolean algebra

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### Context
#### $(0,1)$-Category theory
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[[!include (0,1)-category theory - contents]]
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# Boolean algebras
* table of contents
{: toc}

## Idea

A _Boolean algebra_ or _Boolean [[lattice]]_ is an algebraic structure which models classical [[propositional calculus]], roughly the fragment of the logical calculus which deals with the basic logical connectives "and", "or", "implies", and "not".

## Definitions

### General

There are many known ways of defining a __Boolean algebra__ or __Boolean lattice__. Here are just a few:

* A Boolean algebra is a [[complement|complemented]] [[distributive lattice]].

* A Boolean algebra is a [[Heyting algebra]] $H$ satisfying the [[law of excluded middle]], which means either
$$\neg \neg = id: H \to H$$
or (equivalently)
$$\forall_{x \in H} x \vee \neg x = \top$$

* A Boolean algebra is a [[lattice]] $L$ equipped with a function $\neg: L \to L$ satisfying
$$a \wedge b \leq c \qquad iff \qquad a \leq \neg b \vee c$$

### Explicitly

There are even two explicit definitions: [[order theory|order-theoretic]] and algebraic.

A __Boolean lattice__ is a [[poset]] such that:

*  there is an element $\top$ (a [[top element]]) such that $x \leq \top$ always holds;
*  there is an element $\bot$ (a [[bottom element]]) such that $\bot \leq x$ always holds;
*  given elements $a$ and $b$, there is an element $a \wedge b$ (a [[meet]] of $a$ and $b$) such that $x \leq a \wedge b$ holds iff $x \leq a$ and $x \leq b$;
*  given elements $a$ and $b$, there is an element $a \vee b$ (a [[join]] of $a$ and $b$) such that $a \vee b \leq x$ holds iff $a \leq x$ and $b \leq x$;
*  given an element $a$, there is an element $\neg{a}$ (a [[complement]] of $a$) such that $a \wedge \neg{a} \leq \bot$ and $\top \leq a \vee \neg{a}$;
*  given elements $a$, $b$, and $c$, we have $a \wedge (b \vee c) \leq (a \wedge b) \vee (a \wedge c)$.

Although we don\'t say so, we can prove that $\top$, $\bot$, $a \wedge b$, $a \vee b$, and $\neg{a}$ are unique; this makes it more clear what the last two axioms actually mean.

Notice that a poset carries at most one Boolean algebra structure, making it [[stuff, structure, property|property-like structure]]. (The same is true of [[Heyting algebra]] structure.)

Alternatively, a __Boolean algebra__ is a [[set]] equipped with elements $\top$ and $\bot$, binary operations $\wedge$ and $\vee$, and a unary operation $\neg$, satisfying these identities:

1.  $a \wedge \top = a$,
1.  $a \vee \bot = a$,
1.  $a \wedge (b \wedge c) = (a \wedge b) \wedge c$,
1.  $a \vee (b \vee c) = (a \vee b) \vee c$,
1.  $a \wedge b = b \wedge a$,
1.  $a \vee b = b \vee a$,
1.  $a \wedge (a \vee b) = a$,
1.  $a \vee (a \wedge b) = a$,
1.  $a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c)$,
1.  $a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)$,
1.  $a \wedge \neg{a} = \bot$,
1.  $a \vee \neg{a} = \top$.

We can recover the poset structure: $a \leq b$ iff $a \wedge b = a$. There is a certain amount of redundancy or overkill in this axiom list; for example, it suffices to give just axioms 1, 2, 5, 6, 9, 10, 11, 12.

A very distilled algebraic definition was conjectured by Herbert Robbins: any set equipped with a binary operation $\vee$ and a unary operation $\neg$ obeying

1. associativity: $a\vee \left(b\vee c\right)=\left(a\vee b\right)\vee c$
1. commutativity: $a \vee b = b \vee a$
1. the Robbins equation: $\neg \left(\neg \left(a\vee b\right)\vee \neg \left(a\vee \neg b\right)\right)=a $

is a Boolean algebra.  William McCune proved the conjecture in 1996, using the automated theorem prover EQP.  A short proof was found by Allan Mann (see the references).

#### Principle of duality

However it is defined, the theory of Boolean algebras is **self-dual** in the sense that for any sentence stated in the language $(\leq, \wedge, \vee, \bot, \top, \neg)$, the sentence is a theorem in the theory of Boolean algebras iff the dual sentence, obtained by interchanging $\wedge$ and $\vee$, $\bot$ and $\top$, and replacing $\leq$ by the opposite relation $\leq^{op}$, is also a theorem.

This incredibly useful result can be rephrased in several ways; for example, if a poset $B$ is a Boolean algebra, then so is its opposite $B^{op}$.

#### Boolean rings

A [[Boolean ring]] is a [[ring]] (with [[identity]]) for which every element is idempotent: $x^2 = x$. Notice that from

$$x^2 + 1 = x + 1 = (x + 1)^2 = x^2 + 2 x + 1$$

the equation $2 x = 0$ follows. Also notice that commutativity comes for free, since

$$x^2 + y^2 = x + y = (x + y)^2 = x^2 + x y + y x + y^2$$

whence $x y + y x = 0 = x y + x y$.

Parallel to the way free commutative rings are polynomial rings, which are free $\mathbb{Z}$-modules generated from free commutative monoids, the free Boolean ring on $n$ generators may be constructed, à la Beck [[distributive law]]s, as the free $\mathbb{Z}_2$-vector space $\mathbb{Z}_2[M_n]$ generated from the commutative idempotent monoid $M_n$ on $n$ generators. The latter can be identified with the power set on an $n$-element set with multiplication given by intersection, and $\mathbb{Z}_2[M_n]$ therefore has $2^{2^n}$ elements.

The theory of Boolean algebras is equivalent to the theory of Boolean rings in the sense that their categories of models are equivalent. Given a Boolean ring, we define the operation $\wedge$ to be multiplication, and the operation $\vee$ by $x \vee y = x + y + x y$, and the operation $\neg$ by $\neg x = 1 + x$. The relation $x \leq y$ may be defined by the condition $x y = x$. In the other direction, given a Boolean algebra, we may define addition by [[symmetric difference]]: $x + y = (x \vee y) \wedge \neg(x \wedge y)$. According to this equivalence, the free Boolean ring on $n$ generators may be identified with the Boolean algebra $P(2^n)$, the power set on a set with $2^n$ elements.

### Elements of Stone duality

The equivalence of Boolean rings and Boolean algebras was exploited by [[Marshall Stone]] to give his theory of [[Stone duality]], in which every Boolean algebra $B$ is a Boolean algebra of sets; more particularly the Boolean algebra of clopen (closed and open) sets of a topological space $Spec(B)$, the **Stone space** of $B$. The notation intentionally suggests that the Stone space is the underlying space of the [[spectrum (geometry)|spectrum]] of $B$ as Boolean ring, taking "spectrum" in the sense of algebraic geometry.

A Stone space may be characterized abstractly as a [[topological space]] that is [[compact space|compact]], [[Hausdorff space|Hausdorff]], and [[totally disconnected space|totally disconnected]]. Stone duality asserts among other things that every such space is the [[prime spectrum]] of the Boolean algebra of its clopen subsets.

+-- {: .num_lemma #aremax}
###### Lemma
All prime ideals in $B$ are kernels $\phi^{-1}(0)$ of homomorphisms $\phi: B \to \mathbf{2}$ (and thus are maximal ideals, in bijective correspondence with [[ultrafilters]] $\phi^{-1}(1)$ in $B$).
=--

+-- {: .proof}
###### Proof
If $p$ is a prime ideal in a Boolean ring, then $B/p$ is an integral domain in which every element $x$ is idempotent: $x(x-1) = 0$. Hence $B/p = \{0, 1\}$.
=--

(To be continued at some point.)

## Homomorphisms

Any [[lattice]] homomorphism automatically preserves $\neg$ and is therefore a Boolean algebra homomorphism.

Boolean algebras and Boolean algebra homomorphisms form a [[concrete category]] [[BoolAlg]].

## Definition via Lawvere theories

The concrete category $U: BoolAlg \to Set$ is [[monadic functor|monadic]]: the category of Boolean algebras is the [[Eilenberg-Moore category|category of algebras]] for a [[finitary monad]], or equivalently it is the category of algebras for a [[Lawvere theory]]. In this case the Lawvere theory is very easily described.

The Lawvere theory is equivalent to the category opposite to the category of finitely generated free Boolean algebras, or of finitely generated free Boolean rings. As we observed earlier, the free Boolean algebra on $n$ elements is therefore isomorphic to $P(2^n)$, the [[power set]] of a $2^n$-element set. Applying a "toy" form of [[Stone duality]], the opposite of the category of finitely generated free Boolean algebras is equivalent to the category of finite sets of cardinality $2^n$.

Hence the Lawvere theory is identified with the category $Fin_{2^n}$ of finite sets of cardinality $2^n$, and the category of Boolean algebras is equivalent to the category of product-preserving functors

$$Fin_{2^n} \to Set.$$

## Unbiased Boolean algebras

Observe that the [[Cauchy complete category|Cauchy completion]] of $Fin_{2^n}$ is $Fin_+$, the category of nonempty finite sets. (Indeed, every nonempty finite set is the retract of some set with $2^n$ elements.)

+-- {: .un_prop}
######Proposition
Let $C$ be a category with finite products, and let $i: C \hookrightarrow \widebar{C}$ be its Cauchy completion. Then $\widebar{C}$ has finite products, and the category of product-preserving functors $\widebar{C} \to Set$ is equivalent to the category of product-preserving functors $C \to Set$, via restriction along $i$.
=--

By this proposition, the category of Boolean algebras is equivalent to the category of product-preserving functors

$$Fin_+ \to Set$$

We call a product-preserving functor $Fin_+ \to Set$ an **unbiased Boolean algebra**. The idea here is that the usual concrete way of viewing Boolean algebras is inherently biased towards sets of cardinality $2^n$. Passing to the Cauchy completion removes that bias.

## $k$-valued Post algebras
{#PostAlg}

Alternatively, we could apply the previous proposition in reverse and view Boolean algebras as a concrete category in an entirely different way. For example, the Lawvere theory given by the category of finite sets of cardinality $3^n$ has the same Cauchy completion $Fin_+$. Therefore, the category of product-preserving functors

$$X: Fin_{3^n} \to Set$$

is also equivalent to the category of Boolean algebras. Only here, the appropriate underlying set functor sends $X$ to $X(3)$, the value at the generator $3$.

Similarly, for each fixed cardinality $k \gt 1$, there is a Lawvere theory $Fin_{k^n}$, and they all lead to Boolean algebras as the category of algebras for the theory. The difference is in the associated monadic functor, $U_k \colon Prod(Fin_{k^n}, Set) \to Set$. This concrete category is perhaps better known as the category of **$k$-valued Post algebras** (and is better known still when the letter $k$ is replaced by $n$).

A curious phenomenon that holds for each $k \geq 3$ (but **not** for $k = 2$) is as follows. Let $Un_k$ be the Lawvere subtheory of $Fin_{k^n}$ generated by just the unary operations, so that the algebras of $Un_k$ are identified with sets equipped with actions of the monoid $M_k = \hom(k, k)$ (endofunctions of the $k$-element set under composition), aka $M_k$-sets. By restriction of operations, there is an evident forgetful functor

$$BoolAlg \simeq PostAlg_k \to M_k\text{-}Set$$

+-- {: .un_prop}
######Proposition
For each $k \geq 3$, the forgetful functor from $BoolAlg \to$ $M_k$-$Set$ realizes $BoolAlg$ as a _full_ subcategory of $M_k$-Set.
=--

## See also

* [[complete Boolean algebra]]
* [[Boolean algebra object]]
* [[BoolAlg]] - the category of Boolean algebras

## References

* Allan Mann, A complete proof of the Robbins conjecture.  ([pdf](http://math.colgate.edu/~amann/MA/robbins_complete.pdf))

[[!redirects Boolean algebra]]
[[!redirects Boolean algebras]]
[[!redirects boolean algebra]]
[[!redirects boolean algebras]]

[[!redirects Boolean lattice]]
[[!redirects Boolean lattices]]
[[!redirects boolean lattice]]
[[!redirects boolean lattices]]


Boolean algebra object

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#### Relations
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#### Category theory
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#### Limits and colimits
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# Contents
* table of contents
{: toc}

## Idea

The notion of a *Boolean algebra object* is the generalization of that of *[[Boolean algebra]]* as one passes from the ambient [[category of sets]] [[internalization|into]] more general ambient  [[categories]] with suitable properties.

## Definition

In a [[finitely complete category]] $C$, a **Boolean algebra object** is a [[Boolean prealgebra object]] that is also a [[partially ordered object]].

## See also

* [[partially ordered object]]

* [[Boolean prealgebra object]]

* [[Heyting algebra object]]

* [[Boolean algebra]]

[[!redirects Boolean algebra objects]]


Boolean category

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#Contents#
* table of contents
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## Definition

A **Boolean category** is a [[coherent category]] (such as a [[topos]] or [[pretopos]]) in which every [[subobject]] has a complement, i.e., for any [[monomorphism]] $A\hookrightarrow X$ there is a monomorphism $B\hookrightarrow X$ such that $A\cap B$ is [[initial object|initial]] and $A\cup B = X$.  Therefore, the [[subobject lattice]] $Sub(X)$ of any [[object]] $X$ is a [[Boolean algebra]].

## Properties

Any Boolean category is, in particular, a [[Heyting category]] and therefore supports a full [[first-order logic|first-order]] [[internal logic]].  However, unlike that of an arbitrary Heyting category, the internal logic of a Boolean category satisfies the principle of [[excluded middle]]; it is [[first-order logic|first-order]] [[classical logic]].

In addition, every Boolean category $\mathcal{C}$ is a [[first-order hyperdoctrine|first order]] [[Boolean hyperdoctrine]] given by the [[subobject poset]] [[functor]] $\mathrm{sub}:\mathcal{C}^{op} \to BoolAlg$.

## Related entries

* [[pretopos]]
* [[De Morgan Heyting category]]
* [[positive category]]

[[!include categories and logic - table]]

[[!redirects Boolean category]]
[[!redirects Boolean categories]]
[[!redirects boolean category]]
[[!redirects boolean categories]]


Boolean dagger 2-poset

[[!redirects Boolean dagger 2-posets]]

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## Contents ##
* table of contents
{:toc}

## Idea ##

A Boolean dagger 2-poset is a dagger 2-poset whose [[category of maps]] is a [[Boolean category]].

## Definition ##

A **Boolean dagger 2-poset** $C$ is a [[Heyting dagger 2-poset]] where for each objects $A \in Ob(C)$, $B \in Ob(C)$ and [[monic map in a dagger 2-poset|monic map]] $i_{B,A} \in Hom(B,A)$, there is a [[unitary isomorphism]] $j_{B,A} \in A \cong^\dagger B \cup (B \Rightarrow 0)$.

## Properties ##

* The [[isomorphism classes]] of monic maps into every object $A$ is a [[Boolean algebra]]. Since every monic map is a map, the [[category of maps]] is a [[Boolean category]].

## Examples ##

The dagger 2-poset of [[sets]] with [[decidable equality]] and [[relations]] is an Boolean dagger 2-poset.

## See also ##

* [[dagger 2-poset]]
* [[coherent dagger 2-poset]]
* [[geometric dagger 2-poset]]
* [[Heyting dagger 2-poset]]
* [[Boolean dagger 2-poset]]
* [[Boolean topical dagger 2-poset]]