Contents

topos theory

Contents

Idea

Generally, the propositions in a formal logic may be thought of as having an algebraic structure with respect to logical operators like “and” and “or”. Under this identification, different systems of logic correspond to different sets of operators and axioms.

For the intuitionistic propositional calculus, the operators (including nullary operators, i.e. constants) are “and”, “or”, “true”, “false”, and “implies”. The concept of a Heyting algebra captures these operators and their axioms, so that a Heyting algebra is precisely a model of the intuitionistic propositional calculus.

A Heyting algebra where excluded middle holds is a Boolean algebra, a model of classical propositional calculus.

To model quantifiers and variables, i.e. to extend from propositional calculus to first-order aka predicate logic, one forms a hyperdoctrine on Heyting algebras, called a first-order hyperdoctrine.

Definition

Definition

A Heyting algebra is a lattice $L$ which as a poset admits an operation of implication $\Rightarrow: L^{op} \times L \to L$ satisfying the condition (really a universal property)

(1)$(x \wedge a) \leq b \qquad \text{if and only if} \qquad x \leq (a \Rightarrow b)$

In other words, $a \Rightarrow {-}$ must be right adjoint to ${-} \wedge a$.

This is equivalent to the following definition.

Definition

A Heyting algebra is a bicartesian closed poset, that is a poset which (when thought of as a thin category) is

The implication $a\Rightarrow b$ is the exponential object $b^a$.

Remark

Insofar as all these properties of a poset are described by universal properties, being a Heyting algebra is a property-like structure on a poset; a poset can be a Heyting algebra in at most one way.

The definition of Heyting algebra may be recast into purely equational form: add to the equational theory of lattices the inequalities $(x \Rightarrow y) \wedge x \leq y$ and $y \leq x \Rightarrow (y \wedge x)$, writing these inequalities in equational form via the equivalence $a \leq b$ iff $a = a \wedge b$. Hence we can speak of an internal Heyting algebra in any category with products.

Definition

A Heyting algebra homomorphism is a homomorphism of the underlying lattices that preserve $\Rightarrow$. Heyting algebras and their homomorphisms form a concrete category HeytAlg.

If one relaxes the requirement that $\leq$ be antisymmetric, so that instead of a poset $L$ is only a preorder, the result is a Heyting prealgebra.

Properties

Proposition

Any Heyting algebra is a distributive lattice. That is, finite meets distribute over finite joins and vice versa.

Proof

As discussed at distributive lattice, it suffices to prove the nontrivial direction of either of the two (dual) binary distributivity laws, e.g. $x \wedge (y \vee z) \leq (x \wedge y) \vee (x \wedge z)$. This is a straightforward exercise applying the universal property (1).

More abstractly: the distributivity law $x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)$ says precisely that $x \wedge {-}$ preserves binary joins, which it does because it has a right adjoint (namely $x \Rightarrow {-}$) and therefore preserves all colimits.

The relation of modus ponens is immediate from (1): for any $a, b$ in a Heyting algebra,

(2)$a \wedge (a \Rightarrow b) \leq b$

We list some further basic facts which are frequently useful, including in the proofs below:

Proposition

The following hold for any $a, b, c$ in any Heyting algebra:

1. Composition law: $(a \Rightarrow b) \wedge (b \Rightarrow c) \leq a \Rightarrow c$.

2. Currying: $a \wedge b \Rightarrow c = a \Rightarrow (b \Rightarrow c)$.

3. $a \Rightarrow {-}$ is monotone: $a \Rightarrow b \leq a \Rightarrow c$ if $b \leq c$.

4. $a \Rightarrow {-}$ is increasing: $b \leq a \Rightarrow b$.

5. $a \Rightarrow {-}$ is idempotent: $a \Rightarrow (a \Rightarrow b) = a \Rightarrow b$.

6. ${-} \Rightarrow a$ is antimonotone: $c \Rightarrow a \leq b \Rightarrow a$ if $b \leq c$.

7. ${-} \Rightarrow a$ is self-adjoint: ${-} \Rightarrow a: L \rightarrow L^{\mathrm{op}}$ is left adjoint to ${-} \Rightarrow a: L^{\mathrm{op}} \rightarrow L$.

Explicitly, $b \leq (c \Rightarrow a)$ just if $(b \Rightarrow a) \leq^{\mathrm{op}} c$, i.e. just if $c \leq (b \Rightarrow a)$.

8. $({-} \Rightarrow a) \Rightarrow a$ is increasing: $b \leq (b \Rightarrow a) \Rightarrow a$.

9. $({-} \Rightarrow a) \Rightarrow a$ is idempotent: $(b \Rightarrow a) \Rightarrow a = (((b \Rightarrow a) \Rightarrow a) \Rightarrow a) \Rightarrow a$.

Proof

Each of these is a short exercise using the universal property (1) and modus ponens (2).

Most of these facts can be packaged up more abstractly like so:

Proposition

In any Heyting algebra $H$, for any element $a \in H$:

• $a \Rightarrow {-}$ is a monad.

• $({-} \Rightarrow a) \Rightarrow a$ is a monad.

Proof

A monad in a poset (or a preorder, aka (0,1)-category) like $H$ is just a monotone, increasing, idempotent function.

From Proposition it is immediate that this describes both $a \Rightarrow {-}$ and $({-} \Rightarrow a) \Rightarrow a$.

Negation

In any Heyting algebra $H$, we may define a negation operator:

Definition

The negation operator $\neg\colon H^{op} \to H$ is $\neg x = (x \Rightarrow 0)$, where $0$ is the bottom element of the lattice.

Corollary

$\neg\neg\colon H \to H$ is a monad.

Proof

Immediate from .

We collect some further frequently-useful facts:

Proposition

In any Heyting algebra $H$, we have the following for all $x, y \in H$:

1. Negation is antimonotone: $\neg y \leq \neg x$ if $x \leq y$.

2. Currying: $\neg (x \wedge y) = (x \Rightarrow \neg y)$.

3. Triple negation is just negation: $\neg \neg \neg x = \neg x$.

4. Proof by contradiction: $x \leq \neg y$ just if $x \wedge y \leq 0$.

5. De Morgan's law for negation over disjunction:

$\neg (x \vee y) = \neg x \wedge \neg y$.

6. One-way De Morgan’s law for negation over conjunction: $\neg x \vee \neg y \leq \neg (x \wedge y)$.

7. $x \wedge \neg x = 0$

8. $\neg \neg (x \vee \neg x) = 1$ (equivalently, $\neg (x \vee \neg x) = 0$)

9. $\neg x \vee y \leq x \Rightarrow y$

10. (3)$\neg (x \Rightarrow y) = \neg\neg x \wedge \neg y$

Proof

Each of these is a short exercise using the universal property (1), Proposition , and the preceding properties in the list.

Several of the facts in are weakenings of familiar propositional identities from classical logic:

• A Heyting algebra where the remaining De Morgan law $\neg (x \wedge y) = \neg x \vee \neg y$ holds is precisely a De Morgan Heyting algebra.

• A Heyting algebra satisfying any of the following (equivalent) conditions is precisely a Boolean algebra:

• $\forall x.\; \neg\neg x = x$

• $\forall x.\; x \vee \neg x = 1$

• $\forall x, y.\; x \Rightarrow y = \neg x \vee y$

These also imply the De Morgan law.

Double negation

The double-negation map $\neg\neg \colon H \to H$ on a Heyting algebra $H$ is often relevant, particularly in the relationship with Boolean algebras. We saw in that $\neg\neg$ is a monad. Further:

Lemma

Double negation $\neg \neg\colon L \to L$ preserves finite meets.

Proof

Nullary meets are trivial: $\neg \neg 1 = \neg 0 = 1$. For binary meets, the direction $\neg \neg (x \wedge y) \leq (\neg \neg x) \wedge (\neg \neg y)$ holds simply because $\neg \neg$ is monotone.

In the other direction, we show $(\neg \neg x) \wedge (\neg \neg y) \leq \neg \neg (x \wedge y)$ by currying (per ) $\neg (x \wedge y)$ to calculate:

\begin{aligned} (\neg \neg x) \wedge (\neg \neg y) \wedge \neg (x \wedge y) & = (\neg \neg x) \wedge (\neg y \Rightarrow 0) \wedge (x \Rightarrow \neg y) \\ & \leq (\neg \neg x) \wedge (x \Rightarrow 0) \\ & = (\neg x \Rightarrow 0) \wedge \neg x \\ & \leq 0 \end{aligned}

where the two inequalities follow from composition (Proposition ) and modus ponens (2).

The following Lemma is important for the double negation translation.

Lemma

Double negation $\neg \neg\colon L \to L$ preserves implication: $\neg\neg( a \Rightarrow b ) = (\neg\neg a \Rightarrow \neg\neg b)$.

Proof

Applying (3) and currying (by ), we have

$\neg\neg (a \Rightarrow b) = \neg ((\neg\neg a) \wedge (\neg b)) = (\neg\neg a \Rightarrow (\neg b \Rightarrow 0)) = (\neg \neg a \Rightarrow \neg \neg b) \; .$

Relation to other concepts

To toposes

An elementary topos is a vertical categorification of a Heyting algebra: the notion of Heyting algebra is essentially equivalent to that of (0,1)-topos. Note that a Grothendieck $(0,1)$-topos is a frame or locale.

In a Heyting category, every subobject poset $Sub(A)$ is a Heyting algebra. In particular, this holds for every topos. Furthermore, in a topos, the power object $\mathcal{P}(A)$ is an internal Heyting algebra that corresponds to the external Heyting algebra $Sub(A)$. In a boolean topos, the internal Heyting algebras are all internal boolean algebras. In general, however, the internal logic of a topos (or other Heyting category) is intuitionistic.

The proof of Lemma can be made purely equational, and is therefore internally valid in any category with products. Applied to the internal Heyting algebra $L = \Omega$ of a topos, that is the subobject classifier, this lemma says exactly that the double negation operator $\neg \neg\colon \Omega \to \Omega$ defines a Lawvere–Tierney topology. Similarly, we get the double negation sublocale of any locale.

To topology

One of the chief sources of Heyting algebras is given by topologies. As a poset, the topology of a topological space $X$ is a complete lattice (it has arbitrary joins and meets, although the infinitary meets are not in general given by intersection), and the implication operator is given by

$(U \Rightarrow V) = int(U^c \vee V)$

where $U, V$ are open sets, $U^c$ is the set-theoretic complement of $U$, and $int(S)$ denotes the interior of a subset $S \subseteq X$.

Somewhat more generally, a frame (a sup-lattice in which finite meets distribute over arbitrary sups) also carries a Heyting algebra structure. In a frame, we may define

$(u \Rightarrow v) = \bigvee_{x \wedge u \leq v} x$

and the distributivity property guarantees that the universal property (1) holds. (The detailed proof is a “baby” application of an adjoint functor theorem.)

Thus frames are extensionally the same thing as complete Heyting algebras. However, intensionally they are quite different; that is, a morphism of frames is not usually a morphism of complete Heyting algebras: they do not preserve the implication operator.

A locale is the same thing as a frame, but again the morphisms are different; they are reversed.

Topologies that are Boolean algebras are the exception rather than the rule; basic examples include topologies of Stone spaces; see Stone duality. Another example is the topology of a discrete space $X$.

There are several ways of passing back and forth between Boolean algebras and Heyting algebras, having to do with the double negation operator. By , double negation $\neg \neg\colon L \to L$ is a monad. Further, by , it preserves finite meets.

Now let $L_{\neg\neg}$ denote the poset of regular elements of $L$, that is, those elements $x$ such that $\neg\neg x = x$. (When $L$ is the topology of a space, an open set $U$ is regular if and only if it is the interior of its closure, that is if it is a regular element of the Heyting algebra of open sets described above.) With the help of Lemma , we may prove

Theorem

The poset $L_{\neg\neg}$ is a Boolean algebra. Moreover, the assignment $L \mapsto L_{\neg\neg}$ is the object part of a functor

$F\colon Heyt \to Bool$

called Booleanization, which is left adjoint to the full and faithful inclusion

$i\colon Bool \hookrightarrow Heyt.$

The unit of the adjunction, applied to a Heyting algebra $L$, is the map $L \to L_{\neg\neg}$ which maps each element $x$ to its regularization $\neg\neg x$.

Proof

To avoid confusion from two different maps called $\neg\neg$ that differ only in their codomain, write $M: L \rightarrow L$ for the monad and $U: L \rightarrow L_{\neg\neg}$ for the left adjoint. Write $\iota: L_{\neg\neg} \rightarrow L$ for the right adjoint, so that $M = \iota \circ U$.

We first show that $L_{\neg\neg}$ is a Heyting algebra and $U$ is a Heyting-algebra map. Because $U$ is surjective (and monotone), it suffices to show that it preserves the Heyting-algebra operators: finite meets, finite joins, and implication.

Because $\iota$ is full, it reflects meets. Therefore because by Lemma $M = \iota \circ U$ preserves finite meets, so too does $U$, and as a left adjoint it preserves joins.

Finally, for $a, b \in L$, we show that $\neg\neg (a \Rightarrow b)$, which by Lemma is the $L$-implication $\neg\neg a \Rightarrow \neg\neg b$, satisfies the universal property (1) also in $L_{\neg\neg}$. For any $x \in L_{\neg\neg}$, $\iota(x) \leq \neg\neg a \Rightarrow_L \neg\neg b$ just if $\iota(x) \wedge_L \neg\neg a \leq \neg\neg b$ by the universal property in $L$; but because $\iota$ reflects meets this is equivalent to $x \wedge_{L_{\neg\neg}} \neg\neg a \leq \neg\neg b$, completing the universal property.

Now because $U$ preserves implication and (the empty join) $0$, it preserves negation. Therefore $\neg\neg$ in $L_{\neg\neg}$ is the identity, so the latter is a Boolean algebra.

Therefore $U = \neg\neg \colon L \to L_{\neg\neg}$ is a Heyting algebra quotient which is the coequalizer of $1, \neg\neg \colon L \stackrel{\to}{\to} L$. It follows that a Heyting algebra map $L \to B$ to any Boolean algebra $B$, i.e. any Heyting algebra where $1$ and $\neg\neg$ coincide, factors uniquely through this coequalizer, and the induced map $L_{\neg \neg} \to B$ is a Boolean algebra map. In other words, $\neg\neg \colon L \to L_{\neg\neg}$ is the universal Heyting algebra map to a Boolean algebra, which establishes the adjunction.

Thus $\neg\neg\colon L \to L_{\neg\neg}$ preserves finite joins and finite meets and implication. In the other direction, we have an inclusion $i\colon L_{\neg\neg} \to L$, and this preserves meets but not joins. It also preserves negations; more generally and perhaps surprisingly, it preserves implications as well.

Regular elements are not to be confused with complemented element?s, i.e., elements $x$ in a Heyting algebra such that $x \vee \neg x = 1$, although it is true that every complemented element is regular. An example of a regular element which is not complemented is given by the unit interval $(0, 1)$ as an element of the topology of $\mathbb{R}$; a complemented element in a Heyting algebra given by a topology is the same as a clopen subset.

Complemented elements furnish another universal relation between Boolean algebras and Heyting algebras: the set of complemented elements in a Heyting algebra $H$ is a Boolean algebra $Comp(H)$, and the inclusion $Comp(H) \to H$ is a Heyting algebra map which is universal among Heyting algebra maps $B \to H$ out of Boolean algebras $B$. In other words, we have the following result.

Theorem

The assignment $H \mapsto Comp(H)$ is the object part of a right adjoint to the forgetful functor $Bool \to Heyt$.

Proof

In a Heyting algebra $H$, the elements $0$ and $1$ are clearly complemented. If $x$ and $y$ are complemented, then so are $x \wedge y$, $x \vee y$, and $x \Rightarrow y$; we leave meet and join as an exercise applying , and demonstrate implication (using (3)):

\begin{aligned} (x \Rightarrow y) \vee \neg(x \Rightarrow y) &= (x \Rightarrow y) \vee (\neg\neg x \wedge \neg y) \\ &= ((x \Rightarrow y) \vee \neg\neg x) \wedge ((x \Rightarrow y) \vee \neg y) \\ &\geq (\neg x \vee x) \wedge (y \vee \neg y) \\ &= 1 \; . \end{aligned}

Thus the complemented elements form a Heyting subalgebra $Comp(H) \hookrightarrow H$. Clearly $Comp(H)$ is a Boolean algebra, and clearly if $B$ is Boolean, then any Heyting algebra map $B \to H$ factors uniquely through $Comp(H) \hookrightarrow H$. This proves the theorem.

Examples

Proposition

For $\mathcal{T}$ a topos and $X \in \mathcal{T}$ any object, the poset $Sub(X)$ of subobjects of $X$ is a Heyting algebra.

In other words, every topos is a Heyting category.

In particular for $X = \Omega$ the subobject classifier, $Sub(\Omega)$ is a Heyting algebra.

In $\mathcal{T} =$ Set for every set $S$ we have that $Sub(S)$ is the Boolean algebra of subset of $S$.

More details and examples are spelled out at internal logic#examples.

Proposition

A frame $L$ is a Heyting algebra.

Proof

By the adjoint functor theorem, a right adjoint $x \Rightarrow -$ to the map $x \wedge -: L \to L$ exists since this map preserves arbitrary joins.

References

The original reference:

• Arend Heyting, Die formalen Regeln der intuitionistischen Logik. I, II, III. Sitzungsberichte der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse (1930) 42-56, 57-71, 158-169

abridged reprint in:

Karel Berka, Lothar Kreiser (eds.), Logik-Texte, De Gruyter (1986) 188-192 [doi:10.1515/9783112645826]

A quick introduction can be found in §1.2 of

Last revised on February 18, 2023 at 14:48:18. See the history of this page for a list of all contributions to it.