Trimble on ETCS III

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_This is Part III of an exposition by [[Todd Trimble]] on [[ETCS]]_.

See also

* Part I.  [[Trimble on ETCS I|ZFC and ETCS]]

* Part II.  [[Trimble on ETCS II|ETCS: Internalizing the logic]]

***

## ETCS : Building joins and coproducts ##

After a long hiatus, I'd like to renew the discussion of axiomatic categorical set theory, more specifically the Elementary Theory of the Category of Sets ([[ETCS]]).  [Last
time](http://topologicalmusings.wordpress.com/2008/09/10/etcs-internalizing-the-logic) I blogged about this, I made some initial forays into "internalizing logic" in [[ETCS]], and described in broad brushstrokes how to use that internal logic to derive a certain amount of the structure one associates with a category of sets.  Today I'd like to begin applying some of the results obtained there to the problem of constructing _colimits_ in a category satisfying the ETCS axioms (an _[[ETCS]] category_, for short).

(If you're just joining us now, and you already know some of the jargon, an _[[ETCS]] category_ is a well-pointed topos that satisfies the axiom of choice and with a natural numbers object.  We are trying to build up some of the elementary theory of such categories from scratch, with a view toward foundations of mathematics.)

But let's see --- where were we?  Since it's been a while, I was tempted to review the philosophy behind this undertaking (why one would go to all the trouble of setting up a categories-based alternative to ZFC, when time-tested ZFC is able to express virtually all of present-day mathematics on the basis of a reasonably short list of axioms?).  But in the interest of time and space, I'll confine myself to a few remarks.

As we said, a chief difference between ZFC and ETCS resides in how ETCS treats the issue of membership.  In ZFC, membership is a global binary relation:  we can take any two "sets" $A, B$ and ask whether $A \in B$.  Whereas in ETCS, membership is a relation between entities of different sorts:  we have "sets" on one side and "elements" on another, and the two are not mixed ( _e.g._, elements are not themselves considered sets).

Further, and far more radical:  in ETCS the membership relation $x \in A$ is a _function_, that is, an element $x$ "belongs" to only one set $A$ at a time.  We can think of this as "declaring" how we are thinking of an element, that is, declaring which set (or which type) an element is being considered as belonging to.  (In the jargon, ETCS is a _typed theory_.)  This reflects a general and useful philosophic principle:  that elements in isolation are considered inessential, that what counts are the aggregates or contexts in which elements are organized and interrelated.  For instance, the numeral "2" in isolation has no meaning;  what counts is the context in which we think of it ( _qua_ rational number or _qua_ complex number, _etc._).  Similarly the set of real numbers has no real sense in isolation;  what counts is which category we view it in.

I believe it is reasonable to grant this principle a foundational status, but: rigorous adherence to this principle completely changes the face of what set theory looks like.  If elements "belong" only to one set at a time, how then do we even _define_ such basic concepts as subsets and intersections?  These are some of the issues we discussed last time.

There are other significant differences between ZFC and ETCS:  stylistically, or in terms of presentation, ZFC is more "top-down" and ETCS is more "bottom-up".  For example, in ZFC, one can pretty much define a subset $\{ x \in X : P \}$ by writing down a first-order formula $P$ in the language;  the [[comprehension]] (or [[axiom of separation|separation]]) axiom scheme is a mighty sledgehammer that takes care of the rest.  In the axioms of ETCS, there is no such sledgehammer:  the closest thing one has to a comprehension scheme in the ETCS axioms is the power set axiom (a single axiom, not an axiom scheme).  However, in the formal development of ETCS, one _derives_ a comprehension scheme as one manually constructs the internal logic, in stages, using the simple tools of adjunctions and universal properties.  We started doing some of that in our last post.  So:  with ZFC it's more as if you can just hop in the car and go;  with ETCS you build the car engine from smaller parts with your bare hands, but in the process you become an expert mechanic, and are not so rigidly attached to a particular make and model ( _e.g._, much of the theory is built just on the axioms of a topos, which allows a lot more semantic leeway than one has with ZF).

But, in all fairness, that is perhaps the biggest obstacle to learning ETCS:  at the outset, the tools available [mainly, the idea of a universal property] are quite simple but parsimonious, and one has to learn how to build some set-theoretic and logical concepts normally taken as "obvious" from the ground up.  (Talk about "foundations"!)  On the plus side, by building big logical machines from scratch, one gains a great deal of insight into the inner workings of logic, with a corresponding gain in precision and control and modularity when one would like to use these developments to design, say, automated deduction systems (where there tend to be strong advantages to using type-theoretic frameworks).

Enough philosophy for now; readers may refer to my [earlier](http://topologicalmusings.wordpress.com/2008/09/01/zfc-and-etcs-elementary-theory-of-the-category-of-sets) [posts](http://topologicalmusings.wordpress.com/2008/09/10/etcs-internalizing-the-logic/) for more.  Let's get to work, shall we?  Our last post was about the structure of (and relationships between) posets of subobjects $Sub(X)$ _relative_ to objects $X$, and now we want to exploit the results there to build some _absolute_ constructions, in particular finite coproducts and coequalizers.  In this post we will focus on coproducts.

: __Note to the experts.__  Most textbook treatments of the formal development of topos theory (as for example Mac Lane–Moerdijk) are efficient but highly technical, involving for instance the slice theorem for toposes and, in the construction of colimits, recourse to Beck's theorem in monad theory applied to the double power-set monad [following the elegant construction of Paré].  The very abstract nature of this style of argumentation (which in the application of Beck's theorem expresses ideas of fourth-order set theory and higher) is no doubt partly responsible for the somewhat fearsome reputation of topos theory.

: In these notes I take a much less efficient but much more elementary approach, based on an arrangement of ideas which I hope can be seen as "natural" from the point of view of naive set theory.  I learned of this approach from Myles Tierney, who was my PhD supervisor, and who with Bill Lawvere co-founded elementary topos theory, but I am not aware of any place where the details of this approach have been written up before now.  I should also mention that the approach taken here is not as "purist" as many topos theorists might want;  for example, here and there I take advantage of the strong extensionality axiom of ETCS to simplify some arguments.

### The Empty Set and Two-Valued Logic ###

We begin with the easy observation that a terminal category, _i.e._, a category $\mathbf{1}$ with just one object and one morphism (the identity), satisfies all the ETCS axioms.  Ditto for any category $C$ equivalent to $\mathbf{1}$ (where every object is terminal).  Such boring ETCS categories are called _degenerate_;  obviously our interest is in the structure of nondegenerate ETCS categories.

Let $\mathbf{E}$ be an ETCS category (see [here](http://topologicalmusings.wordpress.com/2008/09/01/zfc-and-etcs-elementary-theory-of-the-category-of-sets) for the ETCS axioms).  Objects of $\mathbf{E}$ are generally called "sets", and morphisms are generally called "functions" or "maps".

__Proposition 0.__  If an ETCS category $\mathbf{E}$ is a preorder, then $\mathbf{E}$ is degenerate.

__Proof.__  Recall that a preorder is a category in which there is at most one morphism $A \to B$ for any two objects $A, B$.  Every morphism in a preorder is vacuously monic.  If there is a nonterminal set $A$, then the monic $A \to 1$ to any terminal set defines a subset $A \subseteq 1$ distinct from the subset defined by $1 \to 1$, thus giving (in an ETCS category) distinct classifying maps $\chi_A, t : 1 \to P(1)$, contradicting the preorder assumption.  Therefore all objects $A$ are terminal. $\Box$

Assume from now on that $\mathbf{E}$ is a nondegenerate ETCS category.

__Proposition 1.__  There are at least two _truth values_, _i.e._, two elements $1 \to P(1)$, in $\mathbf{E}$.

__Proof.__  By proposition 0, there exist sets $X, Y$ and two distinct functions $f, g : X \to Y$.  By the axiom of strong extensionality, there exists $x : 1 \to X$ such that $f x \neq g x$.  The equalizer $E \to 1$ of the pair $f x, g x : 1 \to Y$ is then a proper subset of $1$, and therefore there are at least two distinct elements $\chi_E, \chi_1 : 1 \to P(1)$.  $\Box$

__Proposition 2.__  There are at most two truth values $1 \to P(1)$;  equivalently, there are at most two subsets of $1$.

__Proof.__  If $U, V \subseteq 1$ are distinct subsets of $1$, then either $U \neq U \cap V$ or $V \neq U \cap V$, say the former.  Then $1_U : U \subseteq U$ and $U \cap V \subseteq U$ are distinct subsets, with distinct classifying maps $\chi_{1_U}, \chi_{U \cap V} : U \to P(1)$.  By strong extensionality, there exists $x : 1 \to U$ distinguishing these classifying maps.  Because $1$ is terminal, we then infer $1 \subseteq U$ and $U \subseteq 1$, so $U = 1$ as subsets of $1$, and in that case only $V$ can be a proper subset of $1$.  $\Box$

By Propositions 1 and 2, there is a unique proper subset of the terminal object $1$.  Let $0 \to 1$ denote this subset.  Its domain may be called an "empty set";  by the preceding proposition, it has no proper subsets.  The classifying map $1 \to P1$ of $0 \subseteq 1$ is the truth value we call "false".

__Proposition 3.__  0 is an initial object, _i.e._, for any $X$ there exists a unique function $0 \to X$.

__Proof.__  Uniqueness:  if $f, g : 0 \to X$ are maps, then their equalizer $x : E \to 0$, which is monic, must be an isomorphism since 0 has no proper subsets.  Therefore $f = g$.  Existence:  there are monos

$$\displaystyle 0 \to 1 \stackrel{t_X}{\to} P X$$

$$\displaystyle X \stackrel{\sigma}{\to} P X$$

where $t_X$ is "global truth" (classifying the subset $X \subseteq X$) on $X$ and $\sigma$ is the "singleton mapping $x \mapsto \{ x \}$" on $X$, defined as the classifying map of the diagonal map $\delta : X \subseteq X \times X$ (last time we saw $\sigma$ is monic).  Take their pullback.  The component of the pullback parallel to $\sigma$ is a mono $P \to 0$ which again is an isomorphism, whence we get a map $0 \cong P \to X$ using the other component of the pullback.  $\Box$

__Remark.__  For the "purists", an alternative construction of the initial set 0 that avoids use of the strong extensionality axiom is to define the subset $0 \subseteq 1$ to be "the intersection all subsets of $1$".  Formally, one takes the extension $\left[\phi\right] \subseteq 1$ of the map

$$\displaystyle \phi = (1 \stackrel{t_{P1}}{\to} PP1 \stackrel{\bigcap}{\to} P1);$$

where the first arrow represents the class of all subsets of $P1$, and the second is the internal intersection operator defined at the end of our last post.  Using formal properties of intersection developed later, this intersection $0 \subseteq 1$ has no proper subsets, and then the proof of proposition 3 carries over verbatim.  $\Box$

__Corollary 1.__  For any $X$, the set $0 \times X$ is initial.

__Proof.__  By [cartesian closure](http://topologicalmusings.wordpress.com/2008/09/10/etcs-internalizing-the-logic), maps $0 \times X \to Y$ are in bijection with maps of the form $0 \to Y^X$, and there is exactly one of these since 0 is initial.  $\Box$

__Corollary 2.__  If there exists $f : X \to 0$, then $X$ is initial.

__Proof.__  The composite of $\langle f, 1_X \rangle : X \to 0 \times X$ followed by $\pi_2 : 0 \times X \to X$ is $1_X$, and $\pi_2$ followed by $\langle f, 1_X \rangle : X \to 0 \times X$ is also an identity since $0 \times X$ is initial by Corollary 1.  Hence $X$ is isomorphic to an initial object $0 \times X$.  $\Box$

By Corollary 2, for any object $Y$ the arrow $0 \to Y$ is vacuously monic, hence defines a subset.

__Proposition 4.__  If $X \not\cong 0$, then there exists an element $x : 1 \to X$.

__Proof.__  Under the assumption, $X$ has at least two distinct subsets:  $0 \subseteq X$ and $1_X : X \subseteq X$.  By strong extensionality, their classifying maps $\chi_0, \chi_1 : X \to P1$ are distinguished by some element $x : 1 \to X$.

### External Unions and Internal Joins ###

One of the major goals in this post is to construct finite coproducts in an ETCS category.  As in ordinary set theory, we will construct these as disjoint unions.  This means we need to discuss unions first;  as should be expected by now, in ETCS unions are considered locally, _i.e._, we take unions of _subsets of a given set_.  So, let $A, B \subseteq X$ be subsets.

To define the union $A \cup B \subseteq X$, the idea is to take the intersection of all subsets containing $A$ and $B$.  That is, we apply the internal intersection operator (constructed [last
time](http://topologicalmusings.wordpress.com/2008/09/10/etcs-internalizing-the-logic/) ),

$$\displaystyle \bigcap : P P X \to P X,$$

to the element $1 \to P P X$ that represents the set of all subsets of $X$ containing $A$ and $B$;  the resulting element $1 \to P X$ represents $A \cup B$.  The element $1 \to P P X$ corresponds to the intersection of two subsets $\{ C \in P X : A \subseteq C \} \:\cap\: \{ C \in P X : B \subseteq C \} \subseteq P X$.

: __Remark.__  Remember that in ETCS we are using _generalized_ elements:  $C \in P X$ really means a function $C : U \to P X$ over some domain $U$, which in turn classifies a subset $\left[C\right] \subseteq U \times X$.  On the other hand, the $A$ here is a subset $A \subseteq X$.  How then do we interpret the condition "$A \subseteq C$"?  We first pull back $\chi_A : 1 \to P X$ over to the domain $U$ (that is, we form the composite $\displaystyle U \stackrel{!}{\to} 1 \stackrel{\chi_A}{\to} P X$) and consider the condition that this is bounded above by $C : U \to P X$.  (We will write $\chi_A \leq C$, thinking of the left side as constant over $U$.)  Externally, in terms of subsets, this corresponds to the condition $U \times A \subseteq \left[C\right]$.

We need to construct the subsets $\{ C \in P X : A \subseteq C \}, \{ C \in P X : B \subseteq C \}$.  In ZFC, we could construct those subsets by applying the comprehension axiom scheme, but the axioms of ETCS have no such blanket axiom scheme.  (In fact, as we said earlier, much of the work on "internalizing logic" goes to show that in ETCS, we instead _derive_ a comprehension scheme!)  However, one way of defining subsets in ETCS is by taking loci of equations;  here, we express the condition $A \subseteq C$, more pedantically $A \subseteq \left[C\right]$ or $\chi_A \leq C$, as the equation

$$(\chi_A \Rightarrow C) = t_X$$

where the right side is the predicate "true over $X$".

Thus we construct the subset $\{C \in P X: A \subseteq C\}$ of $P X$ via the pullback:

$$
\begin{matrix}
\{ C : A \le C \} & \longrightarrow & 1 \\
\downarrow & & \downarrow \mathrlap{\scriptsize{t_X}} \\
P X & \underset{\chi_A \Rightarrow -}{\longrightarrow} & P X
\end{matrix}
$$

Let me take a moment to examine what this diagram means exactly.  [Last
time](http://topologicalmusings.wordpress.com/2008/09/10/etcs-internalizing-the-logic) we constructed an internal implication operator

$$\Rightarrow : P1 \times P1 \to P1$$

and now, in the pullback diagram above, what we are implicitly doing is lifting this to an operator

$$\Rightarrow_X : P X \times PX \to P X$$

The easy and cheap way of doing this is to remember the isomorphism $P X \cong P1^X$ we used last time to uncover the cartesian closed structure, and apply this to

$$\displaystyle P1^X \times P1^X \cong (P1 \times P1)^X \stackrel{\Rightarrow^X}{\to} P1^X$$

to define $\Rightarrow_X : P X \times P X \to P X$.  This map classifies a certain subset of $X \times P X \times P X$, which I'll just write down (leaving it as an exercise which involves just chasing the relevant definitions):

$$\{ (x, T, S) \in X \times P X \times P X : x \in_X T \Rightarrow x \in_X S \}$$

__Remark.__  Similarly we can define a meet operator $\wedge_X : P X \times P X \to P X$ by exponentiating the internal meet $P1 \times P 1 \to P 1$.  It is important to know that the general Heyting algebra identities which we established last time for $P1$ lift to the corresponding identities for the operators $\wedge_X, \Rightarrow_X$ on $P X$.  Ultimately this rests on the fact that the functor $(-)^X$, being a right adjoint, preserves products, and therefore preserves any algebraic identity which can be expressed as a commutative diagram of operations between such products.

Hence, for the fixed subset $A \subseteq X$ (classified by $\chi_A : 1 \to P X$), the operator

$$\chi_A \Rightarrow - : P X \to P X$$

classifies the subset

$$\{ (x, S) : x \in_X A \Rightarrow x \in_X S \} \hookrightarrow X \times P X$$

Finally, in the pullback diagram above, we are pulling back the operator $\chi_A \Rightarrow -$ against $t_X$.  But, from [last
time](http://topologicalmusings.wordpress.com/2008/09/10/etcs-internalizing-the-logic), that was exactly the method we used to construct [[universal quantification]].  That is, given a subset

$$R \subseteq X \times Y$$

we defined $\forall_Y R \subseteq X$ to be the pullback of $t_Y : 1 \hookrightarrow PY$ along $\chi_R : X \to PY$.

Putting all this together, the pullback diagram above expresses the definition

$$\{ C \in P X : A \subseteq C \} := \{ C \in P X : \forall_{x \in X} \: x \in_X A \Rightarrow x \in_X C \}$$

that one would expect "naively".

Now that all the relevant constructions are in place, we show that $A \cup B$ is the join of $A$ and $B$ in the poset $Sub(X)$.  There is nothing intrinsically difficult about this, but as we are still in the midst of _constructing_ the internal logic, we will have to hunker down and prove some logic things normally taken for granted or zipped through without much thought.  For example, the internal intersection operator was defined with the help of internal universal quantification, and we will need to establish some formal properties of that.

Here is a useful general principle for doing internal logic calculations.  Let $\chi_R : X \to PY$ be the classifying map of a subset $R \subseteq X \times Y$, and let $f : U \to X$ be a function.  Then the composite $\chi_R f : U \to PY$ classifies the subset

$$(f \times 1_Y)^{-1}(R) \subseteq U \times Y$$

so that one has the general identity $\chi_R f = \chi_{(f \times 1)^{-1}(R)}$.  In passing back and forth between the external and internal viewpoints, the general principle is to try to render "complicated" functions $U \to PY$ into a form $\chi_R f$ which one can more easily recognize.  For lack of a better term, I'll call this the "pullback principle".

__Lemma 1.__  Given a relation $R \hookrightarrow X \times Y$ and a constant $c : 1 \to Y$, there is an inclusion

$$\forall_Y R \subseteq (1_X \times c)^{-1}(R)$$

as subsets of $X$.  (In traditional logical syntax, this says that for any element $c : 1 \to Y$,

$$\forall_{y : Y} R(x, y) \quad implies \quad R(x, c)$$

as predicates over elements $x \in X$.  This is the type of thing that ordinarily "goes without saying", but which we actually have to prove here!)

__Proof.__  As we recalled above, $\forall_Y R \subseteq X$ was defined to be $\chi_R^{-1}(t_Y)$, the pullback of global truth $t_Y : 1 \to PY$ along the classifying map $\chi_R : X \to PY$.  Hold that thought.

Let

$$Pc : PY \to P1$$

be the map which classifies the subset $\{ S \in PY : c \in_Y S\}$.  Equivalently, this is the map

$$P1^c: P1^Y \to P1^1$$

under the canonical isomorphisms $PY \cong P1^Y$, $P1^1 \cong P1$.  Intuitively, this maps $f \mapsto f(c)$, _i.e._, plugs an element $c \in Y$ into an element $f \in P1^Y$.

Using the adjunction $(- \times Y) \dashv (-)^Y$ of cartesian closure, the composite

$$\displaystyle X \stackrel{\chi_R}{\to} PY \stackrel{Pc}{\to} P1$$

transforms to the composite

$$X \times 1 \stackrel{1_X \times c}{\to} X \times Y \stackrel{\chi_{R}}{\to} P1$$

so by the pullback principle, $(Pc)\chi_R : X \times 1 \to P1$ classifies $(1_X \times c)^{-1}(R) \subseteq X \times 1 \cong X$.

Equivalently,

$$(1_X \times c)^{-1}(R) = \chi_{R}^{-1} (Pc)^{-1}(t) \qquad (1)$$

Also, as subsets of $PY$, we have the inclusion

$$(t_Y : 1 \hookrightarrow PY) \subseteq (Pc)^{-1}(t) \qquad (2)$$

[this just says that $t_Y$ belongs to the subset classified by $Pc$, or equivalently that $c : 1 \to Y$ is in the subset $Y \subseteq Y$].  Applying the pullback operation $\chi_{R}^{-1}$ to (2), and comparing to (1), Lemma 1 follows.  $\Box$

__Lemma 2.__  If $R \subseteq S$ as subsets of $X \times Y$, then $\forall_Y R \subseteq \forall_Y S$.

__Proof.__  From the last post, we have an adjunction:

$$T \times Y \subseteq S \quad if and only if \quad T \subseteq \forall_Y S$$

for any subset of $T \subseteq X$.  So it suffices to show $\forall_Y R \times Y \subseteq S$.  But

$$\forall_Y R \times Y \subseteq R \subseteq S$$

where the first inclusion follows from $\forall_Y R \subseteq \forall_Y R$.  $\Box$

Next, recall from the [last post](http://topologicalmusings.wordpress.com/2008/09/10/etcs-internalizing-the-logic) that the internal intersection of $F : 1 \to P P X$ was defined by interpreting the following formula on the right:

$$\displaystyle \bigcap F = \forall_{S \in P X} (S \in_{P X} F) \Rightarrow (x \in_X S)$$

__Lemma 3.__  If $F \leq G : 1 \to P P X$, then $\displaystyle \bigcap G \leq \bigcap F : 1 \to P X$.

__Proof.__  $F : 1 \to P P X$ classifies the subset $\{ S \in P X : S \in_{P X} F \}$, _i.e._, $F$ is identified with the predicate $S \in_{P X} F$ in the argument $S$, so by hypothesis $(S \in_{P X} F) \leq (S \in_{P X} G)$ as predicates on $S$.  Internal implication $a \Rightarrow b$ is contravariant in the argument $a$ [see the following remark], so

$$\left[(S \in G) \Rightarrow (x \in S)\right] \leq \left[(S \in F) \Rightarrow (x \in S)\right]$$

Now apply Lemma 2 to complete the proof.  $\Box$

: __Remark.__  The contravariance of $- \Rightarrow b$, that is, the fact that $x \leq y$ implies $(y \Rightarrow b) \leq (x \Rightarrow b)$, is a routine exercise using the adjunction [discussed last time] $a \wedge c \leq b$ if and only if $c \leq (a \Rightarrow b).$

: Indeed, we have

$$x \wedge (y \Rightarrow b) \leq y \wedge (y \Rightarrow b) \leq b \qquad (*)$$

where the first inequality follows from the hypothesis $x \leq y$, and the second follows from $y \Rightarrow b \leq y \Rightarrow b$.  By the adjunction, the inequality $(*)$ implies $(y \Rightarrow b) \leq (x \Rightarrow b)$.  $\Box$

__Theorem 1.__  For subsets $A, B$ of $X$, the subset $A \cup B$ is an upper bound of $A$ and $B$, _i.e._, $A, B \subseteq A \cup B$.

__Proof.__  It suffices to prove that $\displaystyle A = \bigcap \{C \in P X : A \subseteq C\}$, since then we need only apply Lemma 3 to the trivially true inclusion

$$\{ C \in P X : A \subseteq C \} \:\cap\: \{ C \in P X : B \subseteq C \} \subseteq \{ C \in P X : A \subseteq C \}$$

to infer $A \subseteq A \cup B$, and similarly $B \subseteq A \cup B$.  (Actually, we need only show $\displaystyle A \subseteq \bigcap \{ C \in P X : A \subseteq C \}$.  We'll do that first, and then show full equality.)

The condition we want,

$$A \subseteq \{ x \in X : \forall_{S \in P X} (A \subseteq S) \Rightarrow (x \in_X S) \},$$

is, by the adjunction $(- \times P X) \dashv \forall_{P X} : Sub(X \times P X) \to Sub(X)$, equivalent to

$$A \times P X \subseteq \{ (x, S) : A \subseteq S \Rightarrow (x \in_X S) \}$$

which, by a $\wedge$-$\Rightarrow$ adjunction, is equivalent to

$$(A \times P X) \cap (X \times \{ S \in P X : A \subseteq S \}) \subseteq \: \in_X \qquad (1)$$

as subsets of $X \times P X$.  So we just have to prove (1).  At this point we recall, from our earlier analysis, that

$$\{ S \in P X : A \subseteq S \} = \{ S \in P X : \forall_{x \in X} x \in_X A \Rightarrow x \in_X S \}$$

Using the adjunction $(X \times -) \dashv \forall_X$, as in the proof of Lemma 2, we have

$$X \times \{ S \in P X : \forall_{x \in X} x \in_X A \Rightarrow x \in_X S \}$$

$$\subseteq \{ (x, S) : x \in_X A \Rightarrow x \in_X S \} := (A \times P X) \Rightarrow \in_X,$$

which shows that the left side of (1) is contained in

$$(A \times P X) \cap ((A \times P X) \Rightarrow \in_X) \subseteq \: \in_X,$$

where the last inclusion uses another $\wedge$-$\Rightarrow$ adjunction.  Thus we have established (1) and therefore also the inclusion

$$\displaystyle A \subseteq \bigcap \{S \in P X : A \subseteq S\}$$

Now we prove the opposite inclusion

$$\displaystyle \bigcap \{ S \in P X : A \subseteq S \} \subseteq A,$$

that is to say

$$\{ x \in X : \forall_{S \in P X} A \subseteq S \Rightarrow x \in_X S \} \subseteq A \qquad (**)$$

Here we just use Lemma 1, applied to the particular element $\chi_A : 1 \to P X$:  we see that the left side of $(**)$ is contained in

$$\{ x \in X : A \subseteq \left[\chi_A\right] \Rightarrow x \in_X A\}$$

which collapses to $\{ x \in X : x \in_X A \} = A$, since $A = \left[\chi_A\right]$.  This completes the proof.  $\Box$

__Theorem 2.__  $A \cup B$ is the least upper bound of $A, B$, _i.e._, if $C \subseteq X$ is a subset containing both $A \subseteq X$ and $B \subseteq X$, then $A \cup B \subseteq C$.

__Proof.__  We are required to show that

$$\{ x \in X : \forall_{S \in P X} \: (A \subseteq S \wedge B \subseteq S) \Rightarrow x \in_X S \} \subseteq C.$$

Again, we just apply Lemma 1 to the particular element $\chi_C : 1 \to P X$: the left-hand side of the claimed inclusion is contained in

$$\{ x \in X : (A \subseteq C \wedge B \subseteq C) \Rightarrow x \in_X C \}$$

but since $(A \subseteq C \wedge B \subseteq C)$ is true by hypothesis (is globally true as a predicate on the implicit variable $x \in X$), this last subset collapses to

$$\{ x \in X : t_X \Rightarrow x \in_X C \} = \{ x \in X : x \in_X C \} = C$$

which completes the proof.  $\Box$

Theorems 1 and 2 show that for any set $X$, the external poset $Sub(X)$ admits joins.  One may go on to show (just on the basis of the topos axioms) that as in the case of meets, the global external operation of taking joins is natural in $X$, so that by the Yoneda principle, it is classified by an internal join operation

$$\vee : P1 \times P1 \to P1,$$

namely, the map which classifies the union of the subsets

$$\displaystyle \left[\pi_1\right] = P1 \times 1 \stackrel{1 \times t}{\hookrightarrow} P1 \times P1$$

$$\displaystyle \left[\pi_2\right] = 1 \times P1 \stackrel{t \times 1}{\hookrightarrow} P1 \times P1$$

and this operation satisfies all the expected identities.  In short, $P 1$ carries an internal Heyting algebra structure, as does $P X \cong P 1^X$ for any set $X$.

We will come back to this point later, when we show (as a consequence of strong extensionality) that $P1$ is actually an internal Boolean algebra.

### Construction of Coproducts ###

Next, we construct coproducts just as we do in ordinary set theory:  as disjoint unions.  Letting $X, Y$ be sets (objects in an ETCS category), a _disjoint union_ of $X$ and $Y$ is a pair of monos

$$i : X \hookrightarrow Z \qquad j : Y \hookrightarrow Z$$

whose intersection is empty, and whose union or join in $Sub(Z)$ is all of $Z$.  We will show that disjoint unions exist and are essentially unique, and that they satisfy the universal property for coproducts. We will use the notation $X + Y$ for a disjoint union.

__Theorem 3.__  A disjoint union of $X$ and $Y$ exists.

+-- {: .proof}
###### Proof
It's enough to embed $X, Y$ disjointly into _some_ set $C$, since the union of the two monos in $Sub(C)$ would then be the requisite $Z$.  The idea now is that if a disjoint union or coproduct $X+Y$ exists, then there's a canonical isomorphism $P(X + Y) \cong P X \times P Y$.  Since the singleton map

$$\sigma : X + Y \to P(X + Y) \cong P X \times P Y$$

is monic, one thus expects to be able to embed $X$ and $Y$ disjointly into $P X \times P Y$.  Since we can easily work out how all this goes in ordinary naive set theory, we just write out the formulas and hope it works out in ETCS.

In detail, define $i_X : X \to P X \times P Y$ to be

$$\displaystyle X \cong X \times 1 \stackrel{\sigma_X \times \chi_0}{\to} P X \times P Y$$

where $\sigma_X$ is the singleton mapping and $\chi_0$ classifies $0 \hookrightarrow Y$;  similarly, define $i_Y :Y \to P X \times P Y$ to be

$$\displaystyle Y \cong 1 \times Y \stackrel{\chi_0 \times \sigma_Y}{\to} P X \times P Y.$$

Clearly $i_X$ and $i_Y$ are monic, so to show disjointness we just have to show that their pullback is empty.  But their pullback is isomorphic to the cartesian product of the pullbacks of the diagrams

$$\displaystyle X \stackrel{\sigma_X}{\to} P X \stackrel{\chi_0}{\leftarrow} 1 \qquad 1 \stackrel{\chi_0}{\to} P Y \stackrel{\sigma_Y}{\leftarrow} Y$$

so it would be enough to show that each (or just one) of these two pullbacks is empty, let's say the first.

Suppose given a map $h: A \to X$ which makes the square

$$\array{
\arrayopts{\rowalign{center axis center}}
\A & \longrightarrow & 1 \\
\mathllap{\scriptsize{h}} \downarrow & & \downarrow \mathrlap{\scriptsize{\chi_0}} \\
X & \underset{\dlap{\sigma_X}}{\longrightarrow} & P X
}$$

commute.  Using the pullback principle, the map $A \to 1 \stackrel{\chi_0}{\to} P X$ classifies

$$0 \times A \hookrightarrow X \times A$$

which is just the empty subset.  This must be the same subset as classified by $\sigma_X h = \chi_{\delta} h$ (where $\delta : X \hookrightarrow X \times X$ is the diagonal), which by the pullback principle is

$$(1_X \times h)^{-1}(\delta) \hookrightarrow X \times A.$$

An elementary calculation shows this to be the equalizer of the pair of maps

$$\pi_1, h\pi_2 : X \times A \stackrel{\to}{\to} X.$$

So this equalizer $E$ is empty.  But notice that $\langle h, 1 \rangle : A \to X \times A$ equalizes this pair of maps.  Therefore we have a map $A \to E \cong 0$.  By Corollary 2 above, we infer $A \cong 0$.  This applies to the case where $A$ is the pullback, so the pullback is empty, as was to be shown.
=--

__Theorem 4.__  Any two disjoint unions of $X, Y$ are canonically isomorphic.

+-- {: .proof}
###### Proof
Suppose $i : X \rightarrow Z \leftarrow Y : j$ is a disjoint union.  Define a map

$$\phi = \langle \phi_1, \phi_2 \rangle : Z \to P X \times P Y$$

where $\phi_1 : Z \to P X$ classifies the subset $\langle 1_X, i \rangle : X \to X \times Z$, and $\phi_2 : Z \to P Y$ classifies the subset $\langle 1_Y, j \rangle : Y \to Y \times Z$.  Applying the pullback principle, the composite $\phi_1 i : X \to P X$ classifies

$$(1_X \times i)^{-1}(\langle 1_X, i \rangle) \hookrightarrow X \times X$$

which is easily seen to be the diagonal on $X$.  Hence $\phi_1 i = \sigma_X$.  On the other hand, $\phi_1 j : Y \to P X$ classifies the subset

$$(1_X \times j)^{-1}(\langle 1_X, i \rangle) \hookrightarrow X \times Y$$

which is empty because $i$ and $j$ are disjoint embeddings, so $\phi_1 j = \chi_0 : Y \to P X$.  Similar
calculations yield

$$\phi_2 i = \chi_0 : X \to P Y \qquad \phi_2 j = \sigma_Y : Y \to P Y.$$

Putting all this together, we conclude that $\phi i = i_X : X \to P X \times P Y$ and $\phi j = i_Y : Y \to P X \times P Y$, where $i_X$ and $i_Y$ were defined in the proof of Theorem 3.

Next, we show that $\phi$ is monic.  If not, then by strong extensionality, there exist distinct elements $c, d : 1 \to Z$ for which $\phi(c) = \phi(d)$;  therefore, $\phi_1 c = \phi_1 d : 1 \to P X$ and $\phi_2 c = \phi_2 d : 1 \to PY$.  By the pullback principle, these equations say (respectively)

$$ c^{-1}(i) = d^{-1}(i) \hookrightarrow 1 \qquad c^{-1}(j) = d^{-1}(j) \hookrightarrow 1.$$

If $c^{-1}(i) = 1$, then both $c, d : 1 \to Z$ factor through the mono $i : X \to Z$.  However, since $\phi i = i_X$ is monic, this would imply that $\phi (c) \neq \phi (d)$, contradiction.  Therefore $c^{-1}(i) = c \cap i = 0$.  By similar reasoning, $c \cap j = 0$.  Therefore

$$i \subseteq \neg c \qquad j \subseteq \neg c$$

where $\neg = (- \Rightarrow 0) : Sub(Z) \to Sub(Z)$ is the negation operator.  But then $i \cup j \subseteq \neg c$.  And since $1 : Z \hookrightarrow Z$ is the union $i \cup j$ by assumption, $\neg c$ must be the top element $\top \in Sub(Z)$, whence $c : 1 \to Z$ is the bottom element 0.  This contradicts the assumption that the topos is nondegenerate.  Thus we have shown that $\phi$ must be monic.

The argument above shows that $\phi : Z \hookrightarrow P X \times P Y$ is an upper bound of $i_X : X \to P X \times P Y$ and $i_Y : Y \to P X \times P Y$ in $Sub(P X \times P Y)$.  It follows that the join $X + Y$ constructed in Theorem 3 is contained in $\phi : Z \to P X \times P Y$, and hence can be regarded as the join of $X$ and $Y$ in $Sub(Z)$.  But $Z$ is their join in $Sub(Z)$ by assumption of being a disjoint union, so the containment $X + Y \subseteq Z$ must be an equality.  The proof is now complete.
=--

__Theorem 5.__  The inclusions $i_X : X \to X + Y$, $i_Y : Y \to X + Y$ exhibit $X + Y$ as the coproduct of $X$ and $Y$.

+-- {: .proof}
###### Proof
Let $f : X \to B$, $g : Y \to B$ be given functions.  Then we have monos

$$\langle 1_X, f \rangle : X \hookrightarrow X \times B \qquad \langle 1_Y, g \rangle : Y \hookrightarrow Y \times B \qquad (1)$$

Now the operation $- \times B : Sub(C) \to Sub(C \times B)$ certainly preserves finite meets, and also preserves finite joins because it is left adjoint to $\forall_B : Sub(C \times B) \to Sub(C)$.  Therefore this operation preserves disjoint unions;  we infer that the monos

$$X \times B \stackrel{i_X \times B}{\to} (X + Y) \times B \qquad Y \times B \stackrel{i_Y \times B}{\to} (X + Y) \times B \qquad (2)$$

exhibit $(X + Y) \times B$ as a disjoint union of $X \times B, Y \times B$.  Composing the monos of (1) and (2), we have disjoint embeddings of $X$ and $Y$ in $(X + Y) \times B$.  Using Theorem 4, $X + Y$ is isomorphic to the join of these embeddings;  this means we have an inclusion

$$\langle 1, h \rangle : X + Y \hookrightarrow (X + Y) \times B$$

whose restriction to $X$ yields $\langle i_X, f \rangle$ and whose restriction to $Y$ yields $\langle i_Y, g \rangle$.  Hence $h : X + Y \to B$ extends $f$ and $g$.  It is the unique extension, for if there were two extensions $h, h'$, then their equalizer would be an upper bound of $X, Y$ in $Sub((X + Y) \times B)$, contradicting the fact that $X + Y$ is the least upper bound.  This completes the proof.
=--

I think that's enough for one day.  I will continue to explore the categorical structure and logic of ETCS next time.


Trimble rewiring

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

So-called _Trimble rewiring_ refers to a certain type of transformation on [[graphs]] which were introduced in [Trimble's thesis](#TT94), intended to refine [[Kelly-Mac Lane graphs]] or classical [[proof nets]] in multiplicative [[linear logic]] so as to capture properties of the [[monoidal category|monoidal unit]] in [[symmetric monoidal category|symmetric]] [[closed monoidal categories]]. These unit-extended proof nets may be interpreted as morphisms in [[free construction|free]] [[symmetric monoidal category|symmetric monoidal]] [[closed category|closed categories]], and it turns out that two unit-extended proof nets are interpretable as the same morphism if and only if they are "rewiring equivalent".

Rewirings can be organized into a confluent and strongly normalizing rewrite system, where a rewrite called "directed rewiring". The normal forms of graphs can then be used to state a full coherence theorem, extending the [[coherence theorem]] of [Kelly and Mac Lane](#KM71) for symmetric monoidal closed categories.

The concept of rewiring equivalence extends to [[weakly distributive category|weakly distributive categories]] and $\ast$-[[star-autonomous category|autonomous]] categories as well, and likewise is used to give [coherence theorems](#BCST96) for these structures.

## Definition

(For now this is just the very rough idea. More details to come.)

Recall from [[proof net]] the notion of (cut-free) proof structure and the notion of (cut-free) proof net (a proof structure of the form $S(\pi)$ where $\pi$ is a sequent proof).

+-- {: .num_defn}
###### Definition
A _unit-extended proof structure_ of type $\Delta \to B$ is a proof structure of type $\Delta \to B$ equipped with a function $u: Unit^-(\Delta^-, B^+) \to Subform(\Delta^-, B^+)$ from the set of negatively signed subformula occurrences of $I$ to the set of subformulas of $\Delta$ and $B$.
=--

We draw such a unit-extended proof structure as a graph: the graph of the proof
structure together with an edge from each occurrence $R$ of $I^-$ to $u(R)$. We think of this unit edge as connoting a subterm of the form $r t$, where $r$ is a variable term of type $I$ (a "scalar"), and $t$ is a term of type $u(R)$.

Let $UStruct(\Delta, B)$ denote the set of unit-extended proof structures of type $\Delta \to B$, and let $Proof(\Delta, B)$ denote the set of sequent proofs of the sequent $\Delta \vdash B$. We define a relation from $Proof(\Delta, B)$ to $UStruct(\Delta, B)$, i.e., a function

$$Proof(\Delta, B) \to P(UStruct(\Delta, B))$$

by induction along the sequent proof, as follows. (Details to be filled in. The most important sequent rule to consider is

$$\Gamma(\frac{\displaystyle \pi: \Delta, \Sigma \vdash A}{\displaystyle \Delta, I, \Sigma \vdash A}\; unit_{-})$$

where a related unit-extended proof net is obtained from a proof net for $\pi$ by introducing a fresh unit edge from the occurrence of $I^-$ introduced in the last step to any subformula of $\Delta, \Sigma \vdash A$. The other rules are straightforward.)

+-- {: .num_defn}
###### Definition
A _unit-extended proof net_ is a proof structure that is related to some sequent deduction.
=--

Theorem on graphical criterion for a unit-extended proof structure to be a unit-extended proof net, similar to the criterion of Danos-Regnier.

+-- {: .num_defn}
###### Definition
Two unit-extended proof nets $\gamma, \gamma'$ of type $\Delta \to B$ are _rewiring-equivalent_ if there is a sequence of unit-extended proof nets $\gamma = \gamma_0, \gamma_1, \ldots, \gamma_n = \gamma'$ of type $\Delta \to B$ such that each successive pair of graphs $\gamma_i, \gamma_{i+1}$ differ only by a single unit edge $u(R)$.
=--

Theorem that two sequent proofs $\pi, \pi'$ produce the same morphism in the free smc category $V[T]$ iff any two of their unit-extended proof nets $\gamma, \gamma'$ are rewiring-equivalent.

Notion of directed rewiring...

## References

* [[Todd Trimble]], _Linear logic, bimodules, and full coherence for autonomous categories. PhD thesis, Rutgers University, 1994
 {#TT94}

* {#KM71} [[Max Kelly]], [[Saunders MacLane]], _Coherence in closed categories_, JPAA 1 (1971), 97-140 ([web](http://www.sciencedirect.com/science/article/pii/0022404971900132))

* [[Richard Blute]], Robin Cockett, [[R. A. G. Seely|Robert Seely]], [[Todd Trimble]], _Natural deduction and coherence for weakly distributive categories_, JPAA 113 (1996), 229-296. ([web](http://www.sciencedirect.com/science/article/pii/002240499500159X))
 {#BCST96}

[[!redirects Trimble rewirings]]


Trimble's notion of weak n-category > history

< [[Trimble's notion of weak n-category]]
[[!redirects Trimble's notion of weak n-category -- history]]


Tring group

#Contents#
* table of contents
{:toc}

## Idea

The higher connected cover of the [[Pin group]].

## Related concepts

[[!include table of orthogonal groups and related]]


trinion

The  3-punctured [[2-sphere]]. The "pair of pants".

## Related concepts

* [[annulus]]

* [[Riemann surface]]

* [[fusion bundle]]

[[!redirects trinions]]
[[!redirects pair of pants]]


triple

## Definition

A **triple** is an [[n-tuple]] for $n = 3$.

Beware that an old word for [[monad]] is also "triple".

## Related concepts

* [[quadruple]]

* [[quintuple]]

[[!redirects triples]]


triple category

# Triple categories

* table of contents
{: toc}

## Idea

A **triple category** is an [[n-fold category]] for $n=3$.  That is, it is an [[internal category]] in [[double categories]].  This concept can be considered at various levels of strictness. However, there is not yet a definition in the literature of a fully weak (i.e. pseudo) triple category.

## Examples

* [[strict triple category]]
* [[intercategory]]
* The [[double category of algebras]] can be enhanced to a triple category
* Just as any category gives rise to a double category of [[commutative squares]], any double category gives rise to a triple category of commutative "cubes" by taking the transversal morphisms to be the same as the vertical morphisms.

[[!redirects triple categories]]


triple membrane junction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Among the possible $\frac{1}{8}$-[[BPS states]] of [[11-dimensional supergravity]]/[[M-theory]] is one where three [[black brane|black]] [[M2-branes]] jointly [[intersecting brane|intersect]] at equal angles ([Tseytlin 96](#Tseytlin96))

$$
  M2 \perp M2 \perp M2
  \,.
$$

This is the lift to [[M-theory]] of 3-string junctions in [[type IIB string theory]] ([Krogh-Lee 97](#KroghLee97), [Ramadevi 99](#Ramadevi99)), see at _[[string network]]_.

## Properties

### Triple generalization of Chan-Paton factors

Since [[black brane|black]] [[M2-branes]] may also end on [[M5-branes]], in an M-theory lift of how [[strings]] end on [[D-branes]] as quanta of [[super Yang-Mills theory]], there was the suggestion that this points to a generalization of [[gauge theories]] where [[matrix Lie algebras]] (matrices of [[Chan-Paton gauge field|Chan-Paton factors]]) are enhanced to some kind of higher or hypermatrix structure ([FLST 10](#FLST10)).

### Gauge enhancement at ADE-Singularities

If one accepts the traditional story of the [[M-theory lift of gauge enhancement on D6-branes]], in terms of [[M2-branes]] [[wrapped brane|wrapping]] [[2-spheres]] that touch along the shape of a [[Dynkin diagram]] [[blowup]] of an [[ADE-singularity]],

> graphics grabbed from [HSS18](http://ncatlab.org/schreiber/show/Equivariant+homotopy+and+super+M-branes)

then such triple membrane junctions must correspond to blowups of the [[dihedral group|dihedral]], [[tetrahedral group|tetrahedral]], [[octahedral group|octahedral]] or [[icosahedral group|icosahedral]] [[ADE-singularities]], since in these cases the [[Dynkin diagram]] has a triple junction node :

## Related concepts

[[!include brane bound states -- table]]

## References

Discussion of triple-membrane junction-shaped [[black brane]]-solutions to [[11-dimensional supergravity]] is in

* {#Tseytlin96} [[Arkady Tseytlin]], _Harmonic superpositions of M-branes_, Nucl. Phys. B475:149-163, 1996 ([arxiv:hep-th/9604035](https://arxiv.org/abs/hep-th/9604035))

Lift of and relation to [[string networks]] is discussed in

* {#KroghLee97} Morten Krogh, Sangmin Lee, _String Network from M-theory_, Nucl. Phys. B516:241-254, 1998 ([arXiv:hep-th/9712050](https://arxiv.org/abs/hep-th/9712050))

* {#Ramadevi99} P. Ramadevi, _Supergravity Solution for Three-String Junction in M-Theory_, JHEP 0006 (2000) 005 ([arXiv:hep-th/9906247](https://arxiv.org/abs/hep-th/9906247))

Possible relation to [[hypermatrices]] is discussed in

* {#FLST10} Yuhan Fang, Shir Levkowitz, [[Hisham Sati]], Daniel Thompson, _Hypermatrix factors for string and membrane junctions_, J. Phys. A43: 505401, 2010 ([arXiv:1001.5166](https://arxiv.org/abs/1001.5166))

[[!redirects triple membrane junctions]]

[[!redirects membrane triple junction]]
[[!redirects membrane triple junctions]]


tripos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A __tripos__ is a [[first-order hyperdoctrine|first-order hyperdoctrine with equality]]  satisfying an additional property that allows it to interpret [[impredicative logic|impredicative]] [[higher-order logic]] as well. In particular, every tripos gives rise to a corresponding [[topos]].

The notion of tripos, due to [[Andrew Pitts]], is useful for giving a unified account of two very different classes of toposes: [[localic topos|localic toposes]] and [[realizability topos|realizability toposes]].

The name 'tripos' is to be credited to [[Peter Johnstone]], who was the thesis advisor of Pitts. It can be read as an acronym for "Topos Representing Indexed Partially Ordered Set", but it is memorable also as a pun, referring to a famous examination at Cambridge University where Pitts, Johnstone, and [[Martin Hyland|Hyland]] work.

## Definition

Recall that the data of a first-order hyperdoctrine $T$ consists of a [[syntactic category|category of terms]] $C_T$, assumed to have [[finite products]], and a [[functor]]

$$Pred_T: C_{T}^{op} \to HeytAlg$$

to the category of [[Heyting algebras]], satisfying suitable properties (including [[quantifiers]], i.e., left and right adjoints to "pullback maps" $Pred_T(f)$)[^fine]. We often abbreviate $Pred_T(f)$ to $f^\ast$, calling it a _pullback map_. If $H$ is a Heyting algebra, we let ${|H|}$ denote the underlying set.

+-- {: .num_defn}
###### Definition
A **tripos** is a first-order hyperdoctrine $T$ together with, for every object $c$ of $C_T$, an object $P c$ and an element $in_c \in Pred_T(c \times P c)$ such that for any object $b$ and element $p \in Pred_T(c \times b)$, there exists a map $\chi \colon b \to P c$ with $Pred_T(1_c \times \chi)(in_c) = p$. (N.B.: such $\chi$ need not be unique.)
=--

Another way of phrasing the definition is that to each object $c$ of $C_T$ there is an assigned object $P(c)$ and an assigned _[[epimorphism]]_

$$\hom(-, P c) \to {|Pred_T|}(c \times -)$$

of set-valued functors; the [[predicate]] $in_c$ is the value of $id_c$ under the assigned epimorphism $\hom(P c, P c) \to {|Pred_T(c \times P c)|}$ (*à la* [[Yoneda lemma]]).

The $in_c$ are called _generic predicates_. In particular, put $P = P 1$ and $in = in_1 \in Pred_T(P)$, with corresponding epimorphism $\hom(-, P) \to {|Pred_T|}$. If $C_T$ is [[cartesian closed category|cartesian closed]], we then have [[epimorphism|epis]]

$$\hom(-, P^c) \cong \hom(-, P) \circ (c \times -)^{op} \to {|Pred_T|} \circ (c \times -)^{op} = {|Pred_T|}(c \times -)$$

and in this way the single generic predicate $(P, in)$ can be used to generate a generic predicate over $c$, as $(P^c, in_c = (eval_c)^\ast(in))$, i.e., a generic predicate over $c$ is obtained by pulling back along the [[evaluation]] map $eval_c \colon c \times P^c \to P$. (N.B.: this need not reproduce the original generic predicates, but the resulting tripos structure may be compared with the original tripos structure in both directions: there are maps $P c \to P^c$ and $P^c \to P c$, with the generic predicates in the two triposes pulling back to each other along these maps.)

In practice, the category $C_T$ will indeed often be [[cartesian closed category|cartesian closed]] (frequently taking $C_T = Set$ in fact), and triposes are frequently formed in the manner just described, starting with a generic predicate over $1$ and pulling it back along the maps $eval_c$ to get a generic predicate over any $c$.

+-- {: .num_remark}
###### Remark
The surjective natural transformation defining the suitably generic predicate in a tripos of course induces a [[preorder]] (but not necessarily [[poset|posetal]]) structure on each $Hom_{C_T}(Y, X)$, the posetal reflection of which gives the Heyting algebra $P(Y)$. Accordingly, we can specify a tripos with category of terms $C_T$ by specifying an object $X \in Ob(C_T)$ and putting [[Heyting prealgebra]] structure on $Hom_{C_T}(-, X)$ in such a way as to yield all the first-order hyperdoctrine structure, with the identity morphism on $X$ playing the role of the suitably generic predicate.
=--

## Examples

### Coming from toposes{#Topos}

Every [[topos]] $E$ gives rise to a tripos in an obvious way, using $C_T = E$ and
$$Pred_T = Sub \colon E^{op} \to HeytAlg,$$
taking an object $e$ to the Heyting algebra of [[subobjects]] $Sub(e)$. Here of course we have not just an epimorphism but an [[isomorphism]]
$$\hom(-, \Omega^c) \to Pred_T(c \times -),$$
obtained by pulling back the generic predicate $t \colon 1 \to \Omega$ over $1$ along $eval \colon c \times \Omega^c \to \Omega$.

### Coming from complete Heyting algebras{#Heyt}

If $H$ is a [[complete Heyting algebra]], we can take $C_T = Set$, and put
$$Pred_T = \hom(-, H) \colon Set^{op} \to HeytAlg.$$
Completeness of $H$ is used to ensure that for each function $f \colon X \to Y$, the Heyting algebra map $Pred_T(f) \colon H^Y \to H^X$ has a left and right adjoint. The identity map $\hom(-, H) \to Pred_T$ is the epimorphism we use to generate the tripos structure, taking of course $(H, 1_H \in H^H)$ as the generic predicate over $1$.

### Coming from partial combinatory algebras{#PCA}

If $A$ is a [[partial combinatory algebra]] (PCA), we can form a tripos in the following manner. Let $C_T = Set$. For each set $X$, one can put a [[preorder]] structure on $P(A)^X$ \[here, the $P(A)$ refers to the actual powerset of $A$ in $Set$\] as follows: given $f, g \in P(A)^X$, let $Hom(f, g)$ be the set of $a$ in $A$ such that for all $x$ in $X$ and $b$ in $f(x)$, $a$ applied to $b$ is defined and an element of $g(x)$. We will of course take $f \leq g$ just in case $Hom(f, g)$ is inhabited. The relation $\leq$ is reflexive and transitive, by functional completeness for PCA's. It is straightforward that for a function $f \colon X \to Y$, the pullback map
$$P(A)^f \colon P(A)^Y \to P(A)^X$$
preserves the [[preorder]] structure. Now define $Pred_T(X)$ to be the poset obtained from the preorder $P(A)^X$ by posetal reflection. With the help of functional completeness, it may be shown that $Pred_T(X)$ is in fact a Heyting algebra, and we get in this way a hyperdoctrine; see [here](http://ncatlab.org/nlab/show/partial+combinatory+algebra#realizability_toposes_11) for details. Notice in this case we have, by construction, an epimorphism between set-valued functors
$$\hom_{Set}(-, P(A)) \to {|Pred_T|}$$
and thus we obtain a tripos, called the _realizability tripos_ associated with the PCA $A$.

## From triposes to toposes

Every first-order hyperdoctrine $T = (C_T, Pred_T)$ gives rise to a [[bicategory of relations]], whose objects are the objects of $C_T$ and where [[1-morphism|1-cells]] from $X$ to $Y$ are triples $(X, Y, r \in Pred_T(X \times Y))$, thus thinking of $r$ as a "relation" $r \colon X \to Y$. Thus $\hom(X, Y) = Pred_T(X \times Y)$, and we define hom-composition in the usual way for relational composition, by expressing the formula $(r \circ s)(x, z) = \exists_y r(x, y) \wedge s(y, z)$ [[internal logic|internally]] in the hyperdoctrine:

$$\array{
Pred_T(X \times Y) \times Pred_T(Y \times Z) & \stackrel{\pi_{X \times Y}^\ast \times \pi_{Y \times Z}^\ast}{\to} & Pred_T(X \times Y \times Y \times Z) \times Pred_T(X \times Y \times Y \times Z) \\
& \stackrel{\wedge}{\to} & Pred_T(X \times Y \times Y \times Z) \\
& \stackrel{(1 \times \delta \times 1)^\ast}{\to} & Pred_T(X \times Y \times Z) \\
& \stackrel{\exists_{1 \times ! \times 1}}{\to} & Pred_T(X \times Z).
}$$

This gives in fact a [[dagger-category]] [[enriched category|enriched]] in [[poset|posets]], where $(-)^\dagger: \hom(X, Y) \to \hom(Y, X)$ is the evident isomorphism $Pred_T(X \times Y) \to Pred_T(Y \times X)$. The _opposite_ of a relation $r \colon X \to Y$ is $r^\dagger \colon Y \to X$, and a relation $r \colon X \to X$ is _symmetric_ if $r^\dagger = r$. In summary, we get an [[allegory]] (bicategories of relations being essentially equivalent to unitary pretabular allegories).

In this context, a _[[partial equivalence relation]]_, or PER, may be defined as a symmetric idempotent map $r \colon X \to X$. (In slightly more detail, a partial equivalence relation means we drop reflexivity from the notion of equivalence relation, so we mean transitivity $r r \leq r$ plus symmetry $r^\dagger = r$. However, for any $r \colon X \to X$, it is a consequence of Freyd's modular law for allegories that $r \leq r r^\dagger r$, so that transitivity and symmetry combined yield
$$r \leq r r^\dagger r = r r r \leq r r$$
in addition to $r r \leq r$, meaning that $r$ is in fact idempotent.)

We will be interested in [[split idempotent|splitting]] not all [[idempotents]] (as in $Set$-based or $Pos$-based [[Cauchy completion]]), but just the symmetric idempotents = PERs, in view of the following theorem.

+-- {: .num_theorem}
###### Theorem
Let $T$ be a tripos, and let $Rel_T$ be the bicategory of relations obtained from $T$. Let $Split_{per}(Rel_T)$ be the bicategory obtained by splitting the PERs. Then the locally discrete bicategory of maps (whose 1-cells are left adjoints or total functional relations) in $Split_{per}(Rel_T)$ is a topos.
=--

In the case of a [tripos induced from a topos $T$](#Topos), this construction yields back $T$, since every partial equivalence relation (as a morphism of $Rel_T$) factors through a suitable subquotient object in $T$.

In the case of a [realizability tripos](#PCA) associated with a PCA $A$, this construction yields the realizability topos of $A$. In the particular case where $A$ is [[Kleene's first algebra]] (the PCA whose elements are natural numbers taken as codes for computer programs taking natural number input and producing natural number output if they halt, with obvious application partial function), this is also called the [[effective topos]].

Before we undertake a conceptual analysis of this theorem, we give in the next section some details for the tripos attached to a complete Heyting algebra, described [here](#Heyt). The description of the resulting topos coincides with that of [[Denis Higgs|Higgs]], in his treatment of localic toposes in terms of Boolean-valued or Heyting-valued set theory (*à la* Scott and Solovay).

### Weak triposes

In [Pitts 2002](#Pitts02) it is observed that a somewhat weaker condition than suffices to ensure that the above construction yields a topos.

+-- {: .num_defn}
###### Definition
A **weak tripos** is a first-order hyperdoctrine $T$ together with, for every object $c$ of $C_T$, an object $P c$ and an element $in_c \in Pred_T(c \times P c)$ such that for any object $b$ and element $p \in Pred_T(c \times b)$, the following sentence is satisfied in the [[internal logic]] of $T$:
$$ \forall i:b. \exists s:P c. \forall x:c. \in_c(x,s) \Leftrightarrow p(x,i). $$
=--

This generalization is not vacuous either.  For instance, it is shown in [Pitts 2002](#Pitts02) that for any infinite [[Boolean algebra]] $B$ there is a weak tripos over $FinSet$, constructed analogously to the tripos attached to a complete Heyting algebra.  (We don't need $B$ to be complete since we have only finite sets in the base; but we do need $B$ to be Boolean rather than just Heyting in order to prove that this is a weak tripos.  The resulting topos has $B$ as its poset of [[subterminal objects]]; it is unknown whether every (not necessarily complete) Heyting algebra can occur as the poset of subterminals in an elementary topos.)

In fact, it also seems unnecessary for the hyperdoctrine $T$ to have equality, since in constructing a topos we equip every object with a *new* partial equivalence relation to become its "equality".  However, the above construction that goes through a bicategory of relations does rely on the existence of equality, as does the statement of the universal property of this construction (since equality in $T$ gives an embedding of $C_T$ as "constant objects" in its associated topos, which is the [[unit of an adjunction]]).

### A "synthetic" view of triposes

Every tripos $P$ over a topos $S$ gives rise to a so-called "constant objects" functor $\Delta_P : S \to E$ where $E$ is the topos $S[P]$ induced by $P$. It sends $I$ to $(I,\mathrm{eq}_I)$ where $\mathrm{eq}_I$ is the equality predicate on $I$ in the sense of $P$. For the constant objects functor $\Delta_P : S \to E$ we know that

1. $\Delta_P$ preserves finite limits

2. every $A$ in $E$ appears as subquotient of $\Delta_P I$ for some $I$ in $S$

3. there is a subobject $t : T \rightarrowtail \Delta_P\Sigma$
   such that every mono $m : X \rightarrowtail \Delta_P I$ appears
   as pullback of $t$ along $\Delta_P f$ for some (typically not unique) map
   $f : I \to \Sigma$ in $S$

and that $P$ is obtained from the subobject fibration of $E$ by change of base along along $\Delta_P$. It follows from observations in Pitts' thesis that this way triposes over $S$ correspond up to equivalence to functors $F$ from $S$ to toposes $E$ such that $F$ validates the above conditions (1)-(3). Moreover, _weak_ triposes over $S$ correspond to $F$ which validate only conditions (1) and (2). Such $F$ are inverse image parts of localic geometric morphisms iff $F$ has a right adjoint.

Somewhat surprisingly the "chaotic" functor $\nabla$ from $\mathbf{Set}$ to
the topos of reflexive graphs is a weak tripos in the above sense. For
every non-empty set $I$ the functor $(-)^I : \mathbf{Set} \to \mathbf{Set}$
is a weak tripos. For non-isomorphic finite non-empty sets $I$ and $J$
the functors $(-)^I$ and $(-)^J$ are not equivalent. Thus, nonequivalent weak triposes over $\mathbf{Set}$ can give rise to equivalent toposes. It is an open question whether this is also possible for genuine non-weak triposes over $\mathbf{Set}$.

## Relation to $H$-valued sets

Consider the tripos $T = T_H$ obtained from a complete Heyting algebra $H$ ([example 2](#Heyt)), or more exactly the bicategory of relations obtained from this. Let us describe explicitly the bicategory obtained by splitting the PERs:

* An object in the PER-splitting completion in this case is a set $X$ equipped with a function $e \colon X \times X \to H$ which is symmetric and transitive in the sense described above. This boils down to having, for all elements $x, y, z \in X$,
$$e(x, y) = e(y, x)$$
$$e(x, y) \wedge e(y, z) \leq e(x, z)$$
This is also known as an _$H$-valued set_. (We do not assume reflexivity, where $\top \leq e(x, x)$ for all $x$.) The function $e$ can be thought of as a measure of equality.

* A morphism from $(X, e_X)$ to $(Y, e_Y)$ is a relation $r \colon X \to Y$, or a function $r \colon X \times Y \to H$, such that $r \circ e_X = r = e_Y \circ r$. (Cf. [[Karoubi envelope]].) The conditions $r \circ e_X \leq r$, $e_Y \circ r \leq r$ mean we have a two-sided "action":
$$e_X(x', x) \wedge r(x, y) \leq r(x', y), \qquad r(x, y) \wedge e_Y(y, y') \leq r(x, y').$$
The condition $r \leq r \circ e_X$ means
$$r(x, y) \leq \bigvee_{x' \in X} e_X(x, x') \wedge r(x', y)$$
but since $e_X$ is symmetric and transitive, we have
$$
\array{
r(x, y) & \leq & \bigvee_{x' \in X} e_X(x, x') \wedge r(x', y)\\
& = & \bigvee_{x'} e_X(x, x') \wedge e_X(x', x) \wedge r(x', y)\\
& \leq & \bigvee_{x'} e_X(x, x') \wedge e(x', x) \leq e(x, x)
}
$$
so that in fact we have $r(x, y) \leq e_X(x, x)$. Similarly we have $r(x, y) \leq e(y, y)$, and these two conditions conversely imply $r \leq r \circ e_X$, $r \leq e_Y \circ r$, as may be easily checked.

Such morphisms $r$ are called _relations between $H$-valued sets_. Such relations in $Split_{per}(Rel_T)$ may be composed just as they are in $Rel_T$, and if $r$ is an $H$-valued relation, so is $r^\dagger$. On the other hand, as described at [[Karoubi envelope]], the identity from $(X, e)$ to itself is given by the relation $e$ (clearly the identity $1_X$ doesn't work!). The result $Split_{per}(Rel_T)$ is also a bicategory of relations (although, as we will eventually see, much more is true).

A relation $r \colon (X, e_X) \to (Y, e_Y)$ between PERs is _functional_ if

* $r \circ r^\dagger \leq 1_{(Y, e_Y)} = e_Y$ ("well-definedness").

* $e_X = 1_{(X, e_X)} \leq r^\dagger \circ r$ ("totality");

The first of these conditions means

$$r^\dagger(y, x) \wedge r(x, y') = r(x, y) \wedge r(x, y') \leq e_Y(y, y')$$

The second condition means

$$e_X(x, x') \leq \bigvee_{y \in Y} r(x, y) \wedge r^\dagger(y, x')$$

which certainly implies $e_X(x, x) \leq \bigvee_y r(x, y)$. Conversely, if $e_X(x, x) \leq \bigvee_y r(x, y)$, we have

$$\array{
e_X(x, x') & \leq & e_X(x, x) \wedge e_X(x, x') \\
 & \leq & \bigvee_y r(x, y) \wedge e_X(x, x') \\
 & \leq & \bigvee_y r(x, y) \wedge r(x', y) \\
 & \leq & \bigvee_y r(x, y) \wedge r^\dagger(y, x').
}$$

Thus totality of $r \colon X \to Y$ is equivalent to

* $e_X(x, x) \leq \bigvee_{y \in Y} r(x, y)$

+-- {: .num_theorem}
###### Theorem (Higgs)
For a complete Heyting algebra $H$, the category of $H$-valued sets and functional relations between $H$-valued sets is equivalent to the topos of [[sheaf|sheaves]] on $H$.
=--

Thus all [[localic topos|localic toposes]] arise as toposes of $H$-valued sets, where $H$ is the Heyting algebra of subobjects of $1$.

Compare the description by Walters of sheaves over $H$ as certain types of Cauchy-complete categories enriched in a bicategory. Notice that in that description, the homs are symmetric ($\hom(x, y) = \hom(y, x)$), so that the Cauchy completion or idempotent-splitting completion is the same as the completion by splitting symmetric idempotents.

+-- {: .num_remark}
###### Remark
The triposes over $Set$ of this form are precisely those for which the preorder imposed on $Hom_{Set}(X, P) \simeq P^X$ is the straightforward $X$-ary product of the preorder imposed on $Hom_{Set}(1, P) \simeq P$, where $P$ is the carrier of the generic predicate; in all other cases, the former is a finer-grained preorder than the latter. Thus, in some sense, triposes over $Set$ are a generalization of the notion of "complete Heyting algebra" taking advantage of the ability to use preorder rather than poset structure to allow for a weakening of the condition of completeness.

An example taking advantage of this generalization is given by [realizability triposes](#PCA). See also [[realizability]].
=--

## Relation to exact completion

As stated above, the topos obtained from a tripos can be described as the category of PERs in the bicategory of relations, and functional relations between them. In different language, the bicategory of PERs or symmetric idempotents and relations between them is a [[power allegory]], and the process of passing to functional relations is just the standard process of passing from power allegories to toposes.

The process of splitting symmetric idempotents in the bicategory of relations can be analyzed into two steps, the first related to taking a regular completion, and the second to taking an [[exact completion]] of a regular category. More exactly, they are analogues on the allegorical side of these two types of completion.

The two steps are as follows, starting with a bicategory of relations (or perhaps preferably a [[framed bicategory]] of relations). Splitting symmetric idempotents can be obtained by

* First splitting the coreflexive morphisms ($r \colon X \to X$ is coreflexive if $r \leq 1_X$). This results in a tabular bicategory of relations or a [[allegory|unitary tabular allegory]], which is essentially the same as obtaining a regular category.

* Then, in the unitary tabular allegory, split equivalence relations. In allegorical language, this results in an effective (unitary tabular) allegory (Freyd-Scedrov, page 213); on the categorical side, it means we pass from a regular category $C$ to its [[regular and exact completions|exact completion]] $C_{ex/reg}$.

## Internal logic

Of course, the topos obtained from a tripos has an [[internal logic]].  But a tripos itself has an internal logic in which the "types" are the objects of the base category and the propositions are those of the fiber category.  In fact, the internal logic of a topos is often presented in a way that actually only makes use of the tripos of subobjects of the topos.  The type constructors of this internal logic correspond to objects with universal properties in the base category; so in a general tripos there would be only [[product types]] and [[power types]].  If the base category is cartesian closed, then the latter would decompose into [[function types]] and a [[type of propositions]].  However, note that these "functions" are less general than the morphisms in the topos constructed from the tripos; in the internal language of the tripos the latter manifest as [[anafunctions]].

##References##

* [[Andrew Pitts]], _The theory of [[triposes]]_, thesis, [pdf](http://www.cl.cam.ac.uk/~amp12/papers/thet/thet.pdf)
 {#PittsThesis}

* [[Andrew Pitts]], _Tripos theory in retrospect_, [pdf](http://www.cl.cam.ac.uk/~amp12/papers/tritr/tritr.pdf)
 {#Pitts02}

* [[Peter Freyd]] and Andre Scedrov, _Categories, Allegories_, North-Holland 1990.

[[!redirects triposes]]

[^fine]: In fact one may "quantify" along any term = morphism in the base category $C_T$, i.e., each $Pred_T(f)$ has a left adjoint and a right adjoint, merely if we assume this of projection and diagonal maps and if we assume an appropriate [Frobenius law](http://ncatlab.org/nlab/show/Frobenius+reciprocity#frobenius_laws_and_frobenius_reciprocity_11), as one generally does for first-order hyperdoctrines. However, the status of corresponding [[Beck-Chevalley conditions]] for [[pullback]] diagrams in $C_T$ is another matter; the original treatment by Pitts assumed more than is actually necessary. Namely, it was assumed that $C_T$ is finitely complete and the Beck-Chevalley condition holds for _all_ pullbacks in $C_T$. In actuality, for tripos theory, it is enough to assume Beck-Chevalley only for certain pullbacks which exist by virtue of the finite product structure of $C_T$.


Tristan Crolard

* [Webpage](http://cedric.cnam.fr/cpr/crolard/publications/index.html)

category:people


Tristan Hübsch

* [webpage](http://tristan.nfshost.com/index.html)

## Selected writings:

On [[Calabi-Yau manifolds]]:

* [[Tristan Hübsch]], _Calabi-Yau Manifolds -- A Bestiary for Physicists_, World Scientific 1992 ([doi:10.1142/1410](https://doi.org/10.1142/1410))

On [[adinkras]]:

* [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], _On Graph-Theoretic Identifications of Adinkras, Supersymmetry Representations and Superfields_, Int. J. Mod. Phys. A22: 869-930, 2007 ([arXiv:math-ph/0512016](https://arxiv.org/abs/math-ph/0512016))

On the relation of [[adinkras]] to Clifford supermodules:

* [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], _Off-shell supersymmetry and filtered Clifford supermodules_ ([arXiv:math-ph/0603012](https://arxiv.org/abs/math-ph/0603012))

kinetic terms:

* [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], _Adinkras and the Dynamics of Superspace Prepotentials_ ([arXiv:hep-th/0605269](https://arxiv.org/abs/hep-th/0605269))

The classification of [[adinkras]] in terms of [[graphs]] and [[linear codes]] is due to:

* {#DoranFauxGatesHubschIgaLandweberMiller11} [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], R. L. Miller, _Codes and Supersymmetry in One Dimension_, Adv. in Th. Math. Phys. 15 (2011) 1909-1970 ([arXiv:1108.4124](https://arxiv.org/abs/1108.4124))

The [[dimensional reduction]] of the standard [[supermultiplets]] of $D = 4, \mathcal{N} = 1$ [[supersymmetry]] to [[adinkraic representations]] of $D = 1, \mathcal{N}=4$:

* {#GatesHubschStiffler14} [[Jim Gates]], [[Tristan Hübsch]], Kory Stiffler, _Adinkras and SUSY Holography_, Int. J. Mod. Phys. A29 no. 7, (2014) 1450041 ([arXiv:1208.5999](https://arxiv.org/abs/1208.5999))

On [[Yukawa couplings]] in [[string phenomenology]] for [[heterotic strings]] on [[Calabi-Yau manifolds]]:

* Giorgi Butbaia, Damián Mayorga Peña, Justin Tan, Per Berglund, [[Tristan Hübsch]], Vishnu Jejjala, Challenger Mishra, *Physical Yukawa Couplings in Heterotic String Compactifications* [[arXiv:2401.15078](https://arxiv.org/abs/2401.15078)]

## Related entries

* [[adinkra]]

category: people

[[!redirects Tristan Hubsch]]


trivial algebra

+-- {: .rightHandSide}
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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

\tableofcontents

## Definition

Every [[algebraic category]] (in just about any sense of the term) has a [[terminal object]] which is usually called the __trivial algebra__, often denoted $0$ or $1$.  Its [[underlying set]] is a [[singleton]], and every algebraic operation $1^n \to 1$ is the unique [[function]] to the singleton set.

## Examples

*  The [[singleton set]] is itself the trivial [[set]].
*  The [[trivial monoid]] is the trivial [[monoid]].
*  The [[trivial group]] is the trivial [[group]].
*  The [[trivial ring]] is the trivial [[ring]].
*  The $0$-dimensional vector space is the trivial [[vector space]].
*  The [[trivial Lie algebra]] is the trivial [[Lie algebra]].
*  The [[trivial category]] is the trivial [[strict category]].

[[!redirects trivial algebras]]


trivial coverage

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

## Definition

+-- {: .num_defn #TrivialCoverage}
###### Definition
**(trivial coverage)**

For $\mathcal{C}$ a [[small category]], the _trivial coverage_ is the [[coverage]] with no [[covering]] families at all, meaning that the [[sheaf|sheaf condition]] over the resulting [[site]] is empty, in that _every_ [[presheaf]] is a [[sheaf]] for this coverage.

=--

Hence the [[category of presheaves]] $[\mathcal{C}^{op},Set]$ over a site $\mathcal{C}_{triv}$ with trivial coverage is already the corresponding [[category of sheaves]], hence the corresponding [[sheaf topos]]:

$$
  Sh\left( \mathcal{C}_{triv}\right)
  \;\simeq\;
  [\mathcal{C}^{op}, Set]
  \,.
$$

[[!redirects trivial coverages]]

[[!redirects trivial Grothendieck topology]]
[[!redirects trivial Grothendieck topologies]]


trivial fiber bundle

+-- {: .rightHandSide}
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#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The __trivial fiber bundle__ over $B$ with [[fibre]] $F$ is simply the [[Cartesian product]] $B \times F$, equipped with the [[projection map]] to $B$ to make it a [[fiber bundle]] over $B$

$$
  \array{
    B \times F
    \\
    {}^{\mathllap{pr_1}}\downarrow
    \\
    B
  }
$$

A fiber bundle which  not the trivial bundle but is _[[isomorphism|isomorphic]]_ to the trivial bundle is called _trivializable_ and the chosen [[isomorphism]] is called the corresponding _trivialization_.

## Examples

* If $F$ is a [[vector space]], then the trivial $F$-[[fiber bundle]] is a _[[trivial vector bundle]]_, etc.

* If a [[field bundle]] is trivial with fiber $F$, then the corresponding [[field theory]] is called a _[[sigma model]]_ with [[target space]] $F$.

[[!redirects trivial fiber bundles]]

[[!redirects trivial fibre bundle]]
[[!redirects trivial fibre bundles]]

[[!redirects trivial bundle]]
[[!redirects trivial bundles]]


trivial group

+-- {: .rightHandSide}
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###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The __trivial group__ is the [[group]] whose underlying [[set]] is the [[singleton]], hence whose only [[element]] is the [[neutral element]].

In the context of [[nonabelian groups]] the trivial group is usually denoted $1$, while in the context of [[abelian groups]] it is usually denoted $0$ (being the [[zero object]]) and also called the *zero group* (notably in [[homological algebra]]).

The trivial group is a [[zero object]] (both [[initial object|initial]] and [[terminal object|terminal]]) of [[Grp]].

## Examples

The trivial group is a [[subgroup]] of any other group, and the corresponding inclusion $1 \hookrightarrow G$ is the unique such group homomorpism.

The [[quotient group]] of any [[group]] $G$ by itself is the trivial group: $G/G = 1$, and the quotient projection $G \to G/G =1$ is the unique such group homomorphism.

It can be nontrivial to decide from a [[group presentation]] whether a group so presented is trivial, and in fact the general problem is [[decidability|undecidable]]. See also [[combinatorial group theory]] and [[word problem]].

## Properties

The trivial group is an example of a [[trivial algebra]].

## Related concepts

* [[Andrews-Curtis conjecture]]

* [[trivial monoid]]

* [[terminal category]]

[[!redirects trivial group]]
[[!redirects trivial groups]]

[[!redirects trivial subgroup]]
[[!redirects trivial subgroups]]
[[!redirects identity subgroup]]
[[!redirects identity subgroups]]

[[!redirects trivial quotient group]]
[[!redirects trivial quotient groups]]

[[!redirects zero group]]
[[!redirects zero groups]]


trivial Lie algebra

The __trivial Lie algebra__ is the $0$-dimensional [[vector space]] interpreted as a [[Lie algebra]] in the only way possible: $[0,0] = 0$.

The trivial Lie algebra is an example of a [[trivial algebra]].

[[!redirects trivial Lie algebra]]
[[!redirects trivial Lie algebras]]


trivial model structure

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#Contents#
* table of contents
{:toc}

## Definition

Every [[bicomplete category|bicomplete]] [[category]] (i.e. having all small [[limits]] and [[colimits]]) becomes a [[model category]] by setting:

* the [[weak equivalences]] are the [[isomorphisms]];

* every morphism is a [[fibration]];

* every morphism is a [[cofibration]].

This model structure regards $C$ as an [[(∞,1)-category]] with only trivial [[k-morphisms]] for $k \geq 2$.

## Related concepts

* [[2-trivial model structure]]

[[!redirects trivial model structures]]


trivial monoid

+-- {: .rightHandSide}
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### Context
#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
=--
=--

\tableofcontents

## Definition

The __trivial monoid__ is [[generalized the|the]] unique (up to [[isomorphism]]) [[monoid]] with a single [[element]] - the [[neutral element]] of the monoid. The trivial monoid is also a [[terminal object]] and an [[initial object]] of [[Mon]], making the trivial monoid a [[zero object]] of [[Mon]].

## See also

* [[trivial group]]

* [[trivial algebra]]

* [[trivial category]]

[[!redirects trivial monoid]]
[[!redirects trivial monoids]]

[[!redirects terminal monoid]]
[[!redirects terminal monoids]]

[[!redirects final monoid]]
[[!redirects final monoids]]

[[!redirects initial monoid]]
[[!redirects initial monoids]]


trivial representation

+-- {: .rightHandSide}
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#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[representation]]/[[action]] $V \times G \longrightarrow V$ is _trivial_ if it is given by the [[projection]] out of the [[product]] onto $V$.

## Properties

+-- {: .num_example }
###### Example
**([[induced representation of the trivial representation]])**

Let $G$ be a [[finite group]] and $H \overset{\iota}{\hookrightarrow} G$ a [[subgroup]]-inclusion. Then the [[induced representation]] in [[representation category|Rep(G)]] of the 1-[[dimension|dimensional]] [[trivial representation]] $\mathbf{1} \in Rep(H)$ is the [[permutation representation]] $k[G/H]$ of the [[coset]] [[G-set]] $G/H$:

$$
  \mathrm{ind}_H^G\big( \mathbf{1}\big)
  \;\simeq\;
  k[G/H]
  \,.
$$

This follows directly as a special case of the general formula for [[induced representations]] of [[finite groups]] ([this Example](induced+representation#InductionOfFiniteDimensionalRepresentationsOfFiniteGroups)).

=--

## Related concepts

* [[scalar]]

* [[alternating representation]]

* [[regular representation]]

[[!include homotopy type representation theory -- table]]

[[!redirects trivial representations]]

[[!redirects trivial action]]
[[!redirects trivial actions]]


trivial ring

+-- {: .rightHandSide}
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=--

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* table of contents
{:toc}

## Definition

The __trivial ring__, or __zero ring__, is the [[ring]] with a single [[element]], which is both $0$ and $1$.

One usually denote the trivial ring as $0$ or $\{0\}$, even though $1$ or $\{1\}$ would make as much sense.

## Properties

* The trivial ring is the only ring in which $0 = 1$, since if that is the case then it follows for all elements $x$ that $x = 1 x = 0 x = 0$.

* The trivial ring is the [[terminal object]] in [[Rings]]. It is both [[terminal object|terminal]] and [[initial object|initial]] (hence a [[zero object]]) in the category of [[nonunital rings]], but it is not initial in [[Rings]] (defined as the category of [[unital rings]] and unital ring homomorphisms).  In fact, there are no unital ring homomorphisms from the trivial ring to any nontrivial ring, hence it is a [[strict terminal object]] there.

* The trivial ring is an example of a [[trivial algebra]].

* Every ring is a [[0-truncated]] [[ring groupoid]], and the trivial ring is the terminal ring groupoid.

* If one takes the definition of a [[field]] from [LombardiQuitté2010](#LombardiQuitté2010), then the [[trivial ring]] is a [[terminal object]] in the [[category]] [[Field]] of [[fields]].

## See also

* [[zero object]]

## References

* {#LombardiQuitté2010} [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))

[[!redirects trivial ring]]
[[!redirects trivial rings]]
[[!redirects zero ring]]
[[!redirects zero rings]]

[[!redirects trivial ring groupoid]]
[[!redirects trivial ring groupoids]]
[[!redirects zero ring groupoid]]
[[!redirects zero ring groupoids]]


trivial subalgebra

The __trivial subalgebra__ of an [[gebra|algebra]] $A$ is the smallest [[subalgebra]] of $A$, the [[intersection]] of all subalgebras of $A$ (assuming that this is a subalgebra, otherwise there is no trivial subalgebra).  In the context of [[universal algebra]], this is the __empty subalgebra__ ---the [[empty subset]] of the [[underlying set]] of $A$--- if there are no constant ($0$-ary) operations; more generally, it\'s the set of all of the constants (in which case there is no empty subalgebra).

For example, the [[trivial subgroup]] of a [[group]] is $\{e\}$ (where $e$ is the [[identity element]] of the group), and the [[trivial subring]] of a [[ring]] is $\{\ldots, -2, -1, 0, 1, 2, \ldots \}$ (the set of all constants in the theory of a ring, not just the two listed in the usual presentation).  The [[trivial ideal]] of a ring, on the other hand, is just $\{0\}$, because here we are treating the ring as a [[module]] over itself.  (This is also the trivial sub-[[rng]].)

[[!redirects trivial subalgebra]]
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trivial topology

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## Idea

In [[sheaf and topos theory]], given any [[small category]], its *trivial topology* is the [[coverage]] whose covering families are the [[identity morphisms]], hence the [[Grothendieck topology]] for which only the [[sieves]] generated by [[identity morphisms]] are [[covering]] (i.e. those containing every morphism with the given [[codomain]]).

The [[sheaves]] for the trivial topology are precisely the [[presheaves]] on the [[underlying]] [[category]]. In this way every [[category of presheaves]] is realized as a [[category of sheaves]] and hence as a [[Grothendieck topos]].

## Related concepts

* [[canonical topology]]

* [[subcanonical topology]]

* [[Grothendieck topology]]

## Literature

* [[Francis Borceux]], Example 3.2.14.a in: *[[Handbook of Categorical Algebra]]*, Vol. 3: *Categories of Sheaves*, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525872](https://doi.org/10.1017/CBO9780511525872)]

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trivial torsor

Probably the easiest example of a [[torsor]] to understand is the trivial torsor in the topological case.

##Definition##

Given a space $B$ and a [[sheaf]] of groups, $G$ on $B$, the sheaf of _sets_ underlying $G$ has a natural left action by $G$, which is a sheaf morphism. This is transitive etc. and so gives a torsor, called the **trivial $G$-torsor**, denoted $T_G$.

It is very important to note that $T_G$ has $T_G(B)$ non-empty (i.e., $T_G$ has a 'global section'), since it is a group so must have an identity element. Conversely any $G$-torsor which has such a 'global section' is isomorphic to $T_G$.


trivial vector bundle

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## Idea

For $X$ a suitable [[space]] then a [[vector bundle]] over $X$ is called _trivial_ if there is no twist in how the [[fibers]] vary over it.

## Definition

For $X$ a [[topological space]], then a [[topological vector bundle]] $E \to X$ over a [[topological field]] $k$ is called _trivial_ if its total space is the [[product topological space]]

$$
  E = X \times k^n \overset{pr_1}{\longrightarrow} X
$$

with the [[topological vector space]] $k^n$, for some $n \in \mathbb{N}$. For $n = 1$, one also speaks of a _trivial [[line bundle]]_.

An [[isomorphism]] of vector bundles over $X$ of the form

$$
  E \longrightarrow X \times \mathbb{R}^n
$$

is called a _trivialization_ of $E$. If $E$ admits such an isomorphism, then it is called a _trivializable vector bundle._

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[[!redirects trivial line bundle]]
[[!redirects trivial line bundles]]

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[[!redirects trivializable topological line bundles]]


Troels Harmark

* [webpage](https://www.nbi.ku.dk/english/staff/?pure=en/persons/19668)

## Selected writings

On among other things, [[M2-M5 brane bound states]]:

* [[Troels Harmark]], Section 3.1 of _Open Branes in Space-Time Non-Commutative Little String Theory_, Nucl.Phys. B593 (2001) 76-98 ([arXiv:hep-th/0007147](https://arxiv.org/abs/hep-th/0007147))

* [[Troels Harmark]],  [[Niels A. Obers]], Section 5.1 of _Phase Structure of Non-Commutative Field Theories and Spinning Brane Bound States_, JHEP 0003 (2000) 024 ([arXiv:hep-th/9911169](https://arxiv.org/abs/hep-th/9911169))

On the *[[blackfold]]*-approach to describing [[black branes]]:

* [[Roberto Emparan]], [[Troels Harmark]], [[Vasilis Niarchos]], [[Niels A. Obers]], *Essentials of Blackfold Dynamics*, JHEP 1003 063 (2010) [[arXiv:0910.1601](https://arxiv.org/abs/0910.1601), doi:10.1007/JHEP03(2010)063]

* [[Roberto Emparan]], [[Troels Harmark]], [[Vasilis Niarchos]], [[Niels A. Obers]], *Blackfolds in Supergravity and String Theory*, JHEP 1108 (2011) 154 [[arXiv:1106.4428](https://arxiv.org/abs/1106.4428)doi:10.1007/JHEP08(2011)154]

## Related $n$Lab entries

* [[M2-brane]], [[M5-brane]]

* [[M2-M5 brane bound state]]

category: people


tropical geometry

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## Idea

Tropical geometry is often thought of as [[algebraic geometry]] over the [[tropical semiring]]. Many central results are combinatorial in nature, with relations to the (geometry and combinatorics of) polyhedra, [[matroid]]s, [[cluster algebra]]s and toric geometry. Recently it found applications in explaining [[mirror symmetry]] at a more fundamental level (see e.g. [Gross 11](#Gross11)).

Tropical algebraic geometry establishes and studies some very general principles to translate algebro-geometric problems into purely combinatorial ones.

*Example*
In algebraic geometry one often work with polynomials. In tropical geometry, these polynomials are
"tropicalized"  and this turns them into piecewise linear functions.

For instance: $f (x, y) = x^2 + y^2-1$.

This tropicalizes to $trop(f ) = x^2 \oplus y^2 \oplus 0 =
min(2x, 2y, 0)$, and this is a piecewise linear curve.

(To see this remember that in the [[tropical semiring]], the sum of two numbers is their minimum, and their product is their sum. $x^2 = x\oplus x$, $y^2 = y\oplus y$ and so $x^2+y^2$  converts to $min(2x,2y)$. The 0 will remain mysterious for the moment.  (If you cannot wait look at the AARMS notes listed below.))

## Related concepts

 * [[tropical semiring]]

##References

###Books and Lecture Notes

Textbook accounts/lecture notes include

*  [[Diane Maclagan]], [[Bernd Sturmfels]] _Introduction to tropical geometry_, [draft book](http://homepages.warwick.ac.uk/staff/D.Maclagan/papers/TropicalBook.pdf)

* {#Gross11} [[Mark Gross]], _Tropical geometry and mirror symmetry_, CBMS regional conf. ser. 114 (2011), based on the CBMS course in Kansas, [AMS book page](http://www.ams.org/bookstore-getitem/item=CBMS-114), [pdf](http://www.math.ucsd.edu/~mgross/kansas.pdf)

* G. Mikhalkin, _Tropical geometry_, book, early draft, [scribd](http://www.scribd.com/doc/47771116/Tropical-geometry-Grigory-Mikhalkin); _Tropical geometry and its applications_, Proc. Intern. Congr. Math.,
V. 2 (Madrid, 2006), Eur. Math. Soc., Zürich, 2006, 827–852 [djvu](http://www.mathunion.org/ICM/ICM2006.2/Main/icm2006.2.0827.0852.ocr.djvu) [pdf](http://www.mathunion.org/ICM/ICM2006.2/Main/icm2006.2.0827.0852.ocr.pdf)

* [[Diane Maclagan]], [AARMS Tropical Geometry](http://homepages.warwick.ac.uk/staff/D.Maclagan/AARMS/AARMSTropical.pdf):lecture notes from a four week graduate summer school on Tropical Geometry held at the University of New Brunswick in July/August 2008 under the auspices of the Atlantic Association for Research in the Mathematical Sciences (AARMS).

* MSRI introductory workshop on tropical geometry [page](http://www.msri.org/web/msri/scientific/workshops/show/-/event/Wm483), Aug 24-28, 2009 (with videos of the lectures)
* Erwan Brugallé, Kristin Shaw, _A bit of tropical geometry _, [arxiv/1311.2360](http://arxiv.org/abs/1311.2360)

### Collections of articles

* (CM580) Tropical geometry and integrable systems, [Contemp. Math. __580__](http://www.ams.org/bookstore-getitem/item=CONM-580), Amer. Math. Soc., Providence, RI, 2012
* _Tropical and idempotent mathematics_, [pdf](http://www.mccme.ru/tropical12/Tropics2012final.pdf), proceedings conf. Moscow 2012

###Papers and Preprints

* [[Dan Abramovich]], _Moduli of algebraic and tropical curves_, [arxiv/1301.0474](http://arxiv.org/abs/1301.0474)
* [[Dan Abramovich]], Lucia Caporaso, Sam Payne, _The tropicalization of the moduli space of curves_, [arxiv/1212.0373](http://arxiv.org/abs/1212.0373)

* I. Itenberg, G. Mikhalkin, _Geometry in the tropical limit_, [arXiv:
1108.3111](http://arxiv.org/abs/1108.3111)

* M. Einsiedler, [[M. Kapranov]], D. Lind, _Non-archimedean amoebas and tropical varieties_, [math.AG/0408311](http://arxiv.org/abs/math/0408311)
* [[Mikhail Kapranov]], _Thermodynamics and the moment map_, [arxiv/1108.3472](http://arxiv.org/abs/1108.3472)
* Walter Gubler, _A guide to tropicalizations_, [arxiv/1108.6126](http://arxiv.org/abs/1108.6126)
* E. Katz, _A tropical toolkit_, [math.AG/0610878](http://arxiv.org/abs/math/0610878) (Expo. Math. 27 (2009), No. 1, 1-36)
* Oleg Viro, _Hyperfields for tropical geometry I. hyperfields and dequantization_, [arxiv/1006.3034](http://arxiv.org/abs/1006.3034); _Tropical geometry and hyperfields_, talk at Mathematics - XXI century. PDMI 70th anniversary, [video](http://www.mathnet.ru/php/presentation.phtml?option_lang=eng&presentid=1286); _On basic concepts of tropical geometry_, Trudy Mat. Inst. Steklova __273__ (2011),  271–303
* Patrick Popescu-Pampu, Dmitry Stepanov, _Local tropicalization_, [arxiv/1204.6154](http://arxiv.org/abs/1204.6154)
* W. Gubler, _Tropical varieties for non-Archimedean analytic spaces_, Invent. Math. __169__ (2007), 321–376.
* David Speyer, [[Bernd Sturmfels]], _Tropical mathematics_, [math.CO/0408099](http://arxiv.org/abs/math/0408099); _The tropical Grassmannian_, Adv. Geom. 4 (2004) 389–411 [math.AG/0304218](https://arxiv.org/abs/math/0304218) [emis](http://emis.ams.org/journals/AG/4-3/4_389.pdf)
* Andreas Gathmann, _Tropical algebraic geometry_, [math.AG/0601322](http://arxiv.org/abs/math/0601322)
* [[Diane Maclagan]], _Polyhedral structures on tropical varieties_, [arXiv:1302.5372](http://arxiv.org/abs/1302.5372)
* Paul Johnson, _Hurwitz numbers, ribbon graphs, and tropicalization_, [arxiv/1303.1543](http://arxiv.org/abs/1303.1543) (pages 55-72 in CM580)
* Brugalle Erwan, Markwig Hannah, _Deformation of tropical Hirzebruch surfaces and enumerative geometry_, [arxiv/1303.1340](http://arxiv.org/abs/1303.1340)
* Qingchun Ren, Steven V Sam, Bernd Sturmfels, _Tropicalization of classical moduli spaces_, [arxiv/1303.1132](http://arxiv.org/abs/1303.1132)

* Martin Ulirsch, _Functorial tropicalization of logarithmic schemes: The case of constant coefficients_, [arxiv/1310.6269](http://arxiv.org/abs/1310.6269)
* Luis Felipe Tabera, _On real tropical bases and real tropical discriminants_, [arxiv/1311.2211](http://arxiv.org/abs/1311.2211)

* Simon Hampe, _Tropical linear spaces and tropical convexity_, [arxiv/1505.02045](http://arxiv.org/abs/1505.02045)

* Tyler Foster, _Introduction to adic tropicalization_, [arxiv/1506.00726](http://arxiv.org/abs/1506.00726)

* G. Mikhalkin, _Quantum indices of real plane curves and refined enumerative geometry _, [arxiv/1505.04338](http://arxiv.org/abs/1505.04338)

> We associate a half-integer number, called the quantum index, to algebraic curves in the real plane satisfying to certain conditions. The area encompassed by the logarithmic image of such curves is equal to $\pi^2$ times the quantum index of the curve. We use the quantum index to refine real enumerative geometry in a way consistent with the Block-G\"ottsche invariants from tropical enumerative geometry.

*  Diane Maclagan, Felipe Rincón, _Tropical ideals_, [arxiv/1609.03838](http://arxiv.org/abs/1609.03838)

* Kiumars Kaveh, Christopher Manon, _Khovanskii bases, Newton-Okounkov polytopes and tropical geometry of projective varieties_, [arxiv/1610.00298](https://arxiv.org/abs/1610.00298)

*  [[Alain Connes]], [[Caterina Consani]], _Geometry of the scaling site_, [arxiv/1603.03191](https://arxiv.org/abs/1603.03191)

* Keyvan Yaghmayi, _Geometry over the tropical dual numbers_, [arxiv/1611.05508](https://arxiv.org/abs/1611.05508)

An alternative algebraic framework for tropical mathematics (not based on semirings), "more compatible with valuation theory" has been proposed in

* Zur Izhakian, Manfred Knebusch, Louis Rowen, _Algebraic structures of tropical mathematics_, [arxiv/1305.3906](http://arxiv.org/abs/1305.3906)

Connections to diophantine integration (involving p-adic integration):

* Eric Katz, Joseph Rabinoff, David Zureick-Brown, _Diophantine and tropical geometry, and uniformity of rational points on curves_, [arxiv/1606.09618](http://arxiv.org/abs/1606.09618)

Mirror symmetry for correlation functions of tropical observables:

* Andrey Losev, Vyacheslav Lysov, _Tropical mirror symmetry: correlation functions_, [arXiv:2301.01687](https://arxiv.org/abs/2301.01687)

###Miscellaneous: MO questions, discussions, etc.

* MathOverflow : [Mikhalkin's tropical schemes versus Durov's tropical schemes](http://mathoverflow.net/questions/72613/mikhalkins-tropical-schemes-versus-durovs-tropical-schemes), [learning-tropical-geometry](http://mathoverflow.net/questions/84629/learning-tropical-geometry)

* [wikipedia page](http://en.wikipedia.org/wiki/Tropical_geometry)

* $n$Cafe: [Tight spans, Isbell completions and semi-tropical modules](http://golem.ph.utexas.edu/category/2013/01/tight_spans_isbell_completions.html)

category: geometry


tropical semiring

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# The tropical semiring
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## Definitions

The __tropical semiring__ is a [[semiring]] $(\mathbb{R}\cup \{\infty\},\oplus,\otimes)$ with addition $x\oplus y = min(x,y)$  and multiplication $x\otimes y = x+y$. (This is semiring in the sense of [[rig]], hence sometimes the __tropical rig__.)

[[tropical geometry|Tropical geometry]] is often thought as the algebraic geometry over the tropical semiring.

## Terminology

The tropical rig is also called the [[min-plus algebra]]. There is a related, in fact isomorphic rig called the [[max-plus algebra]]. (Some authors use the term 'tropical algebra' for the max-plus rather than the min-plus algebra.  The theories, of course, run in parallel, as each is the negative of the other.)

In his survey article, cited below, Pin uses the term for a wide range of similar idempotent semirings. For instance $\mathcal{M} = (\mathbb{N} \cup{\infty}, min, +)$ is a tropical semiring introduced by [[Imre Simon]] in 1978.

## Properties

* The tropical semiring is an example of an [[idempotent semiring]], since for all elements $x$, we have $x\oplus x=x$.

## Examples
 {#Examples}

In the min-plus algebra we have, for instance:

$$
  \begin{aligned}
    (5\oplus 6)\otimes 7
    &
    = min(5,6) + 7
    \\
    & =
    5 + 7
    \\
    & =
    12
    \,.
  \end{aligned}
$$

## Applications

Apart from applications in [[tropical geometry]], the min-plus and max-plus algebras have extensive use in providing algebraic models for simple [[discrete event system]]s related to timed [[Petri nets]].

## References

The use of the tropical algebra in [[discrete event systems]] is handled in many sources, e.g

* [[Jean-Eric Pin]], _Tropical semirings_, Publ. Newton Inst. 11 (1998) 50-69 [[hal:00113779](), [pdf](https://hal.archives-ouvertes.fr/hal-00113779/file/Tropical.pdf)

* [[Stéphane Gaubert]], of INRIA Saclay (1999) [[pdf](http://amadeus.inria.fr/gaubert/PAPERS/POLY12-02-1999.pdf)]

Book collection of articles on idempotent semirings:

* J. Gunawardena (ed.), *Idempotency*, Cambridge University Press, 2001,

An original source:

* [[Imre Simon]], (1978), _Limited Subsets of a Free Monoid_, in Proc. 19th Annual Symposium on Foundations of Computer Science, Piscataway, N.J., Institute of Electrical and Electronics Engineers, 143–150 ([doi:10.1109/SFCS.1978.21](https://doi.org/10.1109/SFCS.1978.21))

See also:

* Wikipedia, *[Tropical semiring](https://en.wikipedia.org/wiki/Tropical_semiring)*

and the first few pages of:

* {#Lawvere92} [[William Lawvere]], *Introduction to Linear Categories and Applications*, course lecture notes (1992) [[pdf](https://github.com/mattearnshaw/lawvere/blob/192dac273e8bf352f307f87b9ec4fe8ef7dc85b9/pdfs/1992-introduction-to-linear-categories-and-applications.pdf), [[Lawvere-LinearCategories.pdf:file]]]

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