Noam Zeilberger imploid (Rev #3)

Contents

Definition

An imploid is a preordered set (X,)(X,\le) equipped with:

  1. a binary operation \multimap (called implication) which is contravariant in its first argument and covariant in its second argument:
    (1)a 2a 1b 1b 2 a 1b 1a 2b 2 \array{ \arrayopts{\rowlines{solid}} a_2 \le a_1 \quad b_1 \le b_2 \\ a_1\multimap b_1 \le a_2\multimap b_2 }

    and which satisfies the composition law:

    (2)bc(ab)(aimpc) b \multimap c \le (a \multimap b) \multimap (a \imp c)

    for all a,b,cXa,b,c \in X; and

  2. an element IXI \in X (called unit) satisfying identity and unit laws:
    (3)Iaa I \le a \multimap a
    (4)Iaa I \multimap a \le a

    for all aXa \in X.

An imploid is said to be undirected if the underlying preorder is symmetric, so that (2)-(4) also hold in the converse direction. On the other hand, the imploid is said to be commutative if it additionally satisfies an exchange law:

(5)a(bc)b(ac) a \multimap (b \multimap c) \le b \multimap (a \multimap c)

for all a,b,cXa,b,c \in X. Finally, in any imploid we have that aba \le b implies IabI \le a \multimap b; we say that the imploid is normalized if the converse is also true.

Examples

From a group

Any group G=(G,,I,() 1)G = (G,\cdot,I,(-)^{-1}) can be seen as an undirected (normalized) imploid, by taking the preorder to be the equality relation and defining ab:=ba 1a \multimap b \mathbin{:=} b\cdot a^{-1}. In this case, the composition law (2) holds because

bc=cb 1=ca 1ab 1=(ab)(ac) b \multimap c = c b^{-1} = c a^{-1} a b^{-1} = (a \multimap b) \multimap (a \multimap c)

while the identity and unit laws (3) and (4) likewise follow immediately from the group axioms.

Relating imploids and dmonoids

(See article dmonoid for the definition.)

Via existence of left/right adjoints

It is well-known that if CC has the structure of a monoidal category and for every object aCa \in C, the tensor functor a:CC-\otimes a : C \to C has a right adjoint a[a,]-\otimes a\dashv [a,-], then CC can also be given the structure of a closed category. But what about the converse? In other words, if CC is a closed category and every functor [a,][a,-] has a left adjoint, is CC also a monoidal category? It turns out that the answer is no, because in general the associator maps

α a,b,c:(ab)ca(bc)\alpha_{a,b,c} : (a \otimes b)\otimes c \to a \otimes (b\otimes c)

are not necessarily invertible, i.e., they only establish associativity up to natural transformation, and not up to natural isomorphism.

Proposition

(cf. Street 2013): If M=(M,,,I)M = (M,\le,\cdot,I) is a dmonoid for which each operation a:MM-\cdot a : M \to M (aMa \in M) has a right adjoint operation a:MMa \multimap - : M \to M, then MM is also an imploid. Conversely, if P=(P,,,I)P = (P,\le,\multimap,I) is an imploid for which each operation a:PPa \multimap - : P \to P has a left adjoint operation a:PP-\cdot a : P \to P, then PP is also a dmonoid.

Dual yoneda embeddings

Proposition

Any dmonoid M=(M,,,I)M = (M,\le,\cdot,I) induces an imploid M M^\downarrow whose elements are the downsets of MM ordered by inclusion, and where the implication and unit are defined by:

AB:={ma.aAmaB} A \multimap B \mathbin{:=} \{\, m \mid \forall a.\, a\in A \Rightarrow m\cdot a \in B\,\}
I :={mmI} I^\downarrow \mathbin{:=} \{\, m \mid m \le I \,\}
Proposition

Any imploid P=(P,,,I)P = (P,\le,\multimap,I) induces a dmonoid P P^\uparrow whose elements are the upsets of PP ordered by reverse inclusion, and where the multiplication and unit are defined by:

AB:={pb.bpAbB} A \cdot B \mathbin{:=} \{\, p \mid \exists b.\, b\multimap p \in A \wedge b\in B\,\}
I :={pIp} I^\uparrow \mathbin{:=} \{\, p \mid I \le p\,\}

References

  • B. J. Day and M. L. Laplaza, On Embedding Closed Categories, Bull. Austral. Math. Soc. 18 (1978), 357-371.

  • Ross Street. Skew-closed categories. Journal of Pure and Applied Algebra 217(6) (June 2013), arXiv:1205.6522

  • Kornel Szlachanyi. Skew-monoidal categories and bialgebroids. arXiv:1201.4981

Revision on May 5, 2017 at 14:02:27 by Noam Zeilberger?. See the history of this page for a list of all contributions to it.