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taut functor

Contents

Contents

Definitions

Definition

Let π’ž\mathcal{C} and π’Ÿ\mathcal{D} be categories with pullbacks. A functor T:π’žβ†’π’ŸT: \mathcal{C} \to \mathcal{D} is taut if TT preserves inverse images, i.e., pullbacks of the form

P β†’ B ↓ ↓ A β†’i C\array{ P & \to & B \\ \downarrow & & \downarrow \\ A & \underset{i}{\to} & C }

where ii is monic. (This implies in particular that TT preserves monos.)

Definition

Let π’Ÿ\mathcal{D} be a category with pullbacks, and let Ξ·:Sβ†’T\eta: S \to T be a natural transformation between functors S,T:π’žβ†’π’ŸS, T: \mathcal{C} \to \mathcal{D}. Then Ξ·\eta is taut if the naturality square

SA β†’Ξ·A TA Si↓ ↓Ti SB β†’Ξ·B TB\array{ S A & \stackrel{\eta A}{\to} & T A \\ \mathllap{S i} \downarrow & & \downarrow \mathrlap{T i} \\ S B & \underset{\eta B}{\to} & T B }

is a pullback for every monomorphism ii in π’ž\mathcal{C}.

A monad (T,m,u)(T, m, u) on a category with pullbacks is taut if TT is a taut functor and m,um, u are taut transformations.

Examples

A number of examples of taut functors can be deduced by applying the following observation.

Proposition

Let T:π’žβ†’π’ŸT: \mathcal{C} \to \mathcal{D} be a functor that preserves weak pullbacks, and assume epis in DD are regular. Then TT is taut.

Proof

In the first place, TT preserves monos. For i:A→Bi: A \to B is monic if and only if

A β†’1 A A 1 A↓ ↓i A β†’i B\array{ A & \stackrel{1_A}{\to} & A \\ \mathllap{1_A} \downarrow & & \downarrow \mathrlap{i} \\ A & \underset{i}{\to} & B }

is a pullback. By hypothesis, the canonical map Ο•:T(A)β†’T(A)Γ— T(B)T(A)\phi: T(A) \to T(A) \times_{T(B)} T(A) is (regular) epic, but it is also monic because its composition with either projection T(A)Γ— T(B)T(A)β†’T(A)T(A) \times_{T(B)} T(A) \to T(A) is the identity. Therefore Ο•\phi is an isomorphism, i.e., applying TT to the displayed pullback is a pullback, and this forces T(i)T(i) to be monic.

Now if

P β†’j B q↓ ↓p A β†’i C\array{ P & \stackrel{j}{\to} & B \\ \mathllap{q} \downarrow & & \downarrow \mathrlap{p} \\ A & \underset{i}{\to} & C }

is a pullback with ii monic, we have that jj and therefore T(j)T(j) is monic. The canonical map Ο•:T(P)β†’T(A)Γ— T(C)T(B)\phi: T(P) \to T(A) \times_{T(C)} T(B) is (regular) epic, but also monic, since the mono T(j)T(j) factors through it. Thus Ο•\phi is an isomorphism, which completes the proof.

A similar proof shows that weakly cartesian natural transformations are also taut.

As a result of this proposition,

  • Any cartesian functor is (trivially) taut.

  • The ultrafilter endofunctor on SetSet is taut. (See here for a proof that the ultrafilter functor preserves weak pullbacks.) In fact, the ultrafilter monad is taut.

  • Similarly, the filter monad on SetSet is taut.

  • The covariant power set monad, whose algebras are sup-lattices, is taut.

  • An analytic endofunctor induced by a species is taut. Furthermore, a morphism of species induces a weakly cartesian transformation between the corresponding analytic functors, thus a fortiori a taut transformation. In particular, an analytic monad is taut.

As an exception, we have

  • The double (contravariant) power set functor P∘P op:Setβ†’SetP \circ P^{op}: Set \to Set is not taut.

Applications

  • Paul Taylor has made tautness of TT a central assumption in his account of induction via well-founded coalgebras over TT. See chapter VI of his book.

  • Tautness assumptions play a role in viewing relational TT-algebras and related structures as generalized multicategories in the sense of Cruttwell-Shulman. In the prototypical case of relational beta-modules, there is a virtual double category of relations. A taut monad TT on SetSet (such as the ultrafilter monad) induces a monad TΒ―\bar{T} on this virtual double category (that is, a monad in an appropriate 2-category of virtual double categories). From there, one can define a horizontal Kleisli construction which is another virtual double category HKl(Rel,TΒ―)HKl(Rel, \bar{T}), and a TΒ―\bar{T}-multicategory in RelRel is by definition a monoid in HKl(Rel,TΒ―)HKl(Rel, \bar{T}). In the special case T=Ξ²T = \beta, the ultrafilter monad, this concept recapitulates Barr’s notion of relational Ξ²\beta-module as synonym of β€œtopological space”. This can be generalized further by working with a virtual double category of β€œVV-matrices” where VV is a completely distributive quantale (RelRel being the case V=2V = \mathbf{2}). Again with TT a taut monad, one can define a virtual double category HKl(V-Mat,TΒ―)HKl(V\text{-}Mat, \bar{T}) and then define generalized multicategories as before. (These were studied in a series of articles by Clementino, Hofmann, Tholen, Seal and others under the name β€œ(T,V)(T, V)-algebras”.)

References

  • Paul Taylor, Practical Foundations of Mathematics, Cambridge University Press (1999).
  • Maria Manuel Clementino, Dirk Hofmann, and Walter Tholen, One Setting for All: Metric, Topology, Uniformity, Approach Structure. (pdf)
  • Gavin J. Seal, Canonical and op-canonical lax algebras, Theory and Applications of Categories, 14 (2005), 221–243. (web)

Last revised on February 28, 2019 at 11:44:28. See the history of this page for a list of all contributions to it.