Contents

# Contents

## Definitions

###### Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be categories with pullbacks. A functor $T: \mathcal{C} \to \mathcal{D}$ is taut if $T$ preserves inverse images, i.e., pullbacks of the form

$\array{ P & \to & B \\ \downarrow & & \downarrow \\ A & \underset{i}{\to} & C }$

where $i$ is monic. (This implies in particular that $T$ preserves monos.)

###### Definition

Let $\mathcal{D}$ be a category with pullbacks, and let $\eta: S \to T$ be a natural transformation between functors $S, T: \mathcal{C} \to \mathcal{D}$. Then $\eta$ is taut if the naturality square

$\array{ S A & \stackrel{\eta A}{\to} & T A \\ \mathllap{S i} \downarrow & & \downarrow \mathrlap{T i} \\ S B & \underset{\eta B}{\to} & T B }$

is a pullback for every monomorphism $i$ in $\mathcal{C}$.

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

## Examples

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

###### Proposition

Let $T: \mathcal{C} \to \mathcal{D}$ be a functor that preserves weak pullbacks, and assume epis in $D$ are regular. Then $T$ is taut.

###### Proof

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

$\array{ A & \stackrel{1_A}{\to} & A \\ \mathllap{1_A} \downarrow & & \downarrow \mathrlap{i} \\ A & \underset{i}{\to} & B }$

is a pullback. By hypothesis, the canonical map $\phi: T(A) \to T(A) \times_{T(B)} T(A)$ is (regular) epic, but it is also monic because its composition with either projection $T(A) \times_{T(B)} T(A) \to T(A)$ is the identity. Therefore $\phi$ is an isomorphism, i.e., applying $T$ to the displayed pullback is a pullback, and this forces $T(i)$ to be monic.

Now if

$\array{ P & \stackrel{j}{\to} & B \\ \mathllap{q} \downarrow & & \downarrow \mathrlap{p} \\ A & \underset{i}{\to} & C }$

is a pullback with $i$ monic, we have that $j$ and therefore $T(j)$ is monic. The canonical map $\phi: T(P) \to T(A) \times_{T(C)} T(B)$ is (regular) epic, but also monic, since the mono $T(j)$ factors through it. Thus $\phi$ 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 $Set$ 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 $Set$ 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 $P \circ P^{op}: Set \to Set$ is not taut.

## Applications

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

• Tautness assumptions play a role in viewing relational $T$-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 $T$ on $Set$ (such as the ultrafilter monad) induces a monad $\bar{T}$ 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 $HKl(Rel, \bar{T})$, and a $\bar{T}$-multicategory in $Rel$ is by definition a monoid in $HKl(Rel, \bar{T})$. In the special case $T = \beta$, the ultrafilter monad, this concept recapitulates Barr’s notion of relational $\beta$-module as synonym of “topological space”. This can be generalized further by working with a virtual double category of “$V$-matrices” where $V$ is a completely distributive quantale ($Rel$ being the case $V = \mathbf{2}$). Again with $T$ a taut monad, one can define a virtual double category $HKl(V\text{-}Mat, \bar{T})$ 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 “$(T, V)$-algebras”.)

• 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)