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
where $i$ is monic. (This implies in particular that $T$ preserves monos.)
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
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.
A number of examples of taut functors can be deduced by applying the following observation.
Let $T: \mathcal{C} \to \mathcal{D}$ be a functor that preserves weak pullbacks, and assume $D$ has pullbacks. Then $T$ is taut.
In the first place, $T$ preserves monos. For $i: A \to B$ is monic if and only if
is a pullback. By hypothesis, the canonical map $\phi: T(A) \to T(A) \times_{T(B)} T(A)$ is a split epimorphism (see here). 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
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 split 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
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 Barrβs notion of relational $\beta$-module as synonym of βtopological spaceβ. This can be generalized further by working with a virtual double category of β$V$-matricesβ 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 β$(T, V)$-algebrasβ.)
Last revised on June 6, 2024 at 18:37:42. See the history of this page for a list of all contributions to it.