Information about winter 2022 seminars can be found here.
Abstract:
Actions of a group on a set can be categorified to actions of a
group on a category. The fixed point set becomes a fixed point category. I
will give examples coming from representation theory, topology and physics.
If time permits, I will sketch a generalization to actions of 2-groups on
2-categories.
Abstract:
A distributive law, in the sense of JM Beck, of a monad $M$ over a monad $A$, is a
2-cell
$MA \to AM$, subject to some equations ensuring precisely that the composite $AM$
carries
an obvious monad structure depending on the structures of $M$ and of $A$. The
notion is not symmetric in $M$ and $A$. In particular, an object $k$ carries an
$AM$-algebra structure if and only if $k$ carries an $A$-algebra structure $a:Ak \to k$;
$k$ carries an $M$-algebra ​structure $m:Mk \to k$; and $a:Ak \to k$ is a homomorphism of
$M$-algebras.
These very general ideas are applicable to 2-monads on the 2-category $\mathrm{CAT}$,
whereon some
monads, known as KZ-monads, have the property that structures are left adjoint
to units, while other monads, known as coKZ-monads, have the property that
structures are right adjoint to units. It follows that a category $K$ either
carries a [co]KZ-structure or it doesn't.
In broad strokes, the algebras for a KZ-monad $A$ [coKZ-monad $M$] are categories
having certain colimits, we could call them $A$-colimits [certain limits, we
could call them $M$-limits].
If there is a distributive law $MA \to AM$ (there either is one or there isn't
one) then a category
$K$ is an $AM$-algebra iff $K$ has $A$-colimits; $K$ has $M$-limits; and $AK \to K$ (left
adjoint to the unit $K \to AK$) preserves $M$-limits. Preservation of $M$-limits by
$AK \to K$ provides the definition of
"$M$-limits distribute over $A$-colimits".
In other broad strokes, the KZ-monads live in the construction $K \mapsto \mathrm{CAT}(K^
{\mathrm{op}},\mathbf{Set})$ and the coKZ-monads live in the construction $K \mapsto \mathrm{CAT}(K,\mathbf{Set})^{\mathrm{op}}$.
These constructions are not monads on $\mathrm{CAT}$. However, a locally small category $K$
is said to be total(ly cocomplete) if the Yoneda functor $Y \to \mathrm{CAT}(K^{\mathrm{op}}, \mathbf{Set})$
has a left adjoint $X$ and totally distributive if there is an $F \dashv X \dashv Y$. The point
of these talks is to explain why totally distributive categories warrant the
name, given the orthodoxy of the first two paragraphs, and reach towards a
characterization of such. By the way, for any small category $S$, $\mathbf{Set}^{S^{\mathrm{op}}}$ is
totally distributive but $({\mathbf{Set}^S})^{\mathrm{op}}$, while total, is not totally
distributive.
Abstract:
In the first talk I have tried to explain why a locally small category $K$, with
Yoneda functor $Y:K \to \mathrm{CAT}(K^{\mathrm{op}}, \mathbf{Set})$ and adjunctions
$F \dashv X \dashv Y$, warrants the
name "totally distributive category — TD". Namely: $X$ provides the colimits
for a totally cocomplete catgory, in the sense of Street and Walters; $F(\dashv X)$
ensures that $K$ has the limits of a cototally complete category; and $(F\dashv)X$ tells
us that the provider of total colimits preserves cototal limits. The theory of
monads suggests it is correct to say that such $K$ are those for which cototal
limits distribute over total colimits. In this talk I want to report on efforts
to characterize such $K$.
I think it is clear that totality challenges the orthodoxy of which [co]limits
really merit the name "small [co]limits". The characterization of TD categories
challenges the orthodoxy of which category-like structures also merit the
adjective "small". Koslowski introduced a category-like structure, known as a
"taxon". A taxon ${\bf T}$ admits a hat construction for which $\hat{\bf T}$ is a
category. Several years ago, I showed that every TD $K$ gives rise to a locally
small taxon, $\tilde{K}$, for which there is an equivalence $K \Rightarrow \hat{\tilde{K}}$.
Now since $K$ is locally small, $\hat{\tilde{K}}$ is locally small. In light of a
famous result of Freyd and Street, I propose to say that $\tilde{K}$ is "Freyd/
Street small". Yet, the set of objects of $\tilde{K}$ is the set of objects of $K$.
Rather surprisingly, Freyd/Street smallness appears to be a notion of
smallness that allows one to prove the sort of results that one expects for
small category-like stuctures, directly, — without reference to the size of
the set of objects.
Updated August 18, 2022 by Frank Fu