Zhen Lin Low

Department of Pure Mathematics and Mathematical Statistics
Centre for Mathematical Sciences
Cambridge
CB3 0WB
United Kingdom
Trinity Hall
Cambridge
CB2 1TJ
United Kingdom
E-mail address:

About me

I am a research student of Peter Johnstone. My research is focused on category theory, especially homotopical algebra and categorical logic.


Papers

Cocycles in categories of fibrant objects

Dwyer and Kan developed a homotopical version of the calculus of fractions in order to get a handle on simplicial localisations of categories. We use this to show that, for a category of fibrant objects (in the sense of Brown), Jardine’s cocycle categories functorially compute the homotopy type of the hom-spaces in the simplicial localisation. As an application, we deduce a non-abelian version of Verdier’s hypercovering theorem suggested by Rezk.

From fractions to complete Segal spaces

We show that the Rezk classification diagram of a relative category admitting a homotopical version of the two-sided calculus of fractions is a Segal space up to Reedy-fibrant replacement. This generalizes the result of Rezk and Bergner on the classification diagram of a closed model category, as well as the result of Barwick and Kan on the classification diagram of a partial model category.

Revisiting function complexes and simplicial localisation

In this paper three results are established: firstly, that the homotopy function complexes of Dwyer and Kan can be defined as certain total right derived functors; secondly, that they functorially compute the homotopy type of the hom-spaces in the simplicial localisation; and thirdly, that they can be computed by fibrant replacements in a suitable left Bousfield localisation of the projective model structure on simplicial presheaves.

Internal and local homotopy theory

There is a well-established homotopy theory of simplicial objects in a Grothendieck topos, and folklore says that the weak equivalences are axiomatisable in the geometric fragment of Lω₁, ω. We show that it is in fact a theory of presheaf type, i.e. classified by a presheaf topos. As a corollary, we obtain a new proof of the fact that the local Kan fibrations of simplicial presheaves that are local weak homotopy equivalences are precisely the morphisms with the expected local lifting property.

The heart of a combinatorial model category

We show that small model categories satisfying certain size conditions can be completed to yield a combinatorial model category, and conversely, that every combinatorial model category arises in this way. We also show that these constructions preserve right properness and compatibility with simplicial enrichment.

The homotopy bicategory of (∞, 1)-categories

Evidence is given for the correctness of the Joyal–Riehl–Verity construction of the homotopy bicategory of the (∞, 2)-category of (∞, 1)-categories; in particular, it is shown that the analogous construction using complete Segal spaces instead of quasicategories yields a bicategorically equivalent 2-category.

Universes for category theory

The Grothendieck–Verdier universe axiom asserts that every set is a member of some set-theoretic universe U that is itself a set. One can then work with entities like the category of all U-sets or even the category of all locally U-small categories, where U is an “arbitrary but fixed” universe, all without worrying about which set-theoretic operations one may legitimately apply to these entities. Unfortunately, as soon as one allows the possibility of changing U, one also has to face the fact that universal constructions such as limits or adjoints or Kan extensions could, in principle, depend on the parameter U. We will prove this is not the case for adjoints of accessible functors between locally presentable categories (and hence, limits and Kan extensions), making explicit the idea that "bounded" constructions should not depend on the choice of U.


Slides

The heart of a combinatorial model category

It is well known that the free completion of a small κ-cocomplete category under small κ-filtered colimits is a locally κ-presentable category in the sense of Gabriel and Ulmer. Lurie has proved an analogous theorem in the world of quasicategories, and this leads to a natural question: can every small model category be completed (in some sense) to yield a combinatorial model category, and does every combinatorial model category arise in this fashion? Perhaps unsurprisingly, the answer is affirmative.

Accessible functors and inaccessible cardinals

The Grothendieck–Verdier universe axiom asserts that every set is a member of some set-theoretic universe U that is itself a set. One can then work with entities like the category of all U-sets or even the category of all locally U-small categories, where U is an “arbitrary but fixed” universe, all without worrying about which set-theoretic operations one may legitimately apply to these entities. Unfortunately, as soon as one allows the possibility of changing U, one also has to face the fact that universal constructions such as limits or adjoints or Kan extensions could, in principle, depend on the parameter U. The purpose of this talk is to explain how one can prove that this is not the case, at least in the case of adjoints for accessible functors between locally presentable categories (and hence, limits and Kan extensions), making explicit the idea that “bounded” constructions should not depend on the choice of U.

Toposes as localic and spatial groupoids

It is often said that a Grothendieck topos is like (the category of sheaves on) a generalised space where points can have non-trivial automorphisms – so something like an orbifold. This turned out to be true in a very precise sense, as explained in the [1984] monograph of Joyal and Tierney: every bounded topos is indeed the category of equivariant sheaves on a localic groupoid. A later result of Butz and Moerdijk [1997, 1998] showed that certain Grothendieck toposes even admit representations as topological groupoids.