# Zhen Lin Low

## Thesis

###### Categories of spaces built from local models

Many of the classes of objects studied in geometry are defined by first choosing a class of nice spaces and then allowing oneself to glue these local models together to construct more general spaces. The most well-known examples are manifolds and schemes. The main purpose of this thesis is to give a unified account of this procedure of constructing a category of spaces built from local models and to study the general properties of such categories of spaces. The theory developed here will be illustrated with reference to examples, including the aforementioned manifolds and schemes.

For concreteness, consider the passage from commutative rings to schemes. There are three main steps: first, one identifies a distinguished class of ring homomorphisms corresponding to open immersions of schemes; second, one defines the notion of an open covering in terms of these distinguished homomorphisms; and finally, one embeds the opposite of the category of commutative rings in an ambient category in which one can glue (the formal duals of) commutative rings along (the formal duals of) distinguished homomorphisms. Traditionally, the ambient category is taken to be the category of locally ringed spaces, but following Grothendieck, one could equally well work in the category of sheaves for the large Zariski site – this is the so-called ‘functor of points approach’. A third option, related to the exact completion of a category, is described in this thesis.

The main result can be summarised thus: categories of spaces built from local models are extensive categories with a class of distinguished morphisms, subject to various stability axioms, such that certain equivalence relations (defined relative to the class of distinguished morphisms) have pullback-stable quotients; moreover, this construction is functorial and has a universal property.

## Papers

###### Cocycles in categories of fibrant objects

We establish that a category of fibrant objects (in the sense of Brown) admits a Dwyer-Kan homotopical calculus of right fractions. This is done using a homotopical calculus of cocycles, which is an auxiliary structure that can be defined on every category of fibrant objects. As an application, we deduce some non-abelian versions of the Verdier hypercovering theorem.

###### 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
*L*_{ω₁, 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 every small model category that satisfies certain size conditions can be completed to yield a combinatorial model category, and conversely, every combinatorial model category arises in this way. We will also see that these constructions preserve right properness and compatibility with simplicial enrichment. Along the way, we establish some technical results on the index of accessibility of various constructions on accessible categories, which may be of independent interest.

###### 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

###### Categories of spaces built from local models

Many of the classes of objects studied in geometry are defined by first choosing a class of nice spaces and then allowing oneself to glue these local models together to construct more general spaces. The most well-known examples are manifolds and schemes. Such categories of spaces built from local models are extensive categories with a class of distinguished morphisms – local diffeomorphisms in the case of smooth manifolds, local isomorphisms in the case of schemes – subject to various stability axioms, such that certain special equivalence relations (defined relative to the class of distinguished morphisms) have pullback-stable quotients. The construction of such categories essentially amounts to adjoining quotients for special equivalence relations; as such, it is functorial and has a universal property, which I will explain in this talk.

###### Another viewpoint on cartesian theories

As is well known, cartesian theories have essentially the
same expressive power as finite limit sketches, but some details are lost in the
translation: for instance, a cartesian theory has an underlying algebraic
theory, but this disappears after passing to the syntactic category. The gap can
be bridged by introducing the notion of cartesian hyperdoctrine. Such a
structure gives rise to a category of fibrant objects, and in the case of the
cartesian hyperdoctrine generated by a cartesian theory *T*, its homotopy
category is the syntactic category of *T*.

###### Cocycles in categories of fibrant objects

Jardine introduced a very general notion of cocycle in categories with weak equivalences: very simply, a cocycle is a span where one of the legs is a weak equivalence. I will try to explain how 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. This can be interpreted as a non-abelian version of Verdier’s hypercovering theorem.

###### Generalising the functor of points approach

The passage from commutative rings to schemes has three main steps: first, one identifies a distinguished class of ring homomorphisms corresponding to open immersions of schemes; second, one defines the notion of an open covering in terms of these distinguished homomorphisms; and finally, one embeds the opposite of the category of commutative rings in an ambient category in which one can glue (the formal duals of) commutative rings along (the formal duals of) distinguished homomorphisms. Traditionally, the ambient category is taken to be the category of locally ringed spaces, but following [Demazure and Gabriel], one could equally well work in the category of sheaves for the large Zariski site – this is the so-called ‘functor of points approach’.

The three procedures described above can be generalised to
other contexts. The first step essentially amounts to reconstructing the class
of open embeddings from the class of closed embeddings. Once we have a suitable
class of open embeddings, the class of open coverings is a subcanonical
Grothendieck pretopology. We then define a notion of `charted space' in the
category of sheaves. This gives a uniform way of defining locally Hausdorff
spaces, schemes, locally finitely presented *C*^{∞}-schemes
etc. as special sheaves on their respective categories of local models, taking
as input just the class of closed embeddings. We can also get many variations
on manifolds by skipping the first step and working directly with a given class
of open embeddings.

###### 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.