I am currently a PhD student in Homotopy Theory at the Department of Mathematics of The University of Western Ontario, under the supervision of professor Dan Christensen.

The research group in which I work also includes Chris Kapulkin (Assistant Professor), James Richardson (PhD student) and Luis Scoccola (PhD student).

### E-mail: mvergura@uwo.ca

Middlesex College
University of Western Ontario
Room 109
1151 Richmond Street

## Mathematical Interests

My main interests in Mathematics include:
• Homotopy Type Theory

• Model Categories and Abstract Homotopy Theory

• Category Theory and Topos Theory

## Short CV

Here you can find an extended Curriculum Vitae.

## Writings

• Models for $(\infty,1)$-categories. Notes for a talk I gave for a seminar in Higher Category Theory at UWO. I give here a survey of complete Segal spaces, simplicial and relative categories as presentations of $(\infty,1)$-categories.

• A brief survey on Kan Extensions. Notes for a talk I gave in a student learning group at UWO. The title is pretty much self-explanatory.

• Homotopical Structures in Dependent Type Theory. Written project for a course in Categorical Logic I took during Fall 2015 at the University of Western Ontario. Given a dependent type theory with identity types $\mathbb{T}$, we show how to endow the syntactic category of $\mathbb{T}$ with a weak factorization system and with a fibration-category structure. If $\mathbb{T}$ also has rules for a suitable higher inductive context, one can pass this fibration category structure to a pre-model category structure. Our main references are [GG08], [Gar09], [Lum11] and [AKL15].

• A Giraud-Type Theorem for Model Topoi. My Master thesis. Following the unpublished work of C. Rezk, Toposes and homotopy toposes, we present a formulation of the notion of model topos, intended as a model-categorical version of the classical concept of Grothendieck topos. Such a definition is sensible enough to establish a Giraud-type theorem for model topoi. There is a little gap still present in some proofs of Chapter 6 (cfr footnote 2 at page 139 and footnotes from 3 to 5 in the following pages), on which I may try to work further at some point.

• Localization Theory in Triangulated Categories: a (brief, incomplete and probably superficial) overview. A short journey into the subject, coming from an essay I submitted as part of the application for a PhD position. I spend here few words about localization theory in categories and in triangulated categories and give some connections with recollements, torsion pairs and the telescope conjecture. This text is nothing more than a humble, minuscule drop in the mare magnum of the subject, but someone may perhaps find something useful in it.

• Homotopy (Co)limits: a brief survey. Notes for a talk introducing homotopy (co)limits in model categories, given at the Algebraic Topology Student Seminar, organized by the Radboud Topology Group at the Radboud University Nijmegen (Netherlands). We present here a short overview of possible different definitions for the notions of homotopy limits and homotopy colimits in the context of model categories, also providing suitable (and hopefully enough) context to compare them and show their equivalence when this comparison makes sense. We focus in particular on three distinct approaches, namely the one involving derived functor, the homotopical approach and the simplicial version.

• Aggiunzioni in Topologia (in Italian). Notes for a talk on some examples of adjunctions in Topology, given to peer students with little background in category theory.

## Teaching at the University of Western Ontario

### Fall 2016-Winter 2018:

• Teaching Assistant for the course ''Math 1600A: Linear Algebra'' (Fall 2016 course webpage - the course has basically been the same for the other terms too). Tasks for this course include: running and preparing tutorials, marking quizzes and exams.

### Fall 2015:

• Teaching Assistant for the course ''Math 1600A: Linear Algebra I''. Tasks for this course included: running and preparing tutorials, marking quizzes and exams.
• Help Centre: helping first-year undergraduate students with their courses in Mathematics, mainly in calculus and linear algebra.