A1 Regular and Cartesian Closed Categories A2 Toposes - Basic Theory A3 Allegories A4 Geometric Morphisms - Basic Theory B1 Fibrations and Indexed Categories B2 Internal and Locally Internal Categories B3 Toposes over a base B4 BTop/S as a 2-Category

REVIEWS-Sketches of an elephant: A topos theory compendium

/pdf/enriched-categories-internal-categories-and-change-of-base-53se9jn6ja.pdf

Enriched Categories, Internal Categories and Change of Base

We define the notion of exact completion with respect to an existential elementary doctrine. We observe that the forgetful functor from the 2-category of exact categories to existential elementary doctrines has a left biadjoint that can be obtained as a composite of two others. Finally, we conclude how this notion encompasses both that of the exact completion of a regular category as well as that of the exact completion of a category with binary products, a weak terminal object and weak pullbacks.

/pdf/unifying-exact-completions-qi90i57d77.pdf

Unifying Exact Completions

We define an elementary higher topos that simultaneously generalizes an elementary topos and higher topos. Then we show it satisfies classical topos theoretic properties, such being locally Cartesian closed and descent. Finally we show we can classify univalent maps in an elementary higher topos.

A theory of elementary higher toposes

This is the author's PhD thesis. It is a contribution to categorical logic, in particular to the theory of realizability toposes. 
While the tools of categorical logic have proven very successful in analyzing and organizing proof theoretic realizability interpretations, it was remarked by experts (notably Peter Johnstone) that the field of realizability toposes itself was not clearly delineated, and lacked a powerful theory analogous to that of Grothendieck toposes. 
The present work sets out to remedy this situation to a certain extent. We argue that realizability toposes are best understood using Grothendieck fibrations, and develop a framework of fibrational cocompletions, which allows to view certain constructions from realizability in precise analogy to constructions of presheaf and sheaf toposes. Using these techniques, and a class of posetal fibrations that we call uniform preorders, we are able to give an extensional characterization of partial combinatory algebras and of the realizability toposes that are constructed from these algebras. 
Striving to develop the analogy to Grothendieck toposes further, we outline how to apply our techniques on arbitrary base toposes, and give a decomposition theorem for constant objects functors induced by triposes, analogous to the known decompositions of geometric morphisms. Finally, we sketch an approach of how to find a unified framework for Grothendieck toposes and realizability toposes, based on the observation that uniform preorders can be identified with preorders internal to a category of sheaves.

A fibrational study of realizability toposes

Despite the progress in legged robotic locomotion, autonomous navigation in unknown environments remains an open problem. Ideally, the navigation system utilizes the full potential of the robots' locomotion capabilities while operating within safety limits under uncertainty. The robot must sense and analyze the travers ability of the surrounding terrain, which depends on the hardware, locomotion control, and terrain properties. It may contain information about the risk, energy, or time consumption needed to traverse the terrain. To avoid hand-crafted traversability cost functions we propose to collect traversability information about the robot and locomotion policy by simulating the traversal over randomly generated terrains using a physics simulator. Thousand of robots are simulated in parallel controlled by the same locomotion policy used in reality to acquire 57 years of real-world locomotion experience equivalent. For deployment on the real robot, a sparse convolutional network is trained to predict the simulated traversability cost, which is tailored to the deployed locomotion policy, from an entirely geometric representation of the envi-ronment in the form of a 3D voxel-occupancy map. This rep-resentation avoids the need for commonly used elevation maps, which are error-prone in the presence of overhanging obstacles and multi-floor or low-ceiling scenarios. The effectiveness of the proposed travers ability prediction network is demonstrated for path planning for the legged robot ANY mal in various indoor and natural environments.

Locomotion Policy Guided Traversability Learning using Volumetric Representations of Complex Environments

We characterize the tripos-to-topos construction of Hyland, Johnstone and Pitts as a biadjunction in a bicategory enriched category of equipment-like structures These abstract concepts are necessary to handle the presence of oplax constructs --- the construction is only oplax functorial on certain classes of cartesian functors between triposes A by-product of our analysis is the decomposition of the tripos-to-topos construction into two steps, the intermediate step being a weakened version of quasitoposes

A 2-Categorical Analysis of the Tripos-to-Topos Construction

We construct a category of fibrant objects $\mathbb{C}\langle P\rangle$ in the sense of Brown from any regular hyperdoctrine $P : \mathbb{C}^{\mathsf{op}}\to\mathbf{Pos}$, and show that its homotopy category is the Barr-exact category $\mathbb{C}[P]$ of partial equivalence relations and compatible functional relations. We give criteria for the the existence of left and right derived functors to functors $\mathbb{C}\langle\Phi\rangle : \mathbb{C}\langle P\rangle\to \mathbb{C}\langle Q\rangle$ induced by finite-meet-preserving transformations ${\Phi} : P \to Q$.

Categories of partial equivalence relations as localizations

Characterizing partitioned assemblies and realizability toposes

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