The numerical limit of perception

In this paper I will try to determine the numerical limits of perception and observation in general. Unlike most philosophers who wrote on perception, I will treat perception from a quantitative point of view and not discuss its qualitative features. What I mean is that instead of discussing qualitative aspects of perception, like its accuracy, I will discuss the quantitative aspects of perception, namely its numerical limits. As it turns out, the number of objects one is able perceive is finite, while the number of objects our mind can imagine might be infinite. Thus there must be a level of infinity by which the ‘number’ of objects our mental world can host is bounded. I will use both philosophical assumptions and observations, and mathematical analysis in order to get an estimate of the ‘number’ of objects we could possibly perceive, which surprisingly turns out to be the first level of infinity or the number of natural numbers. I will start by discussing the nature of concrete objects and the way we access them via perception. I will talk about mathematics as well and its relation with perception in order to justify myself for using mathematics as a tool in this paper. After some sections of discussions of various aspects of perception and imagination, I will finally be ready to make a counting and determine what I called “the numerical limits of perception”

Colimits of categories, zig-zags and necklaces

Given a diagram of small categories $F : J \rightarrow \textbf{Cat}$, we provide a combinatorial description of its colimit in terms of the indexing category $J$ and the categories and functors in the diagram $F$. We introduce certain double categories of zig-zags in order to keep track of the necessary identifications. We found these double categories necessary, but also explanatory. When applied pointwise in the simplicially enriched setting, our constructions offer a shorter proof of the necklace theorem of Dugger and Spivak by direct computation.Comment: 29 pages, comments and corrections are welcom

Higher Correspondences, Simplicial Maps and Inner Fibrations

In this essay we propose a realization of Lurie's claim that inner fibrations $p: X \rightarrow C$ are classified by $C$-indexed diangrams in a "higher category" whose objects are $\infty$-categories, morphisms are correspondences between them and higher morphisms are higher correspondences. We will obtain this as a corollary of a more general result which classifies all simplicial maps. Correspondences between $\infty$-categories, and simplicial sets in general, are a generalization of the concept of profunctor (or bimodule) for categories. While categories, functors and profunctors are organized in a double category, we will exibit simplicial sets, simplicial maps, and correspondences as part of a simpliclal category. This allows us to make precise statements and proofs. Our main tool is the theory of double colimits.Comment: Questions, comments and corrections are welcom