Table of Contents

## What are the images of a simplicial set?

The simplex category Δ is generated by two particularly important families of morphisms (maps), whose images under a given simplicial set functor are called face maps and degeneracy maps of that simplicial set.

## How are simplicial sets used in higher category theory?

Simplicial sets are used to define quasi-categories, a basic notion of higher category theory. A construction analogous to that of simplicial sets can be carried out in any category, not just in the category of sets, yielding the notion of simplicial objects .

## How are simplicial sets similar to directed multigraphs?

Simplicial sets can be viewed as a higher-dimensional generalization of directed multigraphs. A simplicial set contains vertices (known as “0-simplices” in this context) and arrows (“1-simplices”) between some of these vertices.

## What kind of arrows are in a simplicial set?

A simplicial set contains vertices (known as “0-simplices” in this context) and arrows (“1-simplices”) between some of these vertices. Two vertices may be connected by several arrows, and directed loops that connect a vertex to itself are also allowed. Unlike directed multigraphs, simplicial sets may also contain higher simplices.

## How is a simplicial set used in homotopy theory?

One may view a simplicial set as a purely combinatorial construction designed to capture the notion of a ” well-behaved ” topological space for the purposes of homotopy theory.

## When did Samuel Eilenberg invent the simplicial set?

Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and J. A. Zilber.

## Which is the classical model structure on simplicial sets?

This statement is made precise by the existence of the structure of a model category on sSet, called the classical model structure on simplicial sets that is a presentation for the (infinity,1)-category Top, as well as the Joyal model structure which similarly is a presentation of the (∞, 1) -category (∞, 1)Cat.

## Which is the simplicial subset of k prime?

Simplicial sets and their simplicial mappings form a category, $ \Delta ^ {0} \mathop {m Ens} $. If all the $ f _ {n} $ are imbeddings, then $ K $ is called a simplicial subset of $ K ^ \prime $. In this case, the boundary and degeneracy operators in $ K $ are the restrictions to $ K $ of the corresponding operators in $ K ^ \prime $.

## Which is the smallest subset of a simplicial set?

The smallest simplicial subset of a simplicial set $ K $ containing all its non-degenerate simplices of dimension at most $ n $ is denoted by $ K ^ {n} $ or $ \mathop {m Sk} ^ {n} K $, and is called the $ n $- dimensional skeleton or $ n $- skeleton of $ K $. The standard geometric simplices (cf. Standard simplex )

## How is the geometric realisation of a simplicial complex constructed?

Intuition. The geometric realisation of a simplicial complex, is then constructed by taking, for each abstract -simplex, , a copy, of such a standard topological -simplex, and then ‘gluing’ faces together, so whenever is a face of we identify with the corresponding face of . This space is usually denoted .

## Is the category of simplicial sets a quasitopos?

It follows from this characterization that the category of simplicial complexes is a quasitopos, and in particular is locally cartesian closed. The category of simplicial sets on the other hand is a topos. An abstract simplicial complex is a combinatorial gadget that models certain aspects of a spatial configuration.