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# Homology, Homotopy and Applications

## Volume 24 (2022)

### Number 2

### Exponentials of non-singular simplicial sets

Pages: 307 – 314

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n2.a15

#### Authors

#### Abstract

A simplicial set is non-singular if the representing map of each non-degenerate simplex is degreewise injective. The simplicial mapping set $X^K$ has $n$‑simplices given by the simplicial maps $\Delta [n] \times K \to X$. We prove that $X^K$ is non-singular whenever $X$ is non-singular. It follows that non-singular simplicial sets form a cartesian closed category with all limits and colimits, but it is not a topos.

#### Keywords

non-singular simplicial set, exponential ideal, cartesian closed category

#### 2010 Mathematics Subject Classification

18D15, 55U10

Received 18 August 2021

Received revised 8 October 2021

Accepted 11 October 2021

Published 24 August 2022