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Revisiting colimits in $\mathbf{Cat}$ and homotopy category

Varinderjit Mann

Abstract

In this paper, we justify and make precise an elementary approach that establishes the existence of (co)limits in $\mathbf{Cat}$. This approach, while conceptually evident, has not been made fully explicit or systematically described in the literature. We first demonstrate an equivalence between the existence of the homotopy category functor $h : \mathbf{sSet} \rightarrow \mathbf{Cat}$ and the existence of a specific class of weighted colimits in $\mathbf{Cat}$. We then construct these weighted colimits explicitly by using certain properties of simplicial sets and the nerve functor. Consequentially, the embedding $N : \mathbf{Cat} \hookrightarrow \mathbf{sSet}$ is reflective, and can be used to infer the (co)completeness of $\mathbf{Cat}$. Finally, we use this approach to reformulate the construction of coequalizers and localizations in $\mathbf{Cat}$.

Revisiting colimits in $\mathbf{Cat}$ and homotopy category

Abstract

In this paper, we justify and make precise an elementary approach that establishes the existence of (co)limits in . This approach, while conceptually evident, has not been made fully explicit or systematically described in the literature. We first demonstrate an equivalence between the existence of the homotopy category functor and the existence of a specific class of weighted colimits in . We then construct these weighted colimits explicitly by using certain properties of simplicial sets and the nerve functor. Consequentially, the embedding is reflective, and can be used to infer the (co)completeness of . Finally, we use this approach to reformulate the construction of coequalizers and localizations in .
Paper Structure (19 sections, 48 theorems, 58 equations)

This paper contains 19 sections, 48 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results and implications
  4. Overview
  5. Notation and basic results
  6. Preliminaries
  7. Category of elements
  8. Weighted limits
  9. Kan Extensions
  10. Nerve and Realization
  11. Categories and simplicial sets
  12. TEXT
  13. TEXT
  14. Existence of the left adjoint
  15. Weighted colimits and the left adjoint
  16. ...and 4 more sections

Key Result

Proposition 1

The category $\mathbf{Cat}$ is cocomplete iff for every simplicial set $X \in \mathbf{sSet}$ a certain weighted colimit denoted $X \star [-]$ exists.

Theorems & Definitions (96)

  • Proposition
  • Theorem
  • Theorem
  • Definition 2.1.1: Category of elements
  • Lemma 2.1.2: yau2cat2021
  • Example 2.1.3: representable functors
  • Example 2.1.4: category of simplices
  • Definition 2.2.1: Weighted colimit
  • Remark 2.2.2
  • Lemma 2.2.3
  • ...and 86 more