0.1 Reduction

0.1.1

ꢀ_∗is the category of pointed-spaces, modelled as Kan complexes  along with a map

pt . ↪ . This is often thought of as a simplicial model category and an ∞-category.

In what follows it will suﬃce to think of ꢀ_∗as category and remember the deﬁnition of

strong deformation retract in this context.

0.1.2 ꢀ-Reduced Models

A ꢁ-cell ꢂ ∈ ꢃ_ꢁin a pointed space ꢃ is called ꢀ-reduced if ꢄ^∗ꢂ = pt for any ꢄ ∶

[ꢅ] → [ꢁ], with ꢅ < ꢀ. A pointed space, , is called ꢀ-reduced if every cell of  is

ꢀ-reduced: equivalently, if _ꢅ= pt . for 0 ≤ ꢅ < ꢀ. An ꢀ-reduced model for a space ꢃ

is an ꢀ-reduced Kan complex, ꢃ_ꢀ^∼along with an equivalence:

ꢃ_ꢀ^∼≅ ꢃ

Proposition 0.0.1.

1. A space admits an ꢀ-reduced model if and only if it is ꢀ-

connected.

2. There is a co-reﬂection ꢆ ∶ {ꢀ-reduced spaces} → ꢀ_∗∶ ∙^∼∶, where ꢆ is the

ꢀ

forgetful functor.

3. We have ꢆ_ꢀ^∼= id and the counit

ꢃ_ꢀ^∼→ ꢃ

is a strong deformation retract of ꢃ, whenever ꢃ is ꢀ-connected.

Proof.

1. ꢀ-reduced spaces admit non-trivial maps from ꢀ^ꢅ, for ꢅ < ꢀ, so are ꢀ-

connected. (2) and (3) imply the converse of (1).

2. ꢃ_ꢀ^∼consists of the cells, ꢂ ∶ [ꢁ] → ꢃ such that ꢄ^∗ꢂ = pt ., for any ꢄ ∶ [ꢇ] → [ꢁ]

and ꢇ < ꢁ. If ꢈ is ꢀ-reduced then any pointed map ꢈ → ꢃ factors uniquely

through ꢃ_ꢀ^∼, so ꢃ_ꢀ^∼↪ ꢃ is universal from ꢀ-reduced spaces to ꢃ, thus proving

the claim.

3. For ꢀ = 0, there is nothing to prove. If ꢃ is ꢀ-connected for ꢀ > 0 and ꢃ_ꢀ^∼₋₁↪ ꢃ

is a strong deformation retract, then, we may assume that ꢃ is (ꢀ − 1) reduced.

(ꢀ − 1) -cells of ꢃ are then in bijection with maps ꢀ^ꢀ−1→ ꢃ.

The ꢀ-connectivity implies that there exists an assignment associating (ꢄ ∶ ꢀ^ꢀ−1

ꢃ) ↦ ꢄ ∶ ꢉ → ꢃ, where ꢄ is a contraction of ꢄ. This is the underlying data

↦

ꢀ

determining our deformation retract up to homotopy.

If the unique map sk_ꢀ−1ꢃ → sk_ꢀ−1ꢃ_ꢀ^∼has been extended to the ꢀ + ꢅ-skeleton,

ꢊ_ꢀ+ꢅ, along with a homotopy ꢋ_ꢀ+ꢅ∶ id → ꢊ_ꢀ+ꢅ, the extension of both to the

ꢀ + ꢅ + 1 skeleton constructs itself on an a non-degenerate ꢀ + ꢅ + 1-cell, ꢄ, by

extending the map

ꢄ ⨿ ꢌꢋ_ꢀ+ꢅꢌꢄ ∶ [ꢀ]

ꢌ

_{ꢌ[ꢀ],ꢌ[ꢀ]×{0}}[ꢀ] × [1]

ꢋ

_ꢀ+ꢅ+1(ꢄ) ∶ [ꢀ] × 1 → ꢃ

and writing ꢊ_ꢀ+ꢅ+1= (id ×ꢍ₀)^∗(ꢋ_ꢀ+ꢅ+1).