A tower is a diagram of the shape of the poset of natural numbers

X 2X 1X 0. \cdots \to X_2 \to X_1 \to X_0 \,.

A limit over a diagram of this form is called a sequential limit/directed limit. If all connecting morphisms X i+1X iX_{i+1}\to X_i are epimorphism then we sometimes say that the tower is a cofiltration? (though the same term may be sometimes applied more generally for more general index categories, or more restrictively when there is an object in the ambient category such that all X iX_i are its quotients).

Different contexts lead to different notions of morphism of towers, so it is important to consider what category of towers is appropriate for any given use of these objects. In addition to Postnikov towers, and related uses in decomposing homotopy types, towers also occur as a simple type of pro-object in a category. In that situation the morphisms considered between towers are usually pro-morphisms.

In homotopy theory in the presence of (infinity,1)-limits/(infinity,1)-colimits, every tower is a tower of homotopy fibers.

The pro-category of towers


The tower diagram shape ={3210}\mathbb{N}_{\geq} = \{\cdots 3\to 2 \to 1 \to 0\} is evidently a small cofiltered category. As such it makes sense to consider pro-morphisms between tower diagrams:


For any small category 𝒞\mathcal{C}, the full subcategory

i:Tow pro(𝒞)Pro(𝒞)PSh(𝒞) i \;\colon\; Tow_{pro}(\mathcal{C}) \overset{}{\hookrightarrow} Pro(\mathcal{C}) \hookrightarrow PSh(\mathcal{C})

of the category Pro(𝒞)Pro(\mathcal{C}) of pro-objects in 𝒞\mathcal{C} on those that have presentation by formal sequential limits (formal limits over tower diagrams) is the pro-category of towers Tow pro(𝒞)Tow_{pro}(\mathcal{C}) in 𝒞\mathcal{C}.

(e.g. Blanc 96, def. 2.5)

For X : 𝒞X_\bullet \colon \mathbb{N}_{\geq} \longrightarrow \mathcal{C} a tower diagram, we write

lim fX Tow pro(𝒞)Pro(𝒞) \underset{\longleftarrow}{\lim}^f X_\bullet \in Tow_{pro}(\mathcal{C}) \hookrightarrow Pro(\mathcal{C})

for its formal cofiltered limit, i.e.

lim f:Tow(𝒞)Top pro(𝒞). \underset{\longleftarrow}{\lim}^f \;\colon\; Tow(\mathcal{C}) \longrightarrow Top_{pro}(\mathcal{C}) \,.


Notice that generally we have that morphisms between formal cofiltered limits (pro-objects) of shapes 𝒦 1\mathcal{K}_1 and 𝒦 2\mathcal{K}_2, respectively, are represented as cofiltered limits of systems of morphisms indeced on a single cofiltered category 𝒦\mathcal{K} equipped with final functors to 𝒦 1\mathcal{K}_1 and 𝒦 2\mathcal{K}_2, respectively. (See at ind-object this prop.).

The following says that in the case that both shapes are towers, then also 𝒦\mathcal{K} may always be chosen to be of tower shape:


For X X_\bullet and Y Y_\bullet two tower diagrams in 𝒞\mathcal{C}, then every morphism

ϕ:lim fX lim fY \phi \;\colon\; \underset{\longleftarrow}{\lim}^f X_\bullet \longrightarrow \underset{\longleftarrow}{\lim}^f Y_\bullet

between their formal cofiltered limits (every pro-morphism between the diagrams) in Tow pro(𝒞)Pro(𝒞)Tow_{pro}(\mathcal{C}) \hookrightarrow Pro(\mathcal{C}) is represented by component morphisms

ϕ k:X h(k)Y k \phi_{k} \;\colon\; X_{h(k)} \longrightarrow Y_k

making commuting diagrams

X h(k+1) ϕ k+1 Y k+1 X h(k) ϕ k Y k \array{ X_{h(k+1)} &\overset{\phi_{k+1}}{\longrightarrow}& Y_{k+1} \\ \downarrow && \downarrow \\ X_{h(k)} &\underset{\phi_k}{\longrightarrow}& Y_k }

i.e. by a natural transformation ϕ \phi_\bullet between

h X 𝒞 \mathbb{N}_{\geq} \overset{h}{\longrightarrow} \mathbb{N}_{\geq} \overset{X_\bullet}{\longrightarrow} \mathcal{C}


Y 𝒞 \mathbb{N}_{\geq} \overset{Y_\bullet}{\longrightarrow} \mathcal{C}


ϕlim f(ϕ ). \phi \simeq \underset{\longleftarrow}{\lim}^f (\phi_\bullet) \,.

(e.g. Blanc 96, p. 6, Libman, p.4)


The pro-category of towers Tow pro(𝒞)Tow_{pro}(\mathcal{C}) (def. 1) has all finite limits and finite colimits. Moreover these are presented by the degreewise finite limits of any representing diagram of towers, via prop. 1.

(Blanc 96, pop. 2.7)



  • David Blanc, Colimits for the pro-category of towers of simplicial sets, Cahiers de Topologie et Géométrie Différentielle Catégoriques (1996) Volume: 37, Issue: 4, page 258-278 (numdam)

  • Assaf Libman, Tower techniques for cofacial resolutions, Hopf archive (pdf)

Revised on August 30, 2016 11:36:44 by Zoran Škoda (