nLab topologically enriched category



Enriched category theory


topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



In the following we say Top-enriched category and Top-enriched functor etc. for what often is referred to as “topological category” and “topological functor” etc. As discussed there, these latter terms are ambiguous.



Top cgTop Top_{cg} \hookrightarrow Top

for the full subcategory of Top on the compactly generated topological spaces. Under forming Cartesian product

()×():Top cg×Top cgTop cg (-)\times (-) \;\colon\; Top_{cg} \times Top_{cg} \longrightarrow Top_{cg}

and compactly generated mapping spaces

() ():Top cg op×Top cgTop cg (-)^{(-)} \;\colon\; Top_{cg}^{op}\times Top_{cg} \longrightarrow Top_{cg}

this is a cartesian closed category (see at convenient category of topological spaces).


A topologically enriched category 𝒞\mathcal{C} is a Top cgTop_{cg}-enriched category, hence:

  1. a class Obj(𝒞)Obj(\mathcal{C}), called the class of objects;

  2. for each a,bObj(𝒞)a,b\in Obj(\mathcal{C}) a compactly generated topological space

    𝒞(a,b)Top cg, \mathcal{C}(a,b)\in Top_{cg} \,,

    called the space of morphisms or the hom-space between aa and bb;

  3. for each a,b,cObj(𝒞)a,b,c\in Obj(\mathcal{C}) a continuous function

    a,b,c:𝒞(a,b)×𝒞(b,c)𝒞(a,c) \circ_{a,b,c} \;\colon\; \mathcal{C}(a,b)\times \mathcal{C}(b,c) \longrightarrow \mathcal{C}(a,c)

    out of the cartesian product, called the composition operation

  4. for each aObj(𝒞)a \in Obj(\mathcal{C}) a point id a𝒞(a,a)id_a\in \mathcal{C}(a,a), called the identity morphism on aa

such that the composition is associative and unital.


Given a topologically enriched category as in def. , then forgetting the topology on the hom-spaces (along the forgetful functor U:Top kSetU \colon Top_k \to Set) yields an ordinary locally small category with

Hom 𝒞(a,b)=U(𝒞(a,b)). Hom_{\mathcal{C}}(a,b) = U(\mathcal{C}(a,b)) \,.

It is in this sense that 𝒞\mathcal{C} is a category with extra structure, and hence “enriched”.

The archetypical example is the following:


The category Top cgTop_{cg} from def. itself, being a cartesian closed category, canonically obtains the structure of a topologically enriched category, def. , with hom-spaces given by compactly generated mapping spaces

Top cg(X,Y)Y X Top_{cg}(X,Y) \coloneqq Y^X

and with composition

Y X×Z YZ X Y^X \times Z^Y \longrightarrow Z^X

given by the (product\dashv mapping-space)-adjunct of the evaluation morphism

X×Y X×Z Y(ev,id)Y×Z YevZ. X \times Y^X \times Z^Y \overset{(ev, id)}{\longrightarrow} Y \times Z^Y \overset{ev}{\longrightarrow} Z \,.

A topologically enriched functor between two topologically enriched categories

F:𝒞𝒟 F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}

is a Top cgTop_{cg}-enriched functor, hence:

  1. a function

    F 0:Obj(𝒞)Obj(𝒟) F_0 \colon Obj(\mathcal{C}) \longrightarrow Obj(\mathcal{D})

    of objects;

  2. for each a,bObj(𝒞)a,b \in Obj(\mathcal{C}) a continuous function

    F a,b:𝒞(a,b)𝒟(F 0(a),F 0(b)) F_{a,b} \;\colon\; \mathcal{C}(a,b) \longrightarrow \mathcal{D}(F_0(a), F_0(b))

    of hom-spaces

such that this preserves composition and identity morphisms in the evident sense.

A homomorphism of topologically enriched functors

η:FG \eta \;\colon\; F \Rightarrow G

is a Top cgTop_{cg}-enriched natural transformation: for each cObj(𝒞)c \in Obj(\mathcal{C}) a choice of morphism η c𝒟(F(c),G(c))\eta_c \in \mathcal{D}(F(c),G(c)) such that for each pair of objects c,d𝒞c,d \in \mathcal{C} the two continuous functions

η dF():𝒞(c,d)𝒟(F(c),G(d)) \eta_d \circ F(-) \;\colon\; \mathcal{C}(c,d) \longrightarrow \mathcal{D}(F(c), G(d))


G()η c:𝒞(c,d)𝒟(F(c),G(d)) G(-) \circ \eta_c \;\colon\; \mathcal{C}(c,d) \longrightarrow \mathcal{D}(F(c), G(d))


We write [𝒞,𝒟][\mathcal{C}, \mathcal{D}] for the resulting category of topologically enriched functors. This itself naturally obtains the structure of topologically enriched category, see at enriched functor category.

Topologically enriched presheaves


For 𝒞\mathcal{C} any topologically enriched category, def. then a topologically enriched functor

F:𝒞Top cg F \;\colon\; \mathcal{C} \longrightarrow Top_{cg}

to the archetypical topologically enriched category from example may be thought of as a topologically enriched copresheaf, at least if 𝒞\mathcal{C} is small (in that its class of objects is a proper set).

Hence the category of topologically enriched functors

[𝒞,Top cg] [\mathcal{C}, Top_{cg}]

according to def. may be thought of as the (co-)presheaf category over 𝒞\mathcal{C} in the realm of topologically enriched categories.

A functor F[𝒞,Top cg]F \in [\mathcal{C}, Top_{cg}] is equivalently

such that composition is respected, in the evident sense.

For every object c𝒞c \in \mathcal{C}, there is a topologically enriched representable functor, denoted y(c)y(c) or 𝒞(c,)\mathcal{C}(c,-) which sends objects to

y(c)(d)=𝒞(c,d)Top cg y(c)(d) = \mathcal{C}(c,d) \in Top_{cg}

and whose action on morphisms is, under the above identification, just the composition operation in 𝒞\mathcal{C}.

There is a full blown Top cgTop_{cg}-enriched Yoneda lemma. The following records a slightly simplified version.


(topologically enriched Yoneda-lemma)

Let 𝒞\mathcal{C} be a topologically enriched category, def. , write [𝒞,Top cg][\mathcal{C}, Top_{cg}] for its category of topologically enriched (co-)presheaves, and for cObj(𝒞)c\in Obj(\mathcal{C}) write y(c)=𝒞(c,)[𝒞,Top k]y(c) = \mathcal{C}(c,-) \in [\mathcal{C}, Top_k] for the topologically enriched functor that it represents, all according to example . Recall also the Top cgTop_{cg}-tensored functors FXF \cdot X from that example.

For cObj(𝒞)c\in Obj(\mathcal{C}), XTop cgX \in Top_{cg} and F[𝒞,Top cg]F \in [\mathcal{C}, Top_{cg}], there is a natural bijection between

  1. morphisms y(c)XFy(c) \cdot X \longrightarrow F in [𝒞,Top cg][\mathcal{C}, Top_{cg}];

  2. morphisms XF(c)X \longrightarrow F(c) in Top cgTop_{cg}.


Given a morphism η:y(c)XF\eta \colon y(c) \cdot X \longrightarrow F consider its component

η c:𝒞(c,c)×XF(c) \eta_c \;\colon\; \mathcal{C}(c,c)\times X \longrightarrow F(c)

and restrict that to the identity morphism id c𝒞(c,c)id_c \in \mathcal{C}(c,c) in the first argument

η c(id c,):XF(c). \eta_c(id_c,-) \;\colon\; X \longrightarrow F(c) \,.

We claim that just this η c(id c,)\eta_c(id_c,-) already uniquely determines all components

η d:𝒞(c,d)×XF(d) \eta_d \;\colon\; \mathcal{C}(c,d)\times X \longrightarrow F(d)

of η\eta, for all dObj(𝒞)d \in Obj(\mathcal{C}): By definition of the transformation η\eta (def. ), the two functions

F()η c:𝒞(c,d)F(d) 𝒞(c,c)×X F(-) \circ \eta_c \;\colon\; \mathcal{C}(c,d) \longrightarrow F(d)^{\mathcal{C}(c,c) \times X}


η d𝒞(c,)×X:𝒞(c,d)F(d) 𝒞(c,c)×X \eta_d \circ \mathcal{C}(c,-) \times X \;\colon\; \mathcal{C}(c,d) \longrightarrow F(d)^{\mathcal{C}(c,c) \times X}

agree. This means that they may be thought of jointly as a function with values in commuting squares in TopTop of this form:

f𝒞(c,c)×X η c F(c) 𝒞(c,f) F(f) 𝒞(c,d)×X η d F(d) f \;\;\;\; \mapsto \;\;\;\; \array{ \mathcal{C}(c,c) \times X &\overset{\eta_c}{\longrightarrow}& F(c) \\ {}^{\mathllap{\mathcal{C}(c,f)}}\downarrow && \downarrow^{\mathrlap{F(f)}} \\ \mathcal{C}(c,d) \times X &\underset{\eta_d}{\longrightarrow}& F(d) }

For any f𝒞(c,d)f \in \mathcal{C}(c,d), consider the restriction of

η d𝒞(c,f)F(d) 𝒞(c,c)×X \eta_d \circ \mathcal{C}(c,f) \in F(d)^{\mathcal{C}(c,c) \times X}

to id c𝒞(c,c)id_c \in \mathcal{C}(c,c), hence restricting the above commuting squares to

f{id c}×X η c F(c) 𝒞(c,f) F(f) {f}×X η d F(d) f \;\;\;\; \mapsto \;\;\;\; \array{ \{id_c\} \times X &\overset{\eta_c}{\longrightarrow}& F(c) \\ {}^{\mathllap{\mathcal{C}(c,f)}}\downarrow && \downarrow^{\mathrlap{F}(f)} \\ \{f\} \times X &\underset{\eta_d}{\longrightarrow}& F(d) }

This shows that η d\eta_d is fixed to be the function

η d(f,x)=F(f)η c(id c,x) \eta_d(f,x) = F(f)\circ \eta_c(id_c,x)

and this is a continuous function since all the operations it is built from are continuous.

Conversely, given a continuous function α:XF(c)\alpha \colon X \longrightarrow F(c), define for each dd the function

η d:(f,x)F(f)α. \eta_d \colon (f,x) \mapsto F(f) \circ \alpha \,.

Running the above analysis backwards shows that this determines a transformation η:y(c)×XF\eta \colon y(c)\times X \to F.


With Top cgTop_{cg} equipped with the classical model structure on topological spaces, which is a presentation for the archetypical (∞,1)-category ∞Grpd of ∞-groupoids, then the topological functor category

[𝒞,Top cg] [\mathcal{C},Top_{cg}]

(def. , def. ) is a model for the (∞,1)-category of (∞,1)-presheaves on 𝒞 op\mathcal{C}^{op}. This is made precise by the model structure on enriched functors, [𝒞,Top Quillen] proj[\mathcal{C},Top_{Quillen}]_{proj}. See at classical model structure on topological spaces – Model structure on functors for details.

Last revised on December 20, 2023 at 09:34:07. See the history of this page for a list of all contributions to it.