nLab topologically enriched category

Definition

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.

Definition

Write

$$Top_{cg} \hookrightarrow Top$$

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

$$(-)\times (-) \;\colon\; Top_{cg} \times Top_{cg} \longrightarrow Top_{cg}$$

and compactly generated mapping spaces

$$(-)^{(-)} \;\colon\; Top_{cg}^{op}\times Top_{cg} \longrightarrow Top_{cg}$$

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

Definition

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

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

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

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

   called the space of morphisms or the hom-space between $a$ and $b$;

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

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

   out of the cartesian product, called the composition operation

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

such that the composition is associative and unital.

Remark

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

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

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

The archetypical example is the following:

Example

The category $Top_{cg}$ 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

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

and with composition

$$Y^X \times Z^Y \longrightarrow Z^X$$

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

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

Definition

A topologically enriched functor between two topologically enriched categories

$$F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$$

is a $Top_{cg}$-enriched functor, hence:

1. a function

   $$F_0 \colon Obj(\mathcal{C}) \longrightarrow Obj(\mathcal{D})$$

   of objects;

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

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

   of hom-spaces

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

A homomorphism of topologically enriched functors

$$\eta \;\colon\; F \Rightarrow G$$

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

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

and

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

agree.

We write $[\mathcal{C}, \mathcal{D}]$ 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

Example

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

$$F \;\colon\; \mathcal{C} \longrightarrow Top_{cg}$$

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

Hence the category of topologically enriched functors

$$[\mathcal{C}, Top_{cg}]$$

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

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

• a compactly generated topological space $F_a\in Top_{cg}$ for each object $a \in Obj(\mathcal{C})$;

• a continuous function

  $$F_a \times \mathcal{C}(a,b) \longrightarrow F_b$$

  for all pairs of objects $a,b \in Obj(\mathcal{C})$

such that composition is respected, in the evident sense.

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

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

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

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

Proposition

(topologically enriched Yoneda-lemma)

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

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

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

2. morphisms $X \longrightarrow F(c)$ in $Top_{cg}$.

Proof

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

$$\eta_c \;\colon\; \mathcal{C}(c,c)\times X \longrightarrow F(c)$$

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

$$\eta_c(id_c,-) \;\colon\; X \longrightarrow F(c) \,.$$

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

$$\eta_d \;\colon\; \mathcal{C}(c,d)\times X \longrightarrow F(d)$$

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

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

and

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

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

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

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

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

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

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

This shows that $\eta_d$ is fixed to be the function

$$\eta_d(f,x) = F(f)\circ \eta_c(id_c,x)$$

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

Conversely, given a continuous function $\alpha \colon X \longrightarrow F(c)$, define for each $d$ the function

$$\eta_d \colon (f,x) \mapsto F(f) \circ \alpha \,.$$

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

Remark

With $Top_{cg}$ 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

$$[\mathcal{C},Top_{cg}]$$

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

Related concepts

• simplicially enriched category

• model structure on enriched functors

• (infinity,1)-category