What is difference between working with small and large category of spaces?

Question

The following construction have always bugged me. This is p328, Remark 5.1.6.1 in Lurie's Higher Topos Theory. Lurie begins with the following:

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Construction: Let $C$ be a simplicial set. $S$ denote the $\infty$-category of small spaces. $s$ a vertex. Composing the Yoneda embedding $$j:C \rightarrow P(C)$$ by the evaluation map $$e_s: P(C)=Fun(C^{op}, S) \rightarrow Fun(\{ s \}, S)\simeq S$$ We obtain the map $j_s:=e_s\circ j:C \rightarrow S$. Where $j_s$ is referred to as the functor correpresented by $s$.

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The remark states that we should replace $S$ by $\widehat{S}$ the large $\infty$-category of spaces.

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I agree this is what we do 1-categorically when $S$ is replaced with sets.

Q1. But provided this construction makes sense, what's wrong with not replacing it?

Q2. Now suppose we are to work with it $\widehat{S}$. Is it safe to regard $Fun(C,S)$ as a fullsubcategory $Fun(C,\widehat{S})$? What do we know about the inclusion $S \hookrightarrow \widehat{S}$ i.e. if this inclusion is

• limit/colimit preserving
• conservative

?

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(Unfortuantely I am unable to work through the material after this - which may answer the above questions)

Answer 1

1. The Yoneda embedding $y: S \rightarrow P(S)$ is informally given by $y(s)(s^{\prime}) = map_{S}(s^{\prime}, s)$ (some care must be taken if $S$ is not an $\infty$-category in interpreting the mapping space, but Lurie gives a precise definition). If $S$ is a small simplicial set, then $map_{S}(s, s^{\prime})$ is a small space, so the functor does in fact land in presheaves of small spaces, as claimed. This is not necessarily true if $S$ is a large simplicial set, and this is what I believe Lurie's remark is about. What is perhaps slightly confusing is that you only really need $S$ to be locally small (ie. to have small mapping spaces), rather than to be small itself, and that's actually a rather common occurrence.

2. The inclusion $S \hookrightarrow \widehat{S}$ preserves all small limits and colimits. The same will be true for $Fun(C, S) \hookrightarrow Fun(C, \widehat{S})$, because (co)limits in functor $\infty$-categories are pointwise.