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Stream: theory: category theory

Topic: Truncation functor

Jens Hemelaer (Nov 10 2020 at 14:06):

@Reid Barton gave some interesting point of view here on how to recover results in 1-category theory from the corresponding results in $(\infty,1)$-category theory.

I understand that there is some truncation functor $\tau : \mathcal{S} \to \mathbf{Sets}$, where $\mathcal{S}$ is the $(\infty,1)$-category of spaces and $\mathbf{Sets}$ is the $(\infty,1)$-category of sets.

Does this truncation functor preserve colimits and limits? I see here on nlab that it is left adjoint to the inclusion functor the other way?

Similarly, there should be a truncation functor $\tau : (\infty,1)-\mathcal{Cat} \to \mathbf{Cat}$, from the $(\infty,2)$-category (?) of $(\infty,1)$-categories to the $(\infty,2)$-category of 1-categories. Does this functor preserve the same kind of colimits and limits?

If $\mathcal{C}$ and $\mathcal{D}$ are $(\infty,1)$-categories, and $\mathbf{Fun}(\mathcal{C},\mathcal{D})$ is the $(\infty,1)$-category of functors between them, then is there an equivalence of categories $\tau(\mathbf{Fun}(\mathcal{C},\mathcal{D})) \simeq \mathbf{Fun}(\tau(\mathcal{C}),\tau(\mathcal{D}))$?

Rune Haugseng (Nov 10 2020 at 17:46):

Both truncations preserve colimits (since they are left adjoints) and also products, but not limits in general. The truncation of infinity-categories definitely doesn't preserve functor categories though. In fact I believe you can recover an infinity-category C from the homotopy categories (truncations) of Fun(I, C) for all small ordinary categories I - this is more or less the (pre)derivator approach to higher categories.

Jens Hemelaer (Nov 10 2020 at 20:40):

Thank you! I'll have a look at the (pre)derivator approach.

Reid Barton (Nov 10 2020 at 20:45):

To recover 1-categorical results from $(\infty,1)$-categorical ones you should apply the other adjoint, the inclusion functor of sets in spaces. This commutes with limits and exponentials and so, for example, the $(\infty, 1)$-category of functors between two 1-categories is the same as the 1-category of functors between them.

Reid Barton (Nov 10 2020 at 20:47):

Likewise, the condition of a diagram being a (co)limit in a 1-category is the same as it being a (co)limit when it is viewed as an $(\infty,1)$-category, again because limits in sets agree with limits in spaces.

Reid Barton (Nov 10 2020 at 20:50):

If you have a way of building new $(\infty,1)$-categories from old ones which involves colimits then it might take you outside the image of the 1-categories, but that isn't common.

John Baez (Nov 10 2020 at 20:57):

The homotopy colimit of a set acted on by a group is a groupoid; this shows up all over the place in physics and geometry, and this is why @Joachim Kock said:

You can kill symmetries by dividing out by them, but their ghosts will haunt you forever.

John Baez (Nov 10 2020 at 21:06):

And this phenomenon keeps pushing us up the dimensional ladder.

Jens Hemelaer (Nov 11 2020 at 08:57):

John Baez said:

The homotopy colimit of a set acted on by a group is a groupoid; this shows up all over the place in physics and geometry, and this is why Joachim Kock said:

You can kill symmetries by dividing out by them, but their ghosts will haunt you forever.

I would compute the homotopy colimit of a diagram $F : G \to \mathbf{Sets}$ as follows. First take the category of elements $\int_G F$ of this diagram. This has again a $G$-action, and the homotopy colimit is then the 1-categorical colimit of this category under the action of $G$.

But I have no idea if/why this gives the correct answer in general.

Jens Hemelaer (Nov 11 2020 at 09:07):

Suppose $\mathcal{P}(\mathcal{C})$ is the category of presheaves on $\mathcal{C}$ with values in spaces, for $\mathcal{C}$ an ordinary category. Then is $\tau(\mathcal{P}(\mathcal{C})) \simeq \mathbf{PSh}(\mathcal{C})$, where the latter is the ordinary category of presheaves?

I would prove this by saying that $\mathcal{P}(\mathcal{C})$ is the $(\infty,1)$-category freely generated under $(\infty,1)$-colimits, while $\mathbf{PSh}(\mathcal{C})$ has the similar 1-categorical universal property.

Jens Hemelaer (Nov 11 2020 at 09:22):

Jens Hemelaer said:

Suppose $\mathcal{P}(\mathcal{C})$ is the category of presheaves on $\mathcal{C}$ with values in spaces, for $\mathcal{C}$ an ordinary category. Then is $\tau(\mathcal{P}(\mathcal{C})) \simeq \mathbf{PSh}(\mathcal{C})$, where the latter is the ordinary category of presheaves?

Ok this doesn't make much sense... $\tau(\mathcal{P}(\mathcal{C}))$ should have the same objects as $\mathcal{P}(\mathcal{C})$.

Jens Hemelaer (Nov 11 2020 at 11:04):

There is some relevant terminology in the current masterclass by Scholze and Clausen: $\mathcal{P}(\mathcal{C})$ is the "animation" of $\mathbf{PSh}(\mathcal{C})$. So there should be a functor $\pi_0 : \mathcal{P}(\mathcal{C}) \to \mathbf{PSh}(\mathcal{C})$ which is left adjoint to the inclusion.

Reid Barton (Nov 11 2020 at 12:01):

A note on truncation. It's tempting to think that, when 1-categories are regarded as $(\infty,1)$-categories, they are exactly the 0-truncated $(\infty,1)$-categories. This is not correct because categories already peek a bit into the higher world. For $C$ to be a 0-truncated $(\infty,1)$-category means that the space of maps from any $(\infty,1)$-category to $C$ is discrete/a set. In particular, this includes the space of maps from the terminal category to $C$, which can be identified with the "space of objects" of $C$. If $C$ came from a 1-category then this space can be identified with the maximal groupoid contained in $C$, and it is only a set when there are no nontrivial automorphisms of objects in $C$.

Reid Barton (Nov 11 2020 at 12:05):

That said, the inclusion of 1-categories in $(\infty,1)$-categories still has a left adjoint, called the "homotopy category" functor or $\tau_1$. Informally, its effect is to apply $\pi_0$ to each mapping space. (You could think of it as the change of enrichment along the truncation $\pi_0$ from spaces to sets.)

Reid Barton (Nov 11 2020 at 12:06):

By definition, the universal property of the homotopy category is that it has the same functors into any 1-category.

Reid Barton (Nov 11 2020 at 12:07):

As you noted, if you apply this to the category $\mathcal{P}(\mathcal{C})$ of presheaves of spaces on an ordinary category $\mathcal{C}$, you won't get presheaves of sets on $\mathcal{C}$. In fact, it's wrong even for $\mathcal{C}$ the terminal category. Then $\mathcal{P}(\mathcal{C})$ is spaces, and its homotopy category is the classical homotopy category of spaces (e.g., the homotopy category of CW complexes).

Reid Barton (Nov 11 2020 at 12:10):

What you could do instead in this situation is the following. The $(\infty,1)$-category $\mathcal{P}(\mathcal{C})$ is (locally) presentable, so it would be better to consider not arbitrary functors out of it but only colimit-preserving ones to cocomplete categories (i.e., left adjoints). Then, we can ask for a universal 1-category with a colimit-preserving functor from $\mathcal{P}(\mathcal{C})$. This will be the category of presheaves of sets on $\mathcal{C}$.

Reid Barton (Nov 11 2020 at 12:14):

In general, this 1-category can be computed as the tensor product with the presentable $(\infty,1)$-category $\mathbf{Set}$. $\mathbf{Set}$ can be built as a presentable $(\infty,1)$-category by imposing the relation $* \amalg_{* \amalg *} * \xrightarrow{\sim} *$, so tensoring with $\mathbf{Set}$ imposes the relation $A \amalg_{A \amalg A} A \xrightarrow{\sim} A$ for every $A$. Informally this means that "giving one path in the space of maps from $A$ to $X$ is the same as giving two paths", i.e., the space of maps from $A$ to $X$ is actually a set, i.e., we obtain a 1-category.

Reid Barton (Nov 11 2020 at 12:31):

In the case of $\mathcal{P}(\mathcal{C})$ the claim can be verified directly: giving a colimit-preserving functor from $\mathcal{P}(\mathcal{C})$ to a 1-category $\mathcal{D}$ is equivalent to giving an ordinary functor from $\mathcal{C}$ to $\mathcal{D}$, which is in turn equivalent to giving a colimit-preserving functor from the category of presheaves of sets on $\mathcal{C}$ to $\mathcal{D}$.

Jens Hemelaer (Nov 11 2020 at 13:04):

Thanks a lot!

So can you recover the 1-Yoneda Lemma from the $(\infty,1)$-Yoneda Lemma by using the diagram below?

diagram.png

If $\mathcal{F}$ is a sheaf with values in sets, then the composition $\pi_0 \circ \mathrm{Hom}(-,\mathcal{F})$ preserves arbitrary limits, so it factors uniquely through $\mathbf{PSh}(\mathcal{C})$. The triangle on the left commutes by the $(\infty,1)$-Yoneda Lemma, and then the commutation of the outer diagram gives the ordinary Yoneda Lemma.

Reid Barton (Nov 12 2020 at 15:22):

This sounds basically correct but maybe it's better to put $\mathbf{Sets}$ where $\infty-\mathbf{Grpd}$ is, and write the top $\pi_0$ as $(\pi_0)^\mathrm{op}$. You could add a functor from $\mathbf{Sets}$ down to $\infty-\mathbf{Grpd}$. Then all the functors not involving $\mathcal{C}^\mathrm{op}$ are right adjoints.

Reid Barton (Nov 12 2020 at 15:24):

The point is that $\mathrm{Hom}(-, \mathcal{F})$ factors through the inclusion of sets in spaces and is still limit-preserving, and both of those facts are because limits in sets are the same as limits in spaces (and homs into $\mathcal{F}$ can be computed as a limit of the values of $\mathcal{F}$).

Jens Hemelaer (Nov 12 2020 at 18:13):

Thanks!