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Stream: theory: category theory

Topic: Reflective subcategories and colimits

James Deikun (Oct 10 2024 at 15:06):

That's right, colimits. Reflective subcategories are closed under limits, and (in sufficiently rich surroundings) they have colimits, but you normally have to apply the reflector to get back into the subcategory. How can we characterize colimits that land in the right place already? I know there can be some, and sometimes have to be (e.g. sufficiently highly filtered colimits in a presentable host category).

In topos theory we use lex reflectors, and we use them specifically to move colimits. Even in that case, I don't know exactly how to characterize which colimits don't get moved. But now I'm in the wild world of all full reflective subcategories, and I don't even have something well-organized like a coverage to start with, just a left-generated reflective OFS.

Kevin Carlson (Oct 10 2024 at 17:12):

Well, the filtered colimits are certainly a characteristic example, and one thing you can say is that if the right adjoint preserves $\kappa$-filtered colimits then the left adjoint preserves $\kappa$-compact objects, and the converse holds when the $\kappa$-compact objects generate; this is often useful.

Kevin Carlson (Oct 10 2024 at 17:13):

You could generalize this to other doctrines of limits, too.

Rémy Tuyéras (Oct 10 2024 at 18:05):

@James Deikun could you briefly describe what your left-generated reflective OFS is?

James Deikun (Oct 10 2024 at 18:52):

Sure, it's a reflective presentation of an essentially algebraic theory, so it's generated by a bunch of "horn inclusions".

James Deikun (Oct 10 2024 at 19:18):

(There are a countable number of generating morphisms, all between finitely presentable objects.)

Rémy Tuyéras (Oct 10 2024 at 19:23):

Are these lifting properties (wrt to the horn inclusions) meant to be used to describe some sort of descent condition on your models?

James Deikun (Oct 10 2024 at 19:25):

It's very similar to the characterization of nerves of categories by orthogonality to inner horn inclusions in simplices.

James Deikun (Oct 10 2024 at 19:27):

(In fact substituting this exact example for my actual one should probably be sufficient for purposes of discussion.)

Rémy Tuyéras (Oct 10 2024 at 20:20):

My understanding of your situation is that your lifting properties help you construct a pointed endofunctor $\eta:1 \Rightarrow Q$, which can then be used to construct the reflector (as a transfinite colimit), denoted by $L$.

When you apply the reflector $L$ to your colimits, say $X = \mathsf{colim} F$, you essentially compute the transfinite colimit:

$X \to Q(X) \to Q(Q(X)) \to \dots \to Q^n(X) \to \dots$

If you can show that $X \to Q(X)$ is an isomorphism, then the sequence above becomes a transfinite colimit of isomorphisms, implying that $L(X) \cong X$.

Where is the challenge? It lies in how you construct $\eta:1 \Rightarrow Q$, as there are many ways to define such a $\eta$. The variations between these pointed endofunctors come from how the data is quotiented.

For example, some versions of $\eta:1 \Rightarrow Q$ may not quotient all the data necessary for $\eta_X$ to be an isomorphism, even if $L(X) \cong X$. This means that you need to find a suitable $\eta:1 \Rightarrow Q$ that quotients all essential data to make the argument above work.

The reason I am asking about the lifting properties and descent conditions (which, after some reformulation, seem to apply to your case study) is that these elements could help you build the appropriate $\eta:1 \Rightarrow Q$.

Would this approach align with your setting? If so, I would be happy to elaborate on the details.

James Deikun (Oct 10 2024 at 20:33):

I wasn't particularly planning to construct the reflector in this manner, but if it helps to analyze closure under colimits, I'm game. Please tell me the details.

Rémy Tuyéras (Oct 10 2024 at 21:46):

Below, my goal is to develop a formal method/algorithm to detect objects for which the construction $\eta:1 \Rightarrow Q$ either adds new elements or does not. If it does not, the object is a model for the theory. You could potentially apply this detection criterion to your colimits.

Descent condition

My first step will be to translate your setting into a descent-condition-like framework. For this, I will assume that your horn inclusions are defined by arrows of the form

$\Lambda \to \Delta$

where $\Delta$ is a colimit $\Lambda \cup \Delta'$. More formally, you likely have a cospan of the form:

$\Lambda \rightarrow \Delta \leftarrow \Delta'$

Now, for an object $X$ to be a model for your theory, you likely require any arrow $\Lambda \to X$ to factor through your horn inclusion. We can reformulate this using spans of homsets:

$\mathcal{C}(\Lambda,X) \leftarrow \mathcal{C}(\Delta,X) \rightarrow \mathcal{C}(\Delta',X)$

With a slight reformulation of your theory, you can probably express this span as:

$\mathsf{lim}_d X(d) \leftarrow X(d_2) \rightarrow X(d_1)$

Lifting properties

In this context, your lifting property would be reinterpreted as stating that $X(d_2) \to \mathsf{lim}_d X(d)$ is a bijection (or a surjection), while the arrow $X(d_2) \rightarrow X(d_1)$ forces the addition of elements such that the object $X$ becomes a model.

By forcing the arrow $X(d_2) \to \mathsf{lim}_d X(d)$ to be a surjection, you are requesting that arrow to satisfy a right lifting property with respect to the arrow $\mathbf{0} \Rightarrow \mathbf{1}$ in $\mathbf{Set}^2$. Specifically, this arrow is represented by the commutative square going from $\mathbf{0} = (\emptyset \to \{\ast\})$ to $\mathbf{1} = (\{\ast\} \to \{\ast\})$.

To enforce this lifting property in a small-object-argument-like fashion, you would form the pushouts of the resulting spans $\{\ast\} \leftarrow \emptyset \rightarrow X(d_2)$. This process would functorially grow your object $X$ by adding elements, thereby creating a new object $X_1$.

If the arrow $X(d_2) \to \mathsf{lim}_d X(d)$ also needs to be injective (ensuring uniqueness of the lifting), you would apply a similar construction to the commutative square $\delta \Rightarrow \mathbf{1}$, where $\delta = (\{\ast,\ast\} \to \{\ast\})$.

These pushouts would yield your object $X_1 = Q(X)$, and repeating these pushouts would give you the transfinite sequence I mentioned earlier.

Better-behaved quotients

This is only one way to create $Q$, and the construction described above is not entirely effective at forming all the necessary quotients. Indeed, the pushout with respect to the commutative square involving $\mathbf{0} \Rightarrow \mathbf{1}$ introduces too much freedom. This excess freedom would likely be contracted in the second step, $X_2$, when applying the pushout with respect to $\delta \Rightarrow \mathbf{1}$, but it could be addressed earlier.

To see this, note that you can form a pushout in the commutative square $\mathbf{0} \Rightarrow \mathbf{1}$. This pushout will give you (by universality) spans of the form:

$\{\ast\} \leftarrow \{\ast,\ast\} \to \mathsf{lim}_d X(d)$

These spans are designed to correct for the free elements. By incorporating these spans into the pushout construction of $X_1$, the resulting object $X_1 = Q'(X)$ will have more quotients.

Conclusion

The construction above provides a formal framework for reasoning about quotients in colimits. I am not sure exactly how this might apply to your context, but you could potentially reverse-engineer the construction back to your original setting and determine how to detect colimits $X$ that do not add new elements when completed.

For a detailed treatment of the approach I have outlined, including homotopy-like descent conditions, you can refer to this paper.

James Deikun (Oct 10 2024 at 23:07):

Thanks! I appreciate the detailed explanation and it looks promising.