Apologies if this question is inappropriate for MO. It is not a research level question in any of the topics it addresses, I just don't see how a novice can go about answering it alone (I've tried navigating the vast sea of literature).

I'm trying to start learning descent theory, and after seeing how descent along open covers for topological bundles already loosens the sheaf condition and adds to cocycle condition, I'm looking for a conceptual definition with machinery that encodes all the coherence conditions.

This MO question and its answers affirm that descent data is the homotopy limit of the the cosimplicial diagram obtained as the image of the Čech nerve along $\mathcal F$. However, this already leaves me unsure as to what $\mathcal F$ should be in the general setting. For sheaves, it's just a presheaf of sets. For bundles, it seems to be a category fibered in categories (thought of as a presheaf of categories?).

Below are two similar remarks which involve something I don't understand. First, from Zhen Lin's answer to the question linked above:

The category of descent data is indeed the homotopy limit of your cosimplicial diagram. In the case where $F$ actually is fibred in categories (and not higher categories), then you can truncate above degree 2, recovering the classical definition. If $F$ is fibred in sets, then you can even truncate above degree 1, recovering the classical sheaf condition. So, morally, the category of descent data generalises the set of matching elements of a presheaf with respect to a cover.

Second, by David Roberts, in the first comment to this MO question:

If you're interested in $n$-limits, these are more easily treated using homotopy colimits, as we don't understand the $n$-category of $(n−1)$-categories very well, for $n>3$, say. A coequaliser is the homotopy colimit (in a 1-category!) of a simplicial diagram truncated all the way down to a pair of parallel arrows. The "2-" version of this is a slightly less truncated simplicial diagram. Mapping out of these (in an appropriate model structure, in the absence of a good higher category structure) gives the cosimplicial version Street has, and replaces colimits by limits.

The three (main) things I don't understand are:

1. Why can we "truncate"? The remarks suggest that if $\mathcal F$ is a presheaf taking values in $n$-categories (with $n=0$ being sets), then we "can truncate" the cosimplicial diagram above level $n+1$. I take it this means the (proper notion of a) limit of the entire cosimplicial diagram is the same as that of the truncated one, but I don't understand why this is true.
2. Why is 'homotopy limit' the correct term here? I have some vague understanding that homotopy limits in the sense of derived functors of functors between homotopical categories somehow present $(\infty,1)$-limits, but this after choosing weak equivalences! Nowhere in the first question are weak equivalences mentioned, so what are they?
3. The meaning of homotopy limit seems to change depending on context in David Robert's remark: coequalizers, which are just $1$-limits, are said to be homotopy limits of parallel pairs, which suggests the weak equivalences here on $\mathsf{Cat}$ are just taken to be isomorphisms. However, if we're fibered over categories then this no longer makes any sense...

Apologies if this question is inappropriate for MO. It is not a research level question in any of the topics it addresses, I just don't see how a novice can go about answering it alone (I've tried navigating the vast sea of literature).

I'm trying to start learning descent theory, and after seeing how descent along open covers for topological bundles already loosens the sheaf condition and adds to cocycle condition, I'm looking for a conceptual definition with machinery that encodes all the coherence conditions.

This MO question and its answers affirm that descent data is the homotopy limit of the the cosimplicial diagram obtained as the image of the Čech nerve along $\mathcal F$. However, this already leaves me unsure as to what $\mathcal F$ should be in the general setting. For sheaves, it's just a presheaf of sets. For bundles, it seems to be a category fibered in categories (thought of as a presheaf of categories?).

Below are two similar remarks which involve something I don't understand. First, from Zhen Lin's answer to the question linked above:

The category of descent data is indeed the homotopy limit of your cosimplicial diagram. In the case where $F$ actually is fibred in categories (and not higher categories), then you can truncate above degree 2, recovering the classical definition. If $F$ is fibred in sets, then you can even truncate above degree 1, recovering the classical sheaf condition. So, morally, the category of descent data generalises the set of matching elements of a presheaf with respect to a cover.

Second, by David Roberts, in the first comment to this MO question:

If you're interested in $n$-limits, these are more easily treated using homotopy colimits, as we don't understand the $n$-category of $(n−1)$-categories very well, for $n>3$, say. A coequaliser is the homotopy colimit (in a 1-category!) of a simplicial diagram truncated all the way down to a pair of parallel arrows. The "2-" version of this is a slightly less truncated simplicial diagram. Mapping out of these (in an appropriate model structure, in the absence of a good higher category structure) gives the cosimplicial version Street has, and replaces colimits by limits.

The three (main) things I don't understand are:

1. Why can we "truncate"? The remarks suggest that if $\mathcal F$ is a presheaf taking values in $n$-categories (with $n=0$ being sets), then we "can truncate" the cosimplicial diagram above level $n+1$. I take it this means the (proper notion of a) limit of the entire cosimplicial diagram is the same as that of the truncated one, but I don't understand why this is true.
2. Why is 'homotopy limit' the correct term here? I have some vague understanding that homotopy limits in the sense of derived functors of functors between homotopical categories somehow present $(\infty,1)$-limits, but this after choosing weak equivalences! Nowhere in the first question are weak equivalences mentioned, so what are they?
3. The meaning of homotopy limit seems to change depending on context in David Robert's remark: coequalizers, which are just $1$-limits, are said to be homotopy limits of parallel pairs, which suggests the weak equivalences here on $\mathsf{Cat}$ are just taken to be isomorphisms. However, if we're fibered over categories then this no longer makes any sense...

Apologies if this question is inappropriate for MO. It is not a research level question in any of the topics it addresses, I just don't see how a novice can go about answering it alone (I've tried navigating the vast see of literature).

I'm trying to start learning descent theory, and after seeing how descent along open covers for topological bundles already loosens the sheaf condition and adds to cocycle condition, I'm looking for a conceptual definition with machinery that encodes all the coherence conditions.

This MO question and its answers affirm that descent data is the homotopy limit of the the cosimplicial diagram obtained as the image of the Čech nerve along $\mathcal F$. However, this already leaves me unsure as to what $\mathcal F$ should be in the general setting. For sheaves, it's just a presheaf of sets. For bundles, it seems to be a category fibered in categories (thought of as a presheaf of categories?).

Below are two similar remarks which involve something I don't understand. First, from Zhen Lin's answer to the question linked above:

The category of descent data is indeed the homotopy limit of your cosimplicial diagram. In the case where $F$ actually is fibred in categories (and not higher categories), then you can truncate above degree 2, recovering the classical definition. If $F$ is fibred in sets, then you can even truncate above degree 1, recovering the classical sheaf condition. So, morally, the category of descent data generalises the set of matching elements of a presheaf with respect to a cover.

Second, by David Roberts, in the first comment to this MO question:

If you're interested in $n$-limits, these are more easily treated using homotopy colimits, as we don't understand the $n$-category of $(n−1)$-categories very well, for $n>3$, say. A coequaliser is the homotopy colimit (in a 1-category!) of a simplicial diagram truncated all the way down to a pair of parallel arrows. The "2-" version of this is a slightly less truncated simplicial diagram. Mapping out of these (in an appropriate model structure, in the absence of a good higher category structure) gives the cosimplicial version Street has, and replaces colimits by limits.

The three (main) things I don't understand are:

1. Why can we "truncate"? The remarks suggest that if $\mathcal F$ is a presheaf taking values in $n$-categories (with $n=0$ being sets), then we "can truncate" the cosimplicial diagram above level $n+1$. I take it this means the (proper notion of a) limit of the entire cosimplicial diagram is the same as that of the truncated one, but I don't understand why this is true.
2. Why is 'homotopy limit' the correct term here? I have some vague understanding that homotopy limits in the sense of derived functors of functors between homotopical categories somehow present $(\infty,1)$-limits, but this after choosing weak equivalences! Nowhere in the first question are weak equivalences mentioned, so what are they?
3. The meaning of homotopy limit seems to change depending on context in David Robert's remark: coequalizers, which are just $1$-limits, are said to be homotopy limits of parallel pairs, which suggests the weak equivalences here on $\mathsf{Cat}$ are just taken to be isomorphisms. However, if we're fibered over categories then this no longer makes any sense...

Apologies if this question is inappropriate for MO. It is not a research level question in any of the topics it addresses, I just don't see how a novice can go about answering it alone (I've tried navigating the vast sea of literature).

I'm trying to start learning descent theory, and after seeing how descent along open covers for topological bundles already loosens the sheaf condition and adds to cocycle condition, I'm looking for a conceptual definition with machinery that encodes all the coherence conditions.

This MO question and its answers affirm that descent data is the homotopy limit of the the cosimplicial diagram obtained as the image of the Čech nerve along $\mathcal F$. However, this already leaves me unsure as to what $\mathcal F$ should be in the general setting. For sheaves, it's just a presheaf of sets. For bundles, it seems to be a category fibered in categories (thought of as a presheaf of categories?).

Below are two similar remarks which involve something I don't understand. First, from Zhen Lin's answer to the question linked above:

The category of descent data is indeed the homotopy limit of your cosimplicial diagram. In the case where $F$ actually is fibred in categories (and not higher categories), then you can truncate above degree 2, recovering the classical definition. If $F$ is fibred in sets, then you can even truncate above degree 1, recovering the classical sheaf condition. So, morally, the category of descent data generalises the set of matching elements of a presheaf with respect to a cover.

Second, by David Roberts, in the first comment to this MO question:

If you're interested in $n$-limits, these are more easily treated using homotopy colimits, as we don't understand the $n$-category of $(n−1)$-categories very well, for $n>3$, say. A coequaliser is the homotopy colimit (in a 1-category!) of a simplicial diagram truncated all the way down to a pair of parallel arrows. The "2-" version of this is a slightly less truncated simplicial diagram. Mapping out of these (in an appropriate model structure, in the absence of a good higher category structure) gives the cosimplicial version Street has, and replaces colimits by limits.

The three (main) things I don't understand are:

1. Why can we "truncate"? The remarks suggest that if $\mathcal F$ is a presheaf taking values in $n$-categories (with $n=0$ being sets), then we "can truncate" the cosimplicial diagram above level $n+1$. I take it this means the (proper notion of a) limit of the entire cosimplicial diagram is the same as that of the truncated one, but I don't understand why this is true.
2. Why is 'homotopy limit' the correct term here? I have some vague understanding that homotopy limits in the sense of derived functors of functors between homotopical categories somehow present $(\infty,1)$-limits, but this after choosing weak equivalences! Nowhere in the first question are weak equivalences mentioned, so what are they?
3. The meaning of homotopy limit seems to change depending on context in David Robert's remark: coequalizers, which are just $1$-limits, are said to be homotopy limits of parallel pairs, which suggests the weak equivalences here on $\mathsf{Cat}$ are just taken to be isomorphisms. However, if we're fibered over categories then this no longer makes any sense...