# Which tweets ‘deserve’ to be included in news stories?
Chronemics of tweet embedding

Munif Ishad Mujib
Drexel University
<EMAIL_ADDRESS>
Asta Zelenkauskaite
Drexel University
Vilnius Gediminas Technical University
<EMAIL_ADDRESS>
Jake Ryland Williams
Drexel University
<EMAIL_ADDRESS>

###### Abstract

The use and selection of user-generated social media content, specifically
tweets, as a news source has become an integral part of news production
practice. Yet, the mapping and the extent of the nature of the practices in
which news outlets integrate social media use is still lacking. This study
focuses on the pressures of immediacy on the media ecosystems, i.e., as
organizational practices of news outlets that make choices related to social
media content integration. By analyzing a large corpora of news outlets that
have embedded tweets, this study analyzes tweet embedding practices by
specifically focusing on the concept of chronemics, conceptualized here as the
time needed to embed tweets. Temporal constraints are particularly pressing
for journalistic practices, given the continuous pressures of 24/7 news cycle.
We ask two main questions: which types of outlets are quicker to embed tweets,
and which types of users’ tweets are more likely to be embedded quickly?

##  1 Introduction

The digitization of news outlets has had numerous implications for news
production: on one hand, content can be continuously updated ([1]); on the
other hand, it has opened up new opportunities to embed external digital
sources into news stories. Direct inclusion of external sources such as social
media (e.g. tweet embedding) is a growing practice in news organizations
([2]). This study investigates new conditions of journalism today explicated
by scholars like [3] as platformization and the blurring of journalism’s
boundaries. It is based on the premise that the inclusion of user-generated
social media content in mass media creates new ecosystems that are in flux, as
argued by [4]. Among various social networks that are the sources of user-
generated content embedded in news, Twitter holds particular significance for
journalists, not only through the use of Twitter to disseminate information
([5]), but also by using tweets as a type of evidence for a particular story
([6]). In this work, the process and pace of tweet integration in news stories
is treated through an ecosystem-as-structure approach that involves actors,
technology, and value creation through publication at the right time, as in
[7]. The focus here is to analyze a specific practice—tweet selection for
news—elucidating how journalists select from the constant stream of content
being generated by users and embed them as interactive data objects within web
versions of news articles. In order to investigate the tweet integration
phenomenon at a large scale, we designed and implemented a streaming data
acquisition system that collects digital news articles and embedded social
media content, particularly content from Twitter, discovered within those
articles. We call this acquisition system _NewsTweet_. We present a study of
the tweet embedding phenomenon performed by analyzing approximately two years’
worth of data collected by NewsTweet from 2019-2021 ([8]).

###  1.1 Chronemics in online spaces

This study’s primary focus is on the temporal dimension, i.e., the speed with
which tweets are incorporated in news stories. This is theoretically
conceptualized through chronemics in online spaces, defined by the
expectations of immediacy ([9]). The concept of “speed” of tweet integration
arises from assumptions based on the concept of chronemics, which has been
extensively analyzed in studies of computer-mediated communication ([9]). We
suggest that chronemics in online spaces matter because it influences the pace
in which content is distributed and received. Chronemics assumes that there is
a certain temporal “threshold” that makes the content, i.e., a tweet,
temporally relevant for the recipients of that content ([9]). In this study,
we ask an overarching question: how do chronemics play out in social media,
i.e., tweet integration as a source in a given news story? Based on the
existing literature on the topic, there are three approaches through which we
considered chronemics to have an effect on tweet selection for the news story
in this study: 1) the news outlet type; 2) the news outlet overall news
posting pacing (we conceptualized them as frequent or infrequent); and 3) the
author of the tweet.

Assumptions for this study are that the news outlets as media ecosystems have
different capacities (due to rate of news story creation by the outlet and
type), thus resulting in different pace or speeds between tweet posting and
their integration in a given news story. We operate on two assumptions. The
first one relates to the recency of the tweet as a source integration
hypothesis–where more recent sources are more valued by news organizations as
a source. The second is the democratizing assumption of twitter for news:
theoretically, we can assume that due to egalitarian access to Twitter, anyone
who posts on Twitter has the same right to be selected, depending on the
content relevance. Alternative to this premise is the assumption of
"trustworthiness" where traditional sourcing, trusting the source (e.g.,
because of its institutional prominence or celebrity status as argued in [10])
will be observed in the account choices for tweets. This assumes that
journalists ascribe varying levels of immediacy to users’ tweets depending on
their authorship, e.g., institutional standing and journalistic affiliation.
We assume that news outlets treat trustworthiness differently for users, here
referred to as authors of a given tweet, or, the stakeholders.

Considering the tweet author as a stakeholder, we should likely expect that
there are some who are more “trusted” by news organizations. Therefore, we
hypothesize the following: Tweets that are embedded faster are from more
“trusted” sources. Another assumption that drives this work is the outlet’s
capacity to produce news stories backed by sources. Thus, we assume that
outlets that produce more stories will also be using tweets for sourcing,
given the accessibility of tweets.

##  2 The mechanisms of socially-connected news production

This section contextualizes journalistic chronemics practices by describing
how Twitter is embraced in journalism. In particular we study Twitter as “the
_de facto_ tool of newsgathering” ([11]), providing sources akin to the live
eye-witness reporting typical within journalism. Via this functionality—re-
inscribed in social media platforms ([12])—Twitter allows for real-time public
sourcing of raw news material ([13, 6, 14, 15]). Similarly, sociotechnical
configurations of Twitter such as the “retweet” function, which allows one
user to share another’s content with their followers, mimic traditional mass
media “broadcasting” ([16]). We study amalgamated tweet-broadcasts that are
integrated in mass media.

###  2.1 Journalistic practices in using Twitter as a news source

The democratizing hypothesis of twitter starts with the premise that tweets as
an acceptable source have become commonplace as a part of established
journalistic practices. By 2010, well-regarded and established print outlets
such as _The New York Times_ and _The Washington Post_ , along with all major
national television networks, were already using Twitter content as primary
sources in some stories ([17]). Tweets have been used to fulfill the needs of
real-time issues, e.g., reporting breaking news events. Examples of major news
events where Twitter has been instrumental as a source medium in reporting are
the 2010 Haiti earthquake, the 2011 Arab Spring uprisings, and the Black Lives
Matter movement ([13, 18, 19, 20]). While it is also noted that tweets as a
source were more frequently used when reporting focused on “less-serious”
topics, such as art, sports, entertainment, and lifestyle, tweets produced by
former U.S. President Donald Trump (until his eventual ban from the platform)
were frequently reported on as well, often becoming news events unto
themselves ([21]).

Similarly, differences in tweet use for news were found between traditional
print-oriented outlets and web-only outlets ([22]), finding that web-only
outlets are more likely to use tweets as sources of information rather than
opinion or reaction. Thus, in this study we take into account the type of the
news outlet and hypothesize that web-only outlets, due to multimodality of the
content, will incorporate more tweets as a source regardless of the content.

Case studies on political reporting have found that in addition to providing
_vox populi_ commentary, tweets were also being used as sources to shed light
on the personal lives of public figures such as politicians and celebrities
([23]). One of the positive aspects of tweets as a source is that Twitter has
allowed journalists to diversify the range of people and organizations that
can be featured as sources, identifying a lowered barrier to entry for who can
qualify as supplying evidence for a news story, thus, cited as a democratizing
effect ([24]). Thus, we hypothesize that there will be a range of users whose
tweets will be included regardless of their "institutional backing", i.e.,
tweets’ sourcing will not be gathered from just a few accounts, if tweet
embeddings are treated as a democratizing instrument for journalism.

Due to velocity of upload and the recency aspect of the sourcing capacity of
the “crowds”, the use of Twitter content for breaking news has also been
celebrated. However, this is often taken with a grain of salt for how readily
available content impacts the journalistic principle of source verification
([18, 25, 26, 27]). The pressures of the 24-hour news cycle can lead to
difficult choices for journalists in deciding between “being fast” and “being
right” ([28, 15]) or the tension between verification and publication ([19]).

Given the temporal constraints and need to provide trusted sources discussed
above, how do journalists select tweets as sources for news? Various
technologies and automatic systems have been developed over the years in order
to help journalists monitor and report on breaking news events ([11, 29, 30,
31, 32, 33, 34]). Journalistic emergent practices for newsworthy tweets
selection include the use of a combination of Twitter-provided and third-party
tools to monitor the platform for news events and developments, going far
beyond “following” other journalists and potentially newsworthy users. Yet, in
this study if we start with the democratizing hypothesis of tweets in
journalism, embedding speeds will not vary by author of tweet. And as a
result, all users will be considered equally newsworthy to be embedded in the
news story.

###  2.2 Embedding tweets under time pressure

In this study, we investigate whether there are differences in how outlets
negotiate the immediacy considerations of tweet embedding given their
orientations in terms of distribution and content focus. We hypothesize that
there is indeed a race among outlets to embed newsworthy tweets. Our study
investigates whether these differences in orientation translate to outlets
“winning” the race—embedding tweets fastest—more often.

The guiding assumption is that newsrooms today are under immense time pressure
to publish articles and capture web traffic, given the high value of immediacy
([35]). Fast-paced sourcing is critical for the internet audience-driven
immediacy expectations becoming dominant in the traditional newsroom cultures
([36]). At times, many viral stories, or, “low-hanging fruit” ([37, 38]), are
produced by citing, reproducing, or otherwise referring to social media
content. Journalism scholars also warned that multiple points of view in
sourcing is the highest standard of reporting, but it has been found that this
is not always achieved: one study found that single-source stories amounted to
76 percent of the analyzed story sample ([39]).

Accessibility of social media content makes tweets into convenient sources.
When news outlets report on events unfolding in real time, the overwhelming
tendency is to publish and disseminate succinct stories first. However, news
organizations are diverse in size and capacity, and hence ability for
processing ever changing cycles of news. For example, in the early age of
internet-enabled journalism, conventional outlets displayed a tendency to
update websites with new stories at regular, slow intervals (such as once a
day), giving rise to the publication of “shovelware” ([40]). This model of
online publication tended to result in online news lagging behind traditional
channels. As the online news ecosystem has evolved, growing to include
discussion and engagement from users through comments, immediacy has become an
expectation in online news ([41]). Thus, we expect variation between speed of
tweet integration by outlet type.

###  2.3 Filtering in tweet selection: speed and authors

Story noteworthiness in journalism has been historically conceptualized as
being achieved through the process of as gatekeeping as a key practice by the
editor-in-chief who selects or discards stories to be published ([42]).
Journalistic gatekeeping has changed over the years, involving more of the
constituents to make decisions about story filtering ([43]). However, the
notion of gatekeeping as selecting or filtering stories has been expanded and
treated as a complex phenomenon that can be analyzed from various
perspectives. From an institutional perspective, this affects the
determination of newsworthiness for news stories as well as sources and event
selection ([44, 43]). Thus, the gatekeeping framework allows us to view the
selection process as a wide or a narrow filter. While this process arises out
of core journalistic values such as trustworthiness and the quality of
information, the effect of gatekeeping is that in most cases, journalists rely
on established sources, which in traditional journalism are institutional
sources that are seen as credible by default ([10]).

Twitter grants journalists access to a much wider range of potential sources
beyond the traditional, yet, journalists are found to adhere to many of the
gatekeeping tendencies that formed in an earlier age of mass media, e.g.,
print media such as newspapers, even when using tweets as sources, preferring
to cite public figures and prominent organizations that are thus vetted by
their mere reputation as being credible ([17]). However, in specific types of
news categories (often time-sensitive ones), some relaxation of these
tendencies has been observed, leading to some authors suggesting that a
reconceptualization of journalism is slowly taking place in that Twitter, for
example, has been established as an online space where journalists gather the
news and find sources, and then report the news and drive traffic to websites
([19]).

The uniquely public nature of discourse and information sharing on Twitter
leads us to consider it in terms of the Habermasian principles of the public
sphere, in particular, of disregard for status and inclusivity ([45]). While
it is often claimed that Twitter has democratized information and opinion
sharing by constructing a “town square” on the internet, in the context of
newsworthiness, users face considerable gatekeeping from journalists. Source
selection from Twitter by journalists has been demonstrated to favor “elite”
voices ([46]). The selection process for tweet embedding operates on the axes
of both institutional sourcing credibility and “specialty” of content. A
particular dynamic in social embedding is the reproduction of content from
established journalists, giving rise to a community of journalists founded on
credibility that is drawn from when tweets are chosen to be embedded.

##  3 Data Collection and Preprocessing

NewsTweet is an ongoing collection of digital news articles. Articles listed
on eight topical feeds in the Google News aggregator (Business, Entertainment,
Health, Nation, Technology, Sports, World, and Headlines) are automatically
collected, and embedded social media content is extracted. A total of
1,635,684 articles were collected over the collection period of this study
(May 2019–June 2021). Embedded social content was extracted from these
articles through automatic identification of specific HTML tags. This method
of extraction ignored any unstructured references to social content such as
through quotes in the article text. Embedded content volumes by platform are
presented in table 1. Tweet embeds made up nearly two-thirds of all embeds,
and the volume of embedded tweets was more than double the next-most common
embed type (YouTube videos). This firmly establishes Twitter’s prominence in
news over other social platforms. Overall, in the full NewsTweet dataset,
355,755 tweet embeddings were found, appearing across 149,901 articles
representing approximately 9% of all articles. These tweets were extracted
from the articles, and extensive tweet metadata was collected using the
Twitter API. Each article contained a publication timestamp provided by Google
News. The accuracy of these timestamps was examined by sampling 160 random
articles, and it was found that the timestamps were reliable in 91% of the
sampled articles. A set of outlets were identified as sources of unreliably-
timestamped articles and discarded; this set consisted of a number of smaller
international outlets as well as local television and radio station websites.
Articles from these discarded outlets consisted of less than 1% of the total
collection.

Table 1: Volumes of embedded content by platform. Platform | Articles | Embeddings
---|---|---
Twitter | 149,901 (52.9%) | 355,755 (64.4%)
YouTube | 95,059 (33.5%) | 135,723 (24.6%)
Instagram | 34,525 (12.2%) | 55,815 (10.1%)
Facebook | 1,911 (0.7%) | 2,258 (0.4%)
Reddit | 1,177 (0.4%) | 1,264 (0.3%)
TikTok | 766 (0.3%) | 1,640 (0.2%)
Total | 283,339 (100.0%) | 552,455 (100.0%)

##  4 Methods

Our study on the chronemics of tweet embedding was operationalized by the
speed of news outlets to include a given tweet. For each embedded tweet, the
time between tweet creation and article publication (i.e., embedding of the
tweet) was computed and termed the _embed delay_. The median embed delay was
approximately 6 hours. The distribution appeared predictably heavy-tailed,
with 77% of embeddings occurring within 24 hours of tweet publication. It is
worth pointing out that 3.5% of embeds were found with “negative” embed
delays, i.e., tweets were found embedded in articles that were published
_before_ the tweets’ creation times. Given that we treated these embeddings
with negative embed times functioning with a potentially different
institutional logic or practices, we did not include these in subsequent
analyses, and they present an interesting avenue for future study.

Table 2: Top embedding outlets overall.

Outlet | Distribution type | Content focus type | Articles with embeds | Median embed delay (hours)
---|---|---|---|---
yahoo.com | Online-only | General | 6,168 | 5.77
thehill.com | Online-only | General | 5,859 | 3.13
cbssports.com | Traditional (TV) | Sports | 5,550 | 1.25
usatoday.com | Traditional (Print) | General | 4,895 | 3.00
cnn.com | Traditional (TV) | General | 3,623 | 14.68
nbcsports.com | Traditional (TV) | Sports | 3,145 | 9.11
cheatsheet.com | Online-only | Culture and Technology | 2,801 | 84.33
nbcnews.com | Traditional (TV) | General | 2,602 | 9.42
cbsnews.com | Traditional (TV) | General | 2,396 | 14.28
washingtonpost.com | Traditional (Print) | General | 1,976 | 11.83
npr.org | Traditional (Radio) | General | 1,956 | 13.08
espn.com | Traditional (TV) | Sports | 1,828 | 2.33
vox.com | Online-only | General | 1,812 | 19.98
theverge.com | Online-only | Culture and Technology | 1,785 | 9.69
comicbook.com | Online-only | Culture and Technology | 1,772 | 7.35
nypost.com | Traditional (Print) | General | 1,662 | 9.51
cnet.com | Online-only | Culture and Technology | 1,608 | 13.39
washingtonexaminer.com | Traditional (Print) | General | 1,589 | 6.41
huffpost.com | Online-only | General | 1,452 | 11.71
deadline.com | Online-only | Culture and Technology | 1,410 | 3.30
engadget.com | Online-only | Culture and Technology | 1,263 | 6.34
bbc.com | Traditional (TV) | General | 1,232 | 10.92
nesn.com | Traditional (TV) | Sports | 1,181 | 2.35
gamespot.com | Online-only | Culture and Technology | 1,140 | 22.25

Articles containing embedded tweets were collected from a total of 2,089 news
outlets over the collection period. On average, 27% of the articles published
by these outlets included at least one tweet. The average number of articles
with embeds collected from an outlet was 66, while 1% of the outlets (24
outlets, to be exact) published more than 1,000 articles containing embedded
tweets. For outlet-level analysis, only the 361 outlets that published at
least 50 articles with embedded tweets in the collection period were
considered. Based on the observed data, this cut-point could indicate a
substantial threshold that distinguishes outlets that use and did not use
tweets as a source.

A Pearson correlation coefficient was computed to assess if any linear
relationship exists between the total articles published, $N_{total}$, and the
total articles with embedded tweets published, $N_{a}$. This analysis
presented a statistically significant positive correlation between the two
variables, $r(359)=.83$, $p<.001$. Thus, the more prolific the outlet, the
more articles with embedded tweets they produce. To assess the nature of the
articles per news source, the number of articles published with embedded
tweets was used as a measure of outlet prolificity (i.e., their story
production volume) due to this linear relationship. Outlets with $500\leq
N_{a}\leq 2,000$ were considered “medium” volume outlets and those with
$N_{a}\geq 2,000$ were considered “large” volume outlets. These categories
were henceforth utilized for investigating the impact of outlet prolificity on
embedding speed.

For each embedded tweet in the dataset, the outlet that embedded it earliest
in an article was identified. Using this information, for each outlet, the
fraction of first-time embeds was calculated as:

$f_{1st}=\frac{N_{1st}}{N_{e}}$ (1)

where $N_{1st}$ was the total number of tweets embedded first by the outlet
and $N_{e}$ was the total number of embedded tweets by the outlet.

To address the mechanics of tweet use in stories, at least two modes for the
inclusion of tweets in articles were identified: (1) the use of a single tweet
as a source, and (2) the use of multiple tweets as sources. To conceptualize
these mechanics, articles with embeds were divided into two categories:
_single-tweet articles_ , which embed only one tweet, and _multi-tweet
articles_ , which embed multiple tweets. We found that in our sample, $61\%$
of all articles with tweets were single-tweet articles.

Combining these two aspects of tweet embedding, then, helps answer the
question of whether the outlets that are more likely to publish one-tweet
stories are also more likely to publish those stories faster than other
outlets. The fraction of single-tweet articles by outlet was thus computed as:

$f_{s}=\frac{N_{s}}{N_{a}}$ (2)

where $N_{s}$ was the total number of single-tweet articles produced by the
outlet.

The top outlets by embedded article volume ($N_{a}\geq 1,000$) were next
examined to investigate embedding speed variation by outlet type, with the
goal of developing a typology based on embedding speed. Results of this
examination are presented in table 2.

Two typologies were developed and embed delay distributions were compared
between types:

  1. 1.

_Types based on distribution channels_ : outlets were divided into two groups
based on the channels on which they distribute content, online-first outlets,
and traditional outlets with presence in print, television, and radio. One of
the authors manually verified the platforms each outlets maintained a presence
on (print, radio, television, and internet).

  2. 2.

_Types based on areas of content focus_ : outlets were divided into three
groups based on their focus areas, namely sports-focused, culture and
technology-focused, and generalized. One of the authors manually examined
outlet websites and annotated each outlet with one of the three content focus
types.

Figure 1: Distribution of first-time embed fractions by outlet.

The next analyses sought to understand if and how tweets from different groups
of users are embedded with different chronemics, and for this, a set of
prominent users were examined. This set of prominent users was determined as
follows. The distribution of total embeds by user was observed to be heavy-
tailed, indicating that most users are only ever embedded just a handful of
times. To study users in this set, the _presence_ of each was defined and
measured by the number of days when at least one tweet from their timeline was
embedded in an article from the collection period of the dataset. A typology
of the top 50 users by presence was then developed with the goal of
elucidating if different types of users are embedded at different speeds.
Here, users were assigned the type of either “Personality”, “Journalist”,
“Media Outlet”, or “Organization”. These types were assigned by manually
examining the Twitter pages of users. If the account identified itself as
belonging to a journalist, media outlet, or organization, those types were
assigned. Otherwise, for personal accounts, the “Personality” type was
assigned. Differences in embedding chronemics between these categories were
examined, and are discussed below.

##  5 Results

These sections present the results of the above-described analysis, focusing
on the effects of outlet prolificity and type in Secs. 5.1–5.2, as well as
user type in Sec. 5.3.

Table 3: Top 20 outlets with high single-tweet article fractions and high
first-time embed fractions.

Outlet | First time embeds | First time embed fraction | Single tweet articles | Single tweet article fraction | Articles with embeds
---|---|---|---|---|---
yahoo.com | 2,222 | 0.16 | 3,507 | 0.57 | 14,113
thehill.com | 1,401 | 0.16 | 4,197 | 0.72 | 8,943
cbssports.com | 3,996 | 0.12 | 2,323 | 0.42 | 33,629
cnn.com | 708 | 0.13 | 2,672 | 0.74 | 5,310
nbcsports.com | 1,111 | 0.14 | 1,764 | 0.56 | 7,811
nbcnews.com | 476 | 0.13 | 2,073 | 0.80 | 3,591
cbsnews.com | 479 | 0.12 | 1,641 | 0.68 | 3,951
espn.com | 727 | 0.19 | 1,242 | 0.68 | 3,927
theverge.com | 472 | 0.20 | 1,429 | 0.80 | 2,396
nypost.com | 286 | 0.14 | 1,443 | 0.87 | 2,077
cnet.com | 942 | 0.19 | 904 | 0.56 | 5,013
washingtonexaminer.com | 357 | 0.13 | 974 | 0.61 | 2,821
deadline.com | 540 | 0.18 | 881 | 0.62 | 3,070
engadget.com | 210 | 0.14 | 1,109 | 0.88 | 1,476
bbc.com | 216 | 0.12 | 910 | 0.74 | 1,797
gamespot.com | 321 | 0.18 | 856 | 0.75 | 1,743
dexerto.com | 239 | 0.14 | 514 | 0.56 | 1,757
radio.com | 140 | 0.11 | 412 | 0.61 | 1,234
vulture.com | 199 | 0.14 | 414 | 0.64 | 1,443
cbslocal.com | 131 | 0.13 | 408 | 0.68 | 1,022

###  5.1 Outlet prolificity and embedding chronemics

Data-driven approaches were employed to deduce that larger outlets
(conceptualized based on the amounts of stories posted) did, in general, “win”
the race to embed tweets more often. Specifically, these data-driven
approaches were based on the distribution of $f_{1st}$ (Eq. 1), displayed in
figure 1. However, the presence of a number of medium outlets that embedded
and published first more often ($f_{1st}\geq 0.1$) skewed the distribution of
$f_{1st}$ higher. Outlets that covered technology (The Verge, CNET, Engadget,
GameSpot, IGN, TechCrunch), many of them blogs, were prominent among these;
demonstrating that journalists and bloggers specializing in topics of
technology may be more inclined to use tweets as a source and report on
emergent trends found on social media. Sports-focused outlets appeared to
embed tweets the quickest, with prominently large shares observed across three
outlets: CBS Sports (large), NBC Sports (large), and ESPN (medium) all had
$f_{1st}>0.1$.

Table 4: Embed delays by outlet type. Typology | Type | Median embed delay (hours)
---|---|---
Distribution | Online-only | 9.22
| Traditional | 3.21
| Sports | 1.70
Content focus | Culture and Technology | 18.23
| General | 8.27

###  5.2 Outlet type and embedding chronemics

The fraction of single-tweet-articles, $f_{s}$ (Eq. 2), demonstrated a
distribution across outlets roughly concentrated in the range of $0.4\leq
f_{s}\leq 0.9$. Within this range, outlets could be divided into two
categories: outlets with $f_{1st}\geq 0.1$, and outlets with $f_{1st}<0.1$. By
design, the second group of outlets are characterized as having _both_ a high
fraction of single-tweet articles _and_ a high fraction of first-time embeds.
These would be the ultimate “winners” of the embedding race, i.e., more likely
to find newsworthy tweets quickly and report on them. The top 20 outlets from
this group by number of embedded articles are presented in table 3. The three
outlets with lowest $f_{s}$ among these are CBS Sports, Dexerto (an esports-
focused outlet), and NBC Sports; illustrating that sports-themed outlets are
more likely to publish more articles collecting multiple tweets. NBC News,
CNN, BBC, The Hill, and CBS News are all part of this list as well: these
mainstream, prominent “hard news” outlets, when they _do_ report on tweets,
are often quick to embed and more likely to focus on a single tweet.

Table 5: Top 50 users by presence. Username | Type | Unique tweets embedded | Days embedded (presence) | Median embed delay (hours)
---|---|---|---|---
realDonaldTrump | Personality | 2,735 | 567 | 6.64
elonmusk | Personality | 1,094 | 547 | 23.28
RapSheet | Journalist | 871 | 416 | 2.44
AdamSchefter | Journalist | 748 | 411 | 3.68
NFL | Organization | 1,386 | 362 | 3.86
espn | Media Outlet | 541 | 361 | 7.43
SportsCenter | Media Outlet | 495 | 352 | 9.28
BleacherReport | Media Outlet | 520 | 343 | 9.78
WWE | Organization | 2,930 | 327 | 0.92
atrupar | Journalist | 774 | 314 | 11.61
PFF | Organization | 487 | 295 | 21.95
NBA | Organization | 477 | 271 | 2.88
AOC | Personality | 360 | 236 | 11.76
TomPelissero | Journalist | 398 | 235 | 1.71
FieldYates | Journalist | 277 | 218 | 3.34
MLB | Organization | 451 | 212 | 2.01
wojespn | Journalist | 604 | 211 | 2.13
ShamsCharania | Journalist | 371 | 202 | 1.52
steelers | Organization | 588 | 198 | 3.39
UniverseIce | Journalist | 182 | 196 | 25.75
KimKardashian | Personality | 218 | 191 | 18.80
PFF_College | Organization | 336 | 187 | 37.66
thecheckdown | Media Outlet | 321 | 187 | 15.17
Braves | Organization | 375 | 186 | 1.46
ESPNStatsInfo | Media Outlet | 245 | 185 | 5.90
Ravens | Organization | 516 | 174 | 1.90
FortniteGame | Organization | 187 | 170 | 10.80
jamisonhensley | Journalist | 231 | 169 | 19.14
Patriots | Organization | 346 | 164 | 2.40
ufc | Organization | 471 | 164 | 0.50
KingJames | Personality | 166 | 163 | 10.97
thehill | Media Outlet | 187 | 163 | 3.52
TODAYshow | Media Outlet | 203 | 161 | 3.72
Yankees | Organization | 322 | 159 | 2.42
DrewShiller | Journalist | 308 | 159 | 257.28
Cubs | Organization | 255 | 157 | 2.85
JoeBiden | Personality | 168 | 157 | 9.91
CBSNews | Media Outlet | 193 | 155 | 6.71
ABC | Media Outlet | 200 | 155 | 3.53
PGATOUR | Organization | 922 | 154 | 1.87
SpaceX | Organization | 158 | 149 | 14.45
CBSSports | Media Outlet | 356 | 146 | 2.00
NFL_DovKleiman | Journalist | 202 | 146 | 10.17
PitchingNinja | Journalist | 212 | 145 | 3.70
NBCSPhilly | Media Outlet | 318 | 145 | 3.30
NYGovCuomo | Personality | 169 | 141 | 4.50
GMA | Media Outlet | 154 | 141 | 4.67
Chiefs | Organization | 343 | 138 | 1.30
MikeGarafolo | Journalist | 159 | 137 | 2.41
AlbertBreer | Journalist | 164 | 135 | 3.95

A Mann-Whitney test indicated that the difference in embed delays between
online-only and traditional outlets were statistically significant,
$U(N_{online-only}=56,875,N_{traditional}=86,802)=1,866,789,937.5,p<.001$. A
one-way ANOVA was performed to compare the effect of outlet content focus on
embed delays. The one-way ANOVA revealed that there was a statistically
significant difference in embed delays between at least two types of outlets
based on their content focus ($F(2,136,836)=134.92$, $p<.001$). The median
embed delays by outlet type are presented in table 4. Among the types,
traditional outlets (when they chose to do so) generally embedded tweets more
quickly than online-only outlets. Among the content focus-based types, outlets
reporting primarily on sports were much quicker to embed tweets, with the
median embed delay being under 2 hours from tweet publication. This
demonstrated strong evidence for event-driven reporting, where the outlets try
to capture public attention during or close to live events. In a notable
contrast to this, culture and technology-focused outlets were much slower to
embed tweets than both sports-focused and generalized outlets.

###  5.3 User type and embedding chronemics

The 50 users with the highest _presence_ are displayed in table 5. The topmost
two users were extreme outliers in terms of presence, impact, and variety:
former U.S. President Donald J. Trump (@realDonaldTrump) and SpaceX/Tesla
C.E.O. Elon Musk (@elonmusk). Tweets from @realDonaldTrump, in addition to
being embedded nearly every day of the collection period that the account was
active (it was suspended permanently by Twitter on January 8, 2021), were
embedded more than twice on average, while 12% of the tweets they produced
were embedded (the median was 0.1%).

A one-way ANOVA was performed to compare the effect of user type on embed
delays. This test revealed that there was a statistically significant
difference in embed delays between at least two types of users
($F(2,34,635)=163.78$, $p<.001$). Median embed delays by user type are
displayed in table 6. Tweets from the organizations in the top 50 accounts
were embedded the quickest, while tweets from personal accounts were embedded
the slowest. Since most of the organizations in this set were associated with
sports, this result aligns with findings at the outlet level, where sports-
focused outlets were seen to embed quicker than other types of outlets. The
prevalence of journalists in this list of top-embedded accounts, particularly
sports journalists and commentators, indicates the presence of an ecosystem
where sports commentary, predictions, and general updates tweeted by
journalists is quickly picked up by outlets—often the very outlets these
journalists are associated with—and embedded and published within their own
articles.

Table 6: Embed delays by user type. Type | Median embed delay (hours)
---|---
Personality | 10.37
Journalist | 4.54
Media Outlet | 5.08
Organization | 1.81

##  6 Discussion

In line with the tweet embedding as a journalistic practice, we expected that
news outlets that produced larger amounts of stories will use tweets more
often. While this was observed in the analysis, a group of “medium” outlets
(conceptualized based on the amounts of stories created) who were often
quicker than others to embed were also discovered. Many of these were
technology-oriented publications and blogs. This indicated that in certain
thematic areas, medium outlets competed at the same level with large outlets.
The presence of these types of outlets illustrates a dynamic where by
utilizing the relatively low-cost access to social media content (such as
tweets), outlets can leverage immediacy to attract audience attention away
from the more established institutions; shining a favorable light on the oft-
discussed democratizing effects of the online ecosystem. Also, we found that
more than half (i.e., 61 percent) of the stories that contained tweets, were
single-tweet sourcing. Future studies should analyze more in depth, if single-
tweet use was only one type of source or they complemented other types of
sourcing.

We found that when reporting on newsworthy social media content, outlets with
traditional media presence (radio, television, or print) in addition to an
online presence in fact acted faster than online-only outlets. This finding
demonstrates the reconfiguration of journalistic values and incentives for
publication. In the always-online attention economy, established outlets have
come to prioritize immediacy and engagement from the audience. The quick
embedding and publication of newsworthy social media content and stories
driven by such content demonstrate the effects of this prioritization.

Differing speeds of embedding by user types demonstrated a more complex
picture of the tweet selection process. We noted how traditional “credibility”
or newsworthy considerations were employed on Twitter by relying on
institutional backing, consistent with the traditional sourcing of
institutional “noteworthiness” argued in previous studies ([10]). The former
president’s tweets, due to the authority of the presidency as an institution,
were often guaranteed selection. The celebrity status of Elon Musk led to a
similar selection advantage. The institutional backing for tweets as a source
creates new challenges in journalism. The former president’s tweets were
treated as newsworthy, even if not for endorsement reasons; thus, some
scholars argued against institutional backing as the monolithic criterion in
journalism coverage ([47]).

Similarly, prominent figures were also featured in the news and intriguingly,
journalists’ tweets were rather a prominent source. Thus, in relation to
authors of the source, the democratizing premises of Twitter were weaker. We
observe that in spite of the obvious possibilities presented by Twitter for
the inclusion of a wide variety of users as sources, journalistic practices
centering on institutional credibility are re-inscribed in online spaces, as
argued by previous studies where regardless of internet-democratizing
aspirations, it favors “elite” voices ([46]), even if regular users were also
included in the news stories. This finding goes against the Twitter as
democratizing sourcing hypothesis or egalitarian user integration for
newsworthiness. However, it is consistent with the previous findings showing
that when journalists did cite tweets from ordinary people, they are used to
represent the vox populi ([48]). These findings suggest that tweets functioned
in news stories as yet another form of “information subsidies” of the digital
age ([49]) where the internet has not only given citizens the means to
publish, but also has given more diverse channels for the elite to spread
their messages.

##  7 Conclusion and Future Work

This study presents an extensive data collection and empirical analysis of the
tweet embedding phenomenon focused on the temporal mechanics of embedding at
scale. At the outlet level, distinct temporal patterns were observed by outlet
type; similarly, at the user level, speed of embedding was also found to vary
by user type.

In this study, we observed how cross-pollination of tweets between
journalists—for multiple stories—is present in the news ecosystem, as a new
reporting mechanism and impact of tweet embedding, warranting future detailed
investigation. Our work likewise demonstrates the need for further qualitative
examination of this ecosystem. While the results presented in this study were
restricted to a focus on temporality, content-level dynamics of tweet
embedding also demand deeper study: what types of tweets are embedded from the
different types of users and by the different types of outlets? Do journalists
employ additional mechanisms of gatekeeping when selecting certain types of
content, while being more lenient when it comes to others? What are the
barriers that potentially newsworthy users must overcome to be included as
sources? Similarly, given that spatial dimensions, e.g, geospatial proximity
appeared to mark tweet selection in political news coverage ([2]), news
category specificity has to be further explored.

There are several implications and future directions of the study. In this
study we excluded stories with the tweets that were published with "negative
times." Even if our manual evaluation indicated that such instances are rather
rare where articles are updated to include new tweets after publication,
however, these “update” embedding events, where older articles are updated and
embedded with new tweets, constitute a unique and interesting journalistic
phenomenon that needs further investigation. Future studies should analyze
this phenomenon, in relation to the speed and filtering of the sourcing, i.e.,
how tweets are embedded in articles published _after_ tweet creation. In this
study we focused on tweet citations automatically extracted through tags;
future studies should analyze and compare different styles of tweet citation
in the news stories. We also found that some news stories used multiple tweets
for a story, which resembles traditional multi-source journalism practice.
Thus, future studies should analyze this phenomenon with more depth to analyze
types of stories and types of tweets selection practice.

Similarly, social media sourcing has broader societal implications in the
current media landscape, especially for user generated content, that is
plagued by threats of misinformation and disinformation ([50]). While efforts
of detecting such threats are promising ([51]), norms of sourcing should be
adjusted, as studies have shown that institutionally-trustworthy accounts
might not contain information that is trustworthy for citing ([47]). Further
studies along credibility considerations for tweet embedding should be
conducted.

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g=\phi^{\prime}\vartriangleright f\bullet\epsilon^{\prime}$, there exists a
unique 2-cell $\tau:h\Rightarrow h^{\prime}$ such that $\tau\vartriangleright
e\bullet\phi^{\prime}=\phi$.

Note that there is a choice for the direction of the 2-cell $\epsilon:e\cdot
f\Rightarrow e\cdot g$ since we could also have chosen $\epsilon:e\cdot
g\Rightarrow e\cdot f$ or we could have required this 2-cell to be inevrtible.
However, since we only consider bicategories in which all 2-cells are
invertible, this choice does not matter.

######  Problem 32.

Given algebras $A$ and $B$ for $\Sigma$ and algebra morphisms
$f,g:A\rightarrow B$, to construct the inserter of $f$ and $g$ in .

######  Construction 33 (for Problem 32; ).

The inserter of $f$ and $g$ is defined as the total algebra
$\textstyle\int{Y}$ of a displayed algebra $Y$ over $A$. Displayed algebras
were introduced in Definition 4. $Y$ is defined as follows:

  * •

The underlying family $Y$ of 1-types over $A$ is $Y(x):\equiv f(x)=g(x)$.

  * •

For each $x:\mathsf{A}(A)$, we are required to construct a map
$\overline{\mathsf{c}}^{Y}:\overline{\mathsf{A}}(Y)(x)\rightarrow
f(\mathsf{c}^{A}(x))=g(\mathsf{c}^{A}(x))$. Suppose we have
$\overline{x}:\overline{\mathsf{A}}(Y)(x)$. By induction on the polynomial
$\mathsf{A}$, it is possible to derive from $\overline{x}$ a path
$p:\mathsf{A}(f)(x)=\mathsf{A}(g)(x)$. We define
$\overline{\mathsf{c}}^{Y}(\overline{x})$ as the following concatenation of
paths:

$f(\mathsf{c}^{A}(x))\stackrel{{\scriptstyle\mathsf{c}^{f}(x)}}{{=}}\mathsf{c}^{B}(\mathsf{A}(f)(x))\stackrel{{\scriptstyle\mathsf{ap}\
\mathsf{c}^{B}\
p}}{{=}}\mathsf{c}^{B}(\mathsf{A}(g)(x))\stackrel{{\scriptstyle(\mathsf{c}^{g}(x))^{-1}}}{{=}}g(\mathsf{c}^{A}(x))$

  * •

The construction of paths $\overline{\mathsf{p}}^{Y}_{j}$ is relatively
involved, and we refer to the formalization for the details.

  * •

The construction of globes $\overline{\mathsf{h}}^{Y}_{j}$ is straightforward.
These are paths between paths in $Y(x)$, for some point $x$. Since $Y(x)$ is a
set, the required paths exist.

The 1-cell $e$ is the first projection out of the total algebra
$\textstyle\int{Y}$, which is an algebra morphism by construction. The 2-cell
$\epsilon$ is the second projection out of $\textstyle\int{Y}$. The algebra
$\textstyle\int{Y}$ satisfies the required universal property of the inserter
spelled out in Definition 8.2. ∎

###  8.3. Equifier

The equifier is finite limit in a bicategory, corresponding to a higher
version of the equalizer.

Let $\mathsf{B}$ be a bicategory. Let $A$ and $B$ be objects of $\mathsf{B}$,
let $f,g:A\rightarrow B$ and $\beta,\gamma:f\Rightarrow g$. The equifier of
$\beta$ and $\gamma$ is an object $E$ together with a 1-cell $e:E\rightarrow
A$ and a path $e\vartriangleleft\beta=e\vartriangleleft\gamma$. The pair
$(E,e)$ must satisfy the following universal property. Suppose we have

  * •

an object $E^{\prime}$;

  * •

a 1-cell $e^{\prime}:E^{\prime}\rightarrow A$;

  * •

a path $e^{\prime}\vartriangleleft\beta=e^{\prime}\vartriangleleft\gamma$.

Then there exists a 1-cell $h:E^{\prime}\rightarrow E$ together with a 2-cell
$\phi:h\cdot e\Rightarrow e^{\prime}$. The pair $(h,\phi)$ is unique up to
unique 2-cell, which means that given another 1-cell
$h^{\prime}:E^{\prime}\rightarrow E$ and another 2-cell
$\phi^{\prime}:h^{\prime}\cdot e\Rightarrow e^{\prime}$, there exists a unique
2-cell $\tau:h\Rightarrow h^{\prime}$ such that $\tau\vartriangleright
e\bullet\phi^{\prime}=\phi$.

######  Problem 34.

Given algebras $A$ and $B$ for $\Sigma$, given 1-cells $f,g:A\rightarrow B$
and 2-cells $\beta,\gamma:f\Rightarrow g$, to construct the equifier of
$\beta$ and $\gamma$ in .

######  Construction 35 (for Problem 34; ).

Similar to the construction of the inserter, we define the equifier of $\beta$
and $\gamma$ as the total algebra $\textstyle\int{Y}$ of the following
displayed algebra $Y$ over $A$.

  * •

The underlying family $Y$ of 1-types over $A$ is
$Y(x):\equiv\beta(x)=\gamma(x)$.

  * •

For each $x:\mathsf{A}(A)$, we are required to construct a map
$\overline{\mathsf{c}}^{Y}:\overline{\mathsf{A}}(Y)(x)\rightarrow\beta(\mathsf{c}^{A}(x))=\gamma(\mathsf{c}^{A}(x))$.
Assume given $\overline{x}:\overline{\mathsf{A}}(Y)(x)$. By induction on the
polynomial $\mathsf{A}$, it is possible to derive from $\overline{x}$ a path
$p:\mathsf{A}(\beta)(x)=\mathsf{A}(\gamma)(x)$. We define
$\overline{\mathsf{c}}^{Y}(\overline{x})$ as the following concatenation of
paths:

$\displaystyle\beta(\mathsf{c}^{A}(x))$
$\displaystyle=\beta(\mathsf{c}^{A}(x))\,@\,\operatorname{\mathbf{idpath}}(g(\mathsf{c}^{A}(x)))$
$\displaystyle=\beta(\mathsf{c}^{A}(x))\,@\,(\mathsf{c}^{g}(x)\,@\,(\mathsf{c}^{g}(x))^{-1})$
$\displaystyle=(\beta(\mathsf{c}^{A}(x))\,@\,\mathsf{c}^{g}(x))\,@\,(\mathsf{c}^{g}(x))^{-1}$
$\displaystyle=(\mathsf{c}^{f}(x)\,@\,\mathsf{ap}\ \mathsf{c}^{B}\
(\mathsf{A}(\beta)(x)))\,@\,(\mathsf{c}^{g}(x))^{-1}$ (by
$\mathsf{c}^{\beta}(x)$) $\displaystyle=(\mathsf{c}^{f}(x)\,@\,\mathsf{ap}\
\mathsf{c}^{B}\ (\mathsf{A}(\gamma)(x)))\,@\,(\mathsf{c}^{g}(x))^{-1}$ (by
$p$)
$\displaystyle=(\gamma(\mathsf{c}^{A}(x))\,@\,\mathsf{c}^{g}(x))\,@\,(\mathsf{c}^{g}(x))^{-1}$
(by $\mathsf{c}^{\gamma}(x)$)
$\displaystyle=\gamma(\mathsf{c}^{A}(x))\,@\,(\mathsf{c}^{g}(x)\,@\,(\mathsf{c}^{g}(x))^{-1})$
$\displaystyle=\gamma(\mathsf{c}^{A}(x))\,@\,\operatorname{\mathbf{idpath}}(g(\mathsf{c}^{A}(x)))$
$\displaystyle=\gamma(\mathsf{c}^{A}(x))$

  * •

The construction of paths $\overline{\mathsf{p}}^{Y}_{j}$ and globes
$\overline{\mathsf{h}}^{Y}_{j}$ is straightforward. These are respectively
paths and paths between paths in $Y(x)$ for some point $x$. Since $Y(x)$ is a
proposition, these constructions are all trivial.

The 1-cell $e$ is the first projection out of the total algebra
$\textstyle\int{Y}$, which is an algebra morphism by construction. The 2-cell
$\epsilon$ is the second projection out of $\textstyle\int{Y}$. The algebra
$\textstyle\int{Y}$ satisfies the required universal property of the equifier
spelled out in Definition 8.3. ∎

##  9\. The Free Algebra

In this section, we discuss the free algebra for a signature $\Sigma$ and show
that it gives rise to a left biadjoint pseudofunctor from 1-types to algebras
in 1-types for $\Sigma$. From this, we conclude that each signature generates
a pseudomonad on the bicategory of 1-types [Lac00].

We construct the free algebra for $\Sigma$ as a biinitial algebra for a
modified version of $\Sigma$. More specifically, suppose that we have a
signature $\Sigma$ and a 1-type $A$. We first construct another signature,
called the _free signature_ , which has all the constructors of $\Sigma$ and
an additional point constructor with arguments from $A$. Then we define the
free $\Sigma$-algebra on $A$ to be the biinitial algebra for the free
signature.

Let $\Sigma$ be a signature and let $A$ be a 1-type. Define the free signature
$\mathsf{FreeSig}_{\Sigma}(A)$ as follows

  * •

$\mathsf{A}^{\mathsf{FreeSig}_{\Sigma}(A)}:\equiv\mathsf{A}^{\Sigma}+\operatorname{\mathbf{C}}(A)$

  * •

Note that each path endpoint $e:\mathsf{E}_{\mathsf{A}^{\Sigma}}(P,Q)$ gives
rise to another path endpoint
$\widehat{e}:\mathsf{E}_{\mathsf{A}^{\mathsf{FreeSig}_{\Sigma}(A)}}(P,Q)$. We
choose
$\mathsf{J}^{\mathsf{FreeSig}_{\Sigma}(A)}_{\mathsf{P}}:\equiv\mathsf{J}^{\Sigma}_{\mathsf{P}}$
and $\mathsf{S}^{\mathsf{FreeSig}_{\Sigma}(A)}:\equiv\mathsf{S}^{\Sigma}$.
Lastly, we define the endpoints
$\mathsf{l}^{\mathsf{FreeSig}_{\Sigma}(A)}_{j}$ and
$\mathsf{r}^{\mathsf{FreeSig}_{\Sigma}(A)}_{j}$ to be
$\widehat{\mathsf{l}^{\Sigma}_{j}}$ and $\widehat{\mathsf{r}^{\Sigma}_{j}}$
respectively.

  * •

To define the homotopy endpoints of $\mathsf{FreeSig}_{\Sigma}(A)$, we first
note that each
$h:\mathsf{H}_{\mathsf{l}^{\Sigma},\mathsf{r}^{\Sigma},a,b}(s,t)$ gives rise
to a homotopy endpoint
$\widehat{h}:\mathsf{H}_{\mathsf{l}^{\mathsf{FreeSig}_{\Sigma}(A)},\mathsf{r}^{\mathsf{FreeSig}_{\Sigma}(A)},\widehat{a},\widehat{b}}(\widehat{s},\widehat{t})$.
Now we define $\mathsf{p}^{\mathsf{FreeSig}_{\Sigma}(A)}$ and
$\mathsf{q}^{\mathsf{FreeSig}_{\Sigma}(A)}$ to be
$\widehat{\mathsf{p}^{\Sigma}}$ and $\widehat{\mathsf{q}^{\Sigma}}$
respectively.

We define the free $\Sigma$-algebra on $A$ to be the biinitial
$\mathsf{FreeSig}_{\Sigma}(A)$-algebra. The free $\Sigma$-algebra on $A$ is
denoted by $\mathsf{FreeAlg}_{\Sigma}(A)$ and the inclusion is denoted by
$\mathsf{inc}_{\Sigma}:A\rightarrow\mathsf{FreeAlg}_{\Sigma}(A)$.

Note that the free algebra exists by Construction 29. The free signature of
the signature for monoidal objects (Subsection 7.4) is given as follows

⬇

Inductive $\mathsf{FreeMon}(A)$ :=

| $\operatorname{\mathbf{inc}}$ : $A\rightarrow\mathsf{FreeMon}(A)$

| $\operatorname{\mathbf{u}}$ : $\mathsf{FreeMon}(A)$

| $\operatorname{\mathbf{m}}$ :
$\mathsf{FreeMon}(A)\rightarrow\mathsf{FreeMon}(A)\rightarrow\mathsf{FreeMon}(A)$

| $\operatorname{\mathbf{lam}}$ :
$\prod(x:\mathsf{FreeMon}(A)),\operatorname{\mathbf{m}}(\operatorname{\mathbf{u}},x)=x$

| $\operatorname{\mathbf{rho}}$ :
$\prod(x:\mathsf{FreeMon}(A)),\operatorname{\mathbf{m}}(x,\operatorname{\mathbf{u}})=x$

| $\operatorname{\mathbf{al}}$ :
$\prod(x,y,z:\mathsf{FreeMon}(A)),\operatorname{\mathbf{m}}(x,(\operatorname{\mathbf{m}}(y,z)))=\operatorname{\mathbf{m}}(\operatorname{\mathbf{m}}(x,y),z)$

| $\operatorname{\mathbf{tr}}$ : $\prod(x,y:\mathsf{FreeMon}(A)),\mathsf{ap}\
(\lambda\,z.\hskip 1.79997pt plus 0.90005pt\operatorname{\mathbf{m}}(x,z))\
(\operatorname{\mathbf{lam}}(y))=\operatorname{\mathbf{al}}(x,\operatorname{\mathbf{u}},y)\bullet\mathsf{ap}\
(\lambda\,z.\hskip 1.79997pt plus 0.90005pt\operatorname{\mathbf{m}}(z,y))\
(\operatorname{\mathbf{rho}}(x))$

| $\operatorname{\mathbf{pent}}$ : $\prod(w,x,y,z:\mathsf{FreeMon}(A)),$

$\operatorname{\mathbf{al}}(w,x,\operatorname{\mathbf{m}}(y,z))\bullet\operatorname{\mathbf{al}}(\operatorname{\mathbf{m}}(w,x),y,z)$

=

$\mathsf{ap}\ (\lambda\,v.\hskip 1.79997pt plus
0.90005pt\operatorname{\mathbf{m}}(w,v))\
(\operatorname{\mathbf{al}}(x,y,z))\bullet\operatorname{\mathbf{al}}(w,\operatorname{\mathbf{m}}(x,y),z)\bullet\mathsf{ap}\
(\lambda\,v.\hskip 1.79997pt plus 0.90005pt\operatorname{\mathbf{m}}(v,z))\
(\operatorname{\mathbf{al}}(w,x,y))$

Note that this corresponds to the free monoidal groupoid as given by
Piceghello [Pic19]. To construct the desired left biadjoint pseudofunctor, we
first give a more convenient formulation of the biinitiality of
$\mathsf{FreeAlg}_{\Sigma}(A)$. By stating the mapping properties with
algebras for $\Sigma$, we can apply them directly when constructing the
biadjunction.

######  Corollary 36.

Let $\Sigma$ be a signature and let $A$ be a 1-type. Suppose that we have
$Y:\mathsf{Alg}(\Sigma)$ and $\iota_{Y}:A\rightarrow Y$.

  * •

There is an algebra morphism $f:\mathsf{FreeAlg}_{\Sigma}(A)\rightarrow Y$
such that for all $a:A$, we have $f(\mathsf{inc}_{\Sigma}(a))=\iota_{Y}(a)$.

  * •

Given algebra morphisms $f_{1},f_{2}:\mathsf{FreeAlg}_{\Sigma}(A)\rightarrow
Y$ and paths $p_{1}:f_{1}(\mathsf{inc}_{\Sigma}(a))=\iota_{Y}(a)$ and
$p_{2}:f_{2}(\mathsf{inc}_{\Sigma}(a))=\iota_{Y}(a)$ for each $a:A$, there is
a unique 2-cell $\theta:f_{1}\Rightarrow f_{2}$ such that
$\theta(\mathsf{inc}_{\Sigma}(a))\bullet\mathsf{c}^{f}(\operatorname{\mathbf{inr}}(a))=\mathsf{c}^{g}(\operatorname{\mathbf{inr}}(a))$
for each $a:A$.

Now we have enough in place to construct the desired biadjunction.

######  Problem 37.

Given a signature $\Sigma$, to construct a coherent biadjunction
$\mathsf{F}_{\Sigma}\dashv\mathsf{U}_{\Sigma}$ where
$\mathsf{U}_{\Sigma}:\mathsf{Pseudo}(\mathsf{Alg}(\Sigma),1\mbox{-}\mathsf{Type})$
is the forgetful pseudofunctor.

######  Construction 38 (for Problem 37; ).

We only indicate how to construct the left biadjoint. To construct
$\mathsf{F}_{\Sigma}:\mathsf{Pseudo}(1\mbox{-}\mathsf{Type},\mathsf{Alg}(\Sigma))$,
we need to give an algebra $\mathsf{F}_{\Sigma}(A)$ for
$A:1\mbox{-}\mathsf{Type}$. We define $\mathsf{F}_{\Sigma}(A)$ to be the
biinitial algebra for $\mathsf{FreeSig}_{\Sigma}(A)$. Pseudofunctoriality
follows from Corollary 36. ∎

From all of this, we conclude that each signature gives rise to a pseudomonad
on 1-types.

######  Proposition 39 (Proposition 5.1 in [Lac00]).

Each coherent biadjunction gives rise to a pseudomonad.

######  Corollary 40.

Each signature gives rise to a pseudomonad on 1-types.

##  10\. The First Isomorphism Theorem

The first isomorphism theorem is one of the classical results in universal
algebra. While this statement is usually about algebraic structures in sets
[LS19], we look at a generalization to the 1-truncated case. More
specifically, we formulate and prove this theorem for algebras for the
signatures defined in Definition 3.1.

Before stating and proving the isomorphism theorem, we need to generalize
several notions from universal algebra to the 1-truncated case. First of all,
following the approach of Section 8, we define the image of an algebra
morphism using displayed algebras. Second of all, we define congruence
relations and their quotients. The main idea here is to use Construction 24,
where we lifted the groupoid quotient to a biadjunction from
$\mathsf{Alg}_{\mathsf{Grpd}}(\Sigma)$ to $\mathsf{Alg}(\Sigma)$. This result
also indicates how congruence relations will be defined. Basically, a
congruence relation on $X:\mathsf{Alg}(\Sigma)$ is a groupoid structure on $X$
which gives an algebra of groupoids. Once these notions are in place, we can
formulate and prove the first isomorphism theorem similarly to the set-
theoretical version.

###  10.1. The Image

We start by defining the image of an algebra morphism. For each
$f:A\rightarrow B$ we are after a factorization
$A\rightarrow\mathsf{Im}{f}\rightarrow B$.

######  Problem 41.

Given algebras $A$ and $B$ for a signature $\Sigma$ and an algebra morphism
$f:A\rightarrow B$, to construct an algebra $\mathsf{Im}(f)$ and morphisms
$\mathsf{proj}_{f}:A\rightarrow\mathsf{Im}{f}$ and
$\mathsf{inc}_{f}:\mathsf{Im}(f)\rightarrow B$.

######  Construction 42 (for Problem 41; ).

We define $\mathsf{Im}(f)$ to be the total algebra $\textstyle\int{Y}$ of the
following displayed algebra $Y$ over $B$:

  * •

The underlying family of 1-types over $B$ is $Y(x):\equiv\exists a:A.\
f(a)=x$.

  * •

For each $x:\mathsf{A}(A)$, we are required to construct a map
$\overline{\mathsf{c}}^{Y}:\overline{\mathsf{A}}(Y)(x)\rightarrow\exists a:A.\
f(a)=\mathsf{c}^{B}(x)$. Suppose we have
$\overline{x}:\overline{\mathsf{A}}(Y)(x)$. By induction on the polynomial
$\mathsf{A}$, it is possible to derive from $\overline{x}$ an inhabitant of
$\exists y:\mathsf{A}(A).\ \mathsf{A}(f)(y)=x$. By invoking the elimination
principle of propositional truncation, it is sufficient to construct a map
$f:\sum_{y:\mathsf{A}(A)}\hskip 1.99997pt plus 1.00006pt\
\mathsf{A}(f)(y)=x\rightarrow\sum_{a:A}\hskip 1.99997pt plus 1.00006pt\
f(a)=\mathsf{c}^{B}(x)$ in order to define the map
$\overline{\mathsf{c}}^{Y}$. So, assume that we have $y:\mathsf{A}(A)$ and a
path $p:\mathsf{A}(f)(y)=x$. Take $f(y,p)$ to be the pair consisting of
$\mathsf{c}^{A}(y)$ and the path

$f(\mathsf{c}^{A}(y))\stackrel{{\scriptstyle\mathsf{c}^{f}(y)}}{{=}}\mathsf{c}^{B}(\mathsf{A}(f)(y))\stackrel{{\scriptstyle\mathsf{ap}\
\mathsf{c}^{B}\ p}}{{=}}\mathsf{c}^{B}(x)$

  * •

Since $Y(x)$ is a proposition for all $x:B$, the construction of paths
$\overline{\mathsf{p}}^{Y}_{j}$ and globes $\overline{\mathsf{h}}^{Y}_{j}$ is
straightforward.

The map $\mathsf{inc}_{f}$ is the first projection and $\mathsf{proj}_{f}$
sends $x:A$ to
$(f(x),\operatorname{\mathbf{inc}}(x,\operatorname{\mathbf{idpath}}(f(x))))$,
where $\operatorname{\mathbf{inc}}$ is the point constructor of propositional
truncation. ∎

###  10.2. Congruence Relations

Next we define congruence relations and we show that each such relation gives
an algebra in groupoids. The difficulty here is constructing the homotopy
constructor of that algebra. For this reason, we define congruence relations
in two steps, and we start by defining _path congruence relations_ and showing
that these give rise to path algebras in groupoids.

Let $\Sigma$ be a signature and let $X:\mathsf{Alg}(\Sigma)$. A path
congruence relation $R$ on $X$ consists of

  * •

a groupoid structure $R$ on $X$;

  * •

for each $x,y:\mathsf{A}^{\Sigma}(X)$ and $f:\mathsf{A}^{\Sigma}(R)(x,y)$, a
morphism $\mathsf{c}^{R}:R(\mathsf{c}^{X}(x),\mathsf{c}^{X}(y))$ and a proof
that this assignment is functorial;

  * •

a proof that the assignment
$\lambda\,x,\mathsf{idtoiso}(\mathsf{p}^{X}_{j}(x))$ gives a natural
transformation
$\langle\mathsf{l}^{\Sigma}_{j}\rangle\Rightarrow\langle\mathsf{r}^{\Sigma}_{j}\rangle$
for each $j:\mathsf{J}^{\Sigma}_{\mathsf{P}}$.

######  Problem 43.

Given a path congruence relation $R$, to construct
$\widehat{R}:\mathsf{PathAlg}_{\mathsf{Grpd}}(\Sigma)$.

######  Construction 44 (for Problem 43; ).

We only discuss the data involved.

  * •

The carrier $\widehat{R}$ is the groupoid whose type of objects is $X$ and
whose morphisms are given by $R$.

  * •

Next we define a functor
$\langle\mathsf{A}^{\Sigma}\rangle(\widehat{R})\rightarrow\widehat{R}$. It is
defined to be $\mathsf{c}^{X}$ on objects and $\mathsf{c}^{R}$ on morphisms.

  * •

Lastly, we define a natural transformation
$\langle\mathsf{l}^{\Sigma}_{j}\rangle\Rightarrow\langle\mathsf{r}^{\Sigma}_{j}\rangle$
for each $j:\mathsf{J}^{\Sigma}_{\mathsf{P}}$. The component function of this
transformation is $\lambda\,x,\mathsf{idtoiso}(\mathsf{p}^{X}_{j}(x))$. ∎

To show that $\widehat{R}$ is an algebra, we also need to give the homotopy
constructor. Since this constructor has both a point and a path constructor,
we need to check an equality of morphisms of $\widehat{R}$, which depend on
points of $X$ and morphisms of $\widehat{R}$. As a result, we cannot reuse the
homotopy constructor of $X$, because it only depends on points of $X$ and
paths in $X$. This means that to construct the groupoid algebra, the homotopy
constructor needs to be constructed from scratch. This is reflected in the
following definition.

A path congruence relation $R$ is a congruence relation if $\widehat{R}$ is an
algebra in groupoids.

Note that each congruence relation $R$ gives rise to an algebra
$\widehat{R}:\mathsf{Alg}_{\mathsf{Grpd}}(\Sigma)$. By Construction 24, the
groupoid quotient lifts to
$\mathsf{GQuot}_{\mathsf{Alg}}:\mathsf{Pseudo}(\mathsf{Alg}_{\mathsf{Grpd}}(\Sigma),\mathsf{Alg}(\Sigma))$.
Hence, we can construct an algebra
$\mathsf{GQuot}_{\mathsf{Alg}}(\widehat{R}):\mathsf{Alg}(\Sigma)$ from a
congruence relation $R$. Note that the map
$\operatorname{\mathbf{gcl}}:X\rightarrow\mathsf{GQuot}_{\mathsf{Alg}}(\widehat{R})$
is a homomorphism of algebras. Since $\mathsf{GQuot}_{\mathsf{Alg}}$ is a left
biadjoint, we also get a mapping property for
$\mathsf{GQuot}_{\mathsf{Alg}}(\widehat{R})$.

######  Remark 45.

Suppose that we have two algebras $X,Y:\mathsf{Alg}(\Sigma)$ and a congruence
relation $R$ on $X$. From the biadjunction, we get a unique morphism
$\overline{g}:\mathsf{GQuot}_{\mathsf{Alg}}{\widehat{R}}\rightarrow Y$ from a
functor $g:R\rightarrow\mathsf{PathGrpd}_{\mathsf{Alg}}{Y}$ such that
$\overline{g}(\operatorname{\mathbf{gcl}}(x))=g(x)$ and $\mathsf{ap}\
\overline{g}\ (\operatorname{\mathbf{gcleq}}(p))=g(p)$ where $p$ is a morphism
in $R$ from $x$ to $y$.

Using this, we can give conditions for when an algebra morphism
$f:X\rightarrow Y$ factors through
$\mathsf{GQuot}_{\mathsf{Alg}}{\widehat{R}}$. We can unfold the definition to
find the necessary ingredients for the factorization. For example, we need to
show that $f$ lifts to a functor, which means we need to provide a path
$f_{2}:f(x)=f(y)$ for each $x,y:X$ and $p:R(x,y)$, and equalities
$f_{2}(\operatorname{\mathbf{refl}}(x))=\operatorname{\mathbf{refl}}(f_{2}(x))$
and $f_{2}(p\cdot q)=f_{2}(p)\bullet f_{2}(q)$.

###  10.3. The First Isomorphism Theorem

Now let us prove a generalization of the first isomorphism theorem to the
1-truncated case. Note that our proof follows the same steps as the proof of
the first isomorphism theorem for sets. Let us start by characterizing adjoint
equivalences in $\mathsf{Alg}(\Sigma)$.

######  Proposition 46.

Suppose we have a signature $\Sigma$, algebras $X,Y:\mathsf{Alg}(\Sigma)$, and
an algebra morphism $f:X\rightarrow Y$. Then $f$ is an adjoint equivalence in
$\mathsf{Alg}(\Sigma)$ if its carrier is an adjoint equivalence in
$1\mbox{-}\mathsf{Type}$.

Before we state and prove the isomorphism theorem, we define the kernel of
algebra morphisms.

[Kernel] Given a signature $\Sigma$, algebras $X,Y:\mathsf{Alg}(\Sigma)$, and
an algebra morphism $f:X\rightarrow Y$, we define a groupoid algebra
$\mathsf{Ker}(f)$ on $X$, called the kernel of $f$, by setting $R(x,y):\equiv
f(x)=f(y)$.

######  Theorem 47.

Let $\Sigma$ be a signature and suppose we have algebras
$X,Y:\mathsf{Alg}(\Sigma)$ and an algebra morphism $f:X\rightarrow Y$. Then

  * •

there is a unique map
$\overline{f}:\mathsf{GQuot}_{\mathsf{Alg}}(\widehat{\mathsf{Ker}(f)})\rightarrow
Y$ such that we have propositional equalities
$p:\prod(x:X),\overline{f}(\operatorname{\mathbf{gcl}}(x))=f(x)$ and
$\mathsf{ap}\ \overline{f}\ (\operatorname{\mathbf{gcleq}}(r))\bullet
p(y)=p(x)\bullet r$ where $r:f(x)=f(y)$;

  * •

there is an adjoint equivalence
$\widetilde{f}:\mathsf{GQuot}_{\mathsf{Alg}}(\widehat{\mathsf{Ker}(f)})\rightarrow\mathsf{Im}(f)$
such that $\pi_{1}(\widetilde{f}(x))=\overline{f}(x)$.

######  Proof 10.1.

Using Remark 45, we can define the maps
$\overline{f}:\mathsf{GQuot}_{\mathsf{Alg}}(\widehat{\mathsf{Ker}(f)})\rightarrow
Y$ and
$\widetilde{f}:\mathsf{GQuot}_{\mathsf{Alg}}(\widehat{\mathsf{Ker}(f)})\rightarrow\mathsf{Im}(f)$.
Note that from Remark 45, we also get that $\overline{f}$ satisfies the
required equalities and that $\overline{f}$ is unique. To show that
$\overline{f}$ is an adjoint equivalence, we use Proposition 46, and we do
that by showing that the fibers are contractible. Proving that the fibers are
inhabited and propositional is similar to proving the surjectivity and
injectivity for the set-theoretical first isomorphism theorem [LS19].

##  11\. Calculating Fundamental Groups

Let us finish by using Construction 29 to determine the fundamental group of
some HITs. Such results are often proven by encode-decode method [LS13, LF14],
but we take a different approach. We only use encode-decode to determine the
path space of the groupoid quotient. For the HITs considered in this section,
we give a simpler description of the initial groupoid algebra, which fixes the
fundamental group. We start by determining the path space of the groupoid
quotient.

######  Proposition 48.

Let $G$ be a groupoid and let $x$ and $y$ be objects in $G$. Then the types
$\operatorname{\mathbf{gcl}}(x)=\operatorname{\mathbf{gcl}}(y)$ and $G(x,y)$
are equivalent.

As a result, the type of paths between two points in a HIT is the type of
morphisms in the initial groupoid algebra. Since this algebra is unique, we
can determine the fundamental group by finding a simpler description of the
initial groupoid algebra.

###  11.1. Circle

Recall that the circle is defined as the following HIT

⬇

Inductive $S^{1}$ :=

| $\operatorname{\mathbf{base}}_{S^{1}}$ : $S^{1}$

| $\operatorname{\mathbf{loop}}_{S^{1}}$ :
$\operatorname{\mathbf{base}}_{S^{1}}=\operatorname{\mathbf{base}}_{S^{1}}$

We can define a signature $S^{1}$ that represents this HIT. Notice that the
HIT generated by this signature includes also a 1-truncation constructor,
which is superflous in this case since $S^{1}$ is already provably
1-truncated.

Next we construct a groupoid algebra $\mathcal{S}$ of this signature and we
prove that $\mathcal{S}$ is biinitial.

We define a groupoid $\mathcal{S}$ as follows

  * •

the type of objects is the unit type;

  * •

the type of morphisms from $\operatorname{\mathbf{tt}}$ to
$\operatorname{\mathbf{tt}}$ is the type integers.

######  Problem 49.

To construct an $S^{1}$-algebra structure on $\mathcal{S}$.

######  Construction 50 (for Problem 49; ).

To construct the desired algebra structure, we first need to define a functor
$\mathfrak{b}$ from the unit category to $\mathcal{S}$. This functor sends the
unique element to $\operatorname{\mathbf{tt}}$. Furthermore, we need to define
a natural transformation $\mathfrak{l}$ from $\mathfrak{b}$ to $\mathfrak{b}$.
On each component, this transformation is defined to be $1$. ∎

######  Proposition 51.

The groupoid $\mathcal{S}$ is biinitial in
$\mathsf{Alg}_{\mathsf{Grpd}}(S^{1})$.

From Construction 29 and the fact that biinitial objects are unique up to
equivalence, we can deduce that the circle is the groupoid quotient of
$\mathcal{S}$ and that its base point $\operatorname{\mathbf{base}}_{S^{1}}$
is $\operatorname{\mathbf{gcl}}(\operatorname{\mathbf{tt}})$. Since the
morphisms of $\mathcal{S}$ are just the integers, we immediately get the
following from Proposition 48

######  Corollary 52.

The type
$\operatorname{\mathbf{base}}_{S^{1}}=\operatorname{\mathbf{base}}_{S^{1}}$ is
equivalent to the integers.

###  11.2. Torus

Next we look at the torus, which we defined in Example 3.1. We use the same
approach to determine its fundamental group, so we start by giving a simpler
description of the biinitial algebra for $\mathcal{T}^{2}$ in groupoids.

Define a groupoid $\mathcal{T}$ as follows

  * •

the type of objects is the unit type;

  * •

the type of morphisms from $\operatorname{\mathbf{tt}}$ to
$\operatorname{\mathbf{tt}}$ is $\mathbb{Z}\times\mathbb{Z}$.

######  Problem 53.

To construct an $\mathcal{T}^{2}$-algebra structure on $\mathcal{T}$.

######  Construction 54 (for Problem 53; ).

Again the point constructor is given by a functor $\mathfrak{b}$ from the unit
category to $\mathcal{T}$, which sends the unique element to
$\operatorname{\mathbf{tt}}$. The two natural transformations $\mathfrak{l}$
and $\mathfrak{r}$ from $\mathfrak{b}$ to $\mathfrak{b}$ send each element to
$(1,0)$ and $(0,1)$ respectively. For the last component, we need to give an
equality $\mathfrak{s}$ between $(1,0)+(0,1)=(0,1)+(1,0)$. This holds
definitionally. ∎

######  Proposition 55.

The groupoid $\mathcal{T}$ is biinitial in
$\mathsf{Alg}_{\mathsf{Grpd}}(\mathcal{T}^{2})$.

Again we can deduce that the torus is the groupoid quotient of $\mathcal{T}$
and that its base point $\operatorname{\mathbf{base}}$ is
$\operatorname{\mathbf{gcl}}(\operatorname{\mathbf{tt}})$. From Proposition
48, we immediately get

######  Corollary 56.

The type $\operatorname{\mathbf{base}}=\operatorname{\mathbf{base}}$ is
equivalent to $\mathbb{Z}\times\mathbb{Z}$.

###  11.3. Group Quotient

For the remainder of this section, we assume that a group $G$ is given. Our
goal is to show that the fundamental group of the group quotient at its base
point is $G$. Again we do this by determining the biinitial groupoid algebra.
The algebra structure is constructed in a similar fashion to Construction 54.

We define a groupoid $\mathcal{G}$ as follows

  * •

the type of objects is the unit type;

  * •

the type of morphisms from $\operatorname{\mathbf{tt}}$ to
$\operatorname{\mathbf{tt}}$ is $G$.

######  Problem 57.

To construct an $\mathsf{GroupQuot(G)}$-algebra structure on $\mathcal{G}$.

######  Proposition 58.

The groupoid $\mathcal{G}$ is biinitial in
$\mathsf{Alg}_{\mathsf{Grpd}}(\mathsf{GroupQuot(G)})$.

######  Corollary 59.

The fundamental group of the group quotient of $G$ is $G$ itself.

##  12\. Conclusion and Further Work

We showed how to construct finitary 1-truncated higher inductive types using
the propositional truncation, quotient, and the groupoid quotient. This
reduces the existence of a general class of HITs to simpler ones. We needed
the types to be 1-truncated, so that we could use the framework of bicategory
theory, and the HITs we studied had to be finitary to guarantee that the
groupoid quotient commutes with the involved operations [CUV19]. On the way,
we also proved that HITs are unique and we studied universal algebra with our
signatures. We showed that the bicategory of algebras has finite limits and we
proved the first isomorphism theorem for these algebras. Lastly, we used the
way we constructed HITs to calculate fundamental groups.

There are numerous ways to improve on these results. First of all, we only
constructed finite limits of algebras while it should also be possible to
construct finite colimits of algebras. The scheme studied in this paper is not
flexible enough to support these colimits since we do not have a path endpoint
that represents the action of a polynomial on the point constructor. Hence, if
we want to internally construct these colimits, then we need to define a more
permissive signature for higher inductive types.

Secondly, it should be possible to modify our approach to obtain HITs in
directed type theory (DTT) [Nor19]. In the model of DTT provided by North,
types are interpreted as categories and higher inductive types in DTT could be
interpreted as initial algebras. We constructed such algebras in the
bicategory of groupoids and in a similar way, one should be able to construct
the desired algebras in the bicategory of categories.

Lastly, our construction only considers a rather simple scheme of HITs. In
particular, we restrict ourselves to the 1-truncated case. Since untruncated
types correspond to $\infty$-groupoids, generalizing the methods used in this
paper to the untruncated case, requires formalizing notions from
$\infty$-category theory in type theory [AR12, CK18, FM17]. This also requires
finding an $\infty$-dimensional generalization of the groupoid quotient. An
alternative approach to deal with untruncated HITs, pointed out by Ali
Caglayan, would be using wild categories [HHT15, KvR19]. Note that
generalizing Construction 27 would also pose a challenge, because the set
quotient cannot be used. Instead all $n$-morphisms must be freely generated by
the $n$-path constructors. We would also like to extend our scheme to
incorporate both indexed HITs and higher inductive-inductive types [CH19,
KK18].

## Acknowledgments

The authors thank Herman Geuvers, Dan Frumin, Benedikt Ahrens, and Ali
Caglayan for helpful comments and discussions. The authors also thank the
anonymous reviewers for their helpful comments and suggestions. Niccolò Veltri
was supported by the Estonian Research Council grant PSG659 and by the ESF
funded Estonian IT Academy research measure (project 2014-2020.4.05.19-0001).

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