Bump, and pic (this and the last pic were both for this board game2)(it was rushed for a competition and probably terribly unbalanced so don't buy it lmao :'D):
18 days later
I aspire to be you one day, @TheLemmaLlama
11 days later
Well, I might be going too far here, but I've been procrastinating too much. This will be my external motivator
So, without further ado, here is my Artistic License:
Yeah, this is not like me at all, but I'm feeling extremely risky (Zoloft, lol!). Maybe a 90 day temporary ban from the forums will motivate me into getting more of my comic done. Time for regrets is later.
I wish you all well, whatever happens next! (And hopefully I'll see all you fuckers in October!
(Feel free to copy this license and use it however way you like! I don't give a shit! XD)
26 days later
Posting this animation again bc it's the only thing I did since my last bump:
Anyway, I'm going to retire my 'one picture per bump' thing as I really need to work on my thesis, which is due July next year 
Instead, I'm going to use this thread to hold me accountable for actually making progress on my research! (I'll post what I've done here, even if no-one finds it interesting or comprehensible, just bc it helps with accountability XD)(I will probably return to some manner of art/comic progress accountability once I'm done with my degree, one way or another)
Each week, I must come up with concrete (showable, written down) goals for the following week. If I either fail to meet my objective OR fail to come up with a concrete objective for the following week, I must tell you and give one free drawing to the first person to respond :]
So by next Tuesday, I must:
- Formalize the full theory of partially saturated anabicategories, as an essentially algebraic theory
- Write something about how pushouts might look like in the category of models of the above structure and strict model maps between them
- Write something about where to assign which 2-cells get mapped to by the composition and identity anafunctors
Alright, I don't think I can meet the above by the end of today*, so Imma offer one drawing to the first person to make a request after this comment. (Any following requests will be disregarded; you'll have to repost the next time I screw up, so I suggest not posting if you see someone has already posted :])
You can request anything that I'm allowed to post on this forum. It will be coloured and roughly shaded, but messy.
Drawing a cute black cat in white bunny onesie for @DasIstWunderblyat :]
*In part because I bumped into something that I believe made the first goal impossible, and the second goal is also technically impossible without completing the first goal. In future I will specify my goals more clearly to allow for success despite unexpected roadblocks XD (I don't think I would've completed all of these even without the roadblock though; I did procrastinate quite a bit XD Which just means this plan is working, as it got me to actually push myself :P)
By next Tuesday, I must:
- See if one can get uniqueness of the saturation property from the other stuff, OR explore the implications of non-uniqueness. Write something about whether this causes any issues.
- Write at least one other thing related to my topic, but doesn't contribute to the above.
Sorry I have just seen this (notification not working)
This is absolutely adorable! Love the floppy ears! Thank you!
- See if one can get uniqueness of the saturation property from the other stuff, OR explore the implications of non-uniqueness. Write something about whether this causes any issues
Non-uniqueness would cause issues in that there would be many anafunctors that more or less send the same stuff to the same stuff, that would not be isomorphic as essentially algebraic structures, so to speak
- Write at least one other thing related to my topic, but doesn't contribute to the above.
https://drive.google.com/file/d/16-TsvFlXdU14tkPDVmlSUolSSvyZaSbB/view?usp=sharing1
- assigning 2-cell mappings necessarily require the 'partial' operations id' and \comp', which takes us outside functor categories
- so then just do it directly in the full structure? with generating cofibrations for the model structure on bicats, lack first worked in cat-graph though, which is also not a functor cat
By next Tuesday, I must:
- read through it more carefully to see how this helps, and write some concrete obsevation about that, and
- write down 1 concrete property of how I think pushouts would look like in the full structure
helps bc there's a free [bicat] generated by any given [cat-graph] (and presumably it would still be easier to work with than the full structure in some way?)
on the other hand, with the structure I thought, you don't get uniquely determined associators and unitors though. But if we add associators and unitors to the structure, it's already partial anabicategories; only the saturation is missing. So I'm wondering if it's even worth specifying this substructure. but I also kind of see a case for it being the analogue of a graph; like for cats you add all the composites to 'complete it', here you add all the saturations
- adding 0-cells and 2-cells seem like it'll be the same as in bicats
- 1-cells seem like they can be just added without generating extra stuff, as composition is partial on 1-cells
- adding equalities on 2-cells seems like it might mess with id' and comp' in some ways. (only wrt saturation? or even when they're just anafunctors?)
By next Tuesday, I must:
- say something more about how adding equalities on 2-cells would look like in partial sanabicats (e.g. if it only messes with id' and comp' wrt saturation, or does it still do so even when they're just anafunctors)
- say something about how adding identity1cell(&witness) would look like in partial sanabicats
- say something about how adding composite1cell(&witness) would look like in partial sanabicats
- say something more about how adding equalities on 2-cells would look like in partial sanabicats (e.g. if it only messes with id' and comp' wrt saturation, or does it still do so even when they're just anafunctors)
It seems like you can just identify two witnesses when you identify two arrows they correspond to? (what if there are multiple witnesses involved, for e.g. an arrow from A to B that's horizontal composite of different stuff meeting at different midpoints C and C'?) This seems like it should solve most things when saturated; when unsaturated it may in fact cause more problems
- say something about how adding identity1cell(&witness) would look like in partial sanabicats
- say something about how adding composite1cell(&witness) would look like in partial sanabicats
bc of other preexisting arrows, identity/composite1cells(&witness) introduced can have more than an identity 2-cell in its endomorphism set, which all need to be closed under composition w' stuff they're composable with etc
By next Tuesday, I must:
do each of these three things again, but for partial sanabicats with the right lifting property wrt those things, rather than pushouts
[
1. say something more about how adding equalities on 2-cells would look like in partial sanabicats (e.g. if it only messes with id' and comp' wrt saturation, or does it still do so even when they're just anafunctors)
2. say something about how adding identity1cell(&witness) would look like in partial sanabicats
3. say something about how adding composite1cell(&witness) would look like in partial sanabicats
]
Not gonna finish my goals, so time to do a penalty drawing
(EDIT: Drawing Liv for @migxmeg :])
Another change of plans: from this week onwards, I will not specify goals; each week I'll just 'do enough work to "answer 3 questions"', but I don't have to lock the questions in, bc it would be bad if my direction changes - it's not great to stick with old plan when it's shown to not be the best approach.
I will instead specify the 'questions' I'm answering after the fact.
I'll come out of my creepy lurk to penalize you I love your style 🤣. May I request a drawing of my OC, Liv?
I remember Liv! Here you go 
I am flattered you liked my style enough to come out of lurking for this, but also it was like 'the pressure is on' XD I hope I did her justice ^_^
My 3 questions and answers for this week:
what does a non-cofibrant bicat look like?
can't identify 1-cells. so e.g. bicat w/ 3 composable 1-cells f, g, h with (fg)h=f(gh) (that is equal, not merely isomorphic) isn't cofibrant
are all partial sanabicats cofibrant? how does saturation play into this?
reminder to self; just add witness, not witness and (id/comp) 1-cell at the same time. then it at least doesn't fall afoul of the above problem
a [non-cofibrant bicat as above] as a (partial) anabicat can be cofibrant when adding witnesses is cofibration
what does it mean for a strict map between (partial) sanabicats to be 'essentially surjective on objects'?
For all $c \in C(C_0)$, there exists\
- $b \in B(C_0)$\
- $f \in \text{Hom}_C(F(b), c)$\
- $g \in \text{Hom}_C(c, F(b))$\
- $s, t \in C(|\text{id}|)$(witnesses for ids on $F(b)$ and $c$)\
- $u, v \in C(|\circ|)$(witnesses witnessing '$fg$' as 'id${F(b)}$ and '$gf$' as 'id${c}$)\
such that (boundary conditions on e.g. witnesses (sufficient bc saturation))\
- $l_\circ(u)=r_\circ(v)=f$\
- $l_\circ(v)=r_\circ(u)=g$\
- $\text{id}1(s)=c\circ(u)$\
- $\text{id}1(t)=c\circ(u)$
wowwwww thank you!!! She's fantastic!! So sorry didn't mean to lay on any pressure was just super excited to snag the request 
will absolutely be sending to the print&frame to enjoy off-device. She looks so gorgeous in your art style 
Also... LaTeX?
prove that total saturated anabicats closed under taking products
Summary
Let\
- $\mathcal{A}, \mathcal{B}: S \rightarrow \text{Set}$ be total sanabicats. Then $\mathcal{A} \times \mathcal{B}$ is a total sanabicat.
(proof)(sketch)\
Let $(A,B) \in (\mathcal{A} \times \mathcal{B})(C_0)$.\
- As $\mathcal{A}$ and $\mathcal{B}$ are total, there exist $s \in \text{ID}\mathcal{A}(A)$ and $t \in \text{ID}\mathcal{A}(A)$; i.e.\
-- $s \in |\text{id}|_\mathcal{A}$\
-- $t \in |\text{id}|_\mathcal{B}$\
-- $s_1\text{id}_1(s)=A$\
-- $s_1\text{id}_1(t)=B$\
- Then\
-- there exists $(s,t) \in |\text{id}|_{\mathcal{A} \times \mathcal{B}}$\
-- such that $s_1\text{id}1(s,t)=(s1\text{id}1(s), s1\text{id}_1(t))=(A,B)$\
Let\
- $(A,B), (A',B'), (A'', B'') \in (\mathcal{A} \times \mathcal{B})(C_0)$\
- $(f,g): (A,B) \rightarrow (A',B'), (f',g'): (A',B') \rightarrow (A'',B'') \in (\mathcal{A} \times \mathcal{B})(C_1)$\
Then\
- As $\mathcal{A}$ and $\mathcal{B}$ are total, there exist\
-- $u \in |\circ|_\mathcal{A}$\
-- $v \in |\circ|_\mathcal{B}$, such that\
-- $l_\circ(u)=f, r_\circ(u)=f', c_\circ(u)=f''$ for some $f'': A \rightarrow A''$\
-- $l_\circ(v)=g, r_\circ(v)=g', c_\circ(v)=g''$ for some $g'': B \rightarrow B''$\
- Then there exists $(u, v) \in |\circ|_{\mathcal{A} \times \mathcal{B}}$ such that\
-- $l_\circ(u,v)=(l_\circ(u), l_\circ(v))=(f,g)$\
-- $r_\circ(u,v)=(r_\circ(u), r_\circ(v))=(f',g')$\
-- $c_\circ(u,v)=(c_\circ(u), c_\circ(v))=(f'',g'')$\
prove that we don't have to deal with isos wrt associators (and unitors)
Summary
(lemma)\
Let\
- $\mathcal{A}: S \rightarrow \text{Set}$ be a partial sanabicat\
- $A, B \in \mathcal{A}(C_0)$\
- $s \in \text{ID}_\mathcal{A}(A)$\
- $t \in \text{ID}_\mathcal{A}(A)$\
- $f: A \rightarrow B$.\
(i) Then there exists some $u' \in \text{COMP}\mathcal{A}(\text{id}1(s), f, f)$.
(proof)(sketch)\
As $\mathcal{A}$ is total, there exists some $h: A \rightarrow B$ and $u \in \text{COMP}\mathcal{A}(\text{id}1(s), f, h)$.\
- There is an isomorphism $\text{lUtor}(f, s, u): h \implies f$. By saturation (via a theorem by makkai), (i) is true\
Similarly, (ii) there exists some $u' \in \text{COMP}\mathcal{A}(f, \text{id}1(t), f)$.\
(lemma)\
Let\
- $\mathcal{A}: S \rightarrow \text{Set}$ be a partial sanabicat\
- $A, B, C, D \in \mathcal{A}(C_0)$\
- $f: A \rightarrow B$\
- $g: B \rightarrow C$\
- $h: C \rightarrow D$\
- $j: A \rightarrow C$\
- $k: B \rightarrow D$\
- $l: A \rightarrow D$\
- $u \in \text{COMP}_\mathcal{A}(f,g,j)$\
- $v \in \text{COMP}_\mathcal{A}(g, h, k)$\
- $w \in \text{COMP}_\mathcal{A}(f,k,l)$\
(iii) There exists some $z \in \text{COMP}_\mathcal{A}(j,h,l)$.
(proof)(sketch)\
As $\mathcal{A}$ is total, there exists some $l': A \rightarrow D$ and $z \in \text{COMP}_\mathcal{A}(j,h,l')$.\
- There exists an isomorphism $\text{Ator}(v, w, u, z): l \implies l'$\
- By saturation (via a theorem by makkai), (i) is true\
for inverse equivanences between total sanabicats, rigourously define a candidate mapping for 0-cells, 1-cells and 2-cells
Summary
(action on 0-cells)\
$(\forall B \in \mathcal{B}$, let $G(B)=A$ for some $A \in \mathcal{A}$ such that $F(A) \simeq B$\
(action on 1-cells)\
Let $B, B' \in \mathcal{B}(C_1)$ and $f \in \text{Hom}_{\mathcal{B}}(B, B')$.\
- There exist equivalences $e_B: FG(B) \rightarrow B$ and $e_{B'}: FG(B') \rightarrow B'$.\
- As $\mathcal{B}$ is total, there exists $s \in \text{ID}_\mathcal{B}(FG(B'))$.\
- As $e_{B'}$ is an equivalence, there exist $e_{B'}^{-1}: B' \rightarrow FG(B')$ and $u_{FG(B)} \in \text{COMP}\mathcal{B}(e{B'}, e_{B'}^{-1}, s)$\
- As $\mathcal{B}$ is total, there exist\
-- $w \in \text{COMP}\mathcal{B}(eB, f, h)$ for some $h: FG(B) \rightarrow B'$\
-- $w' \in \text{COMP}\mathcal{B}(h, e{B'}^{-1}, g)$ for some $g: FG(B) \rightarrow FG(B')$\
- As $F$ is essentially surjective on 1-cells, there exists some $g': G(B) \rightarrow G(B')$ and isomorphism $\alpha: F(g') \cong g$\
Let $G(f) = g'$. As $\mathcal{B}$ is saturated, there exists\
- $w'' \in \text{COMP}\mathcal{B}(h, e{B'}^{-1}, F(G(f))$ such that $\circ'(\text{id}2(h), \text{id}2(e_{B'}^{-1}), w'', w')=\alpha$\
(just define action on cells for now and worry about 'functoriality' etc later?)
(action on 2-cells)\
Let\
- $f, g \in \text{Hom}_\mathcal{B}(B, B')$\
- $\alpha \in \text{Hom}_\mathcal{B}(f, g)$\
- $\beta := \circ'(\circ'(\text{id}2(eB), \alpha, w, u), \text{id}2((eB')^{-1}), w'', u'')$ for $w, u, w'', u'' \in |\circ|$\
such that $c_{w''} = FG(f)$ and $c_{u''} = FG(g)$.\
- $\beta \in \text{Hom}_\mathcal{B}(FG(f), FG(g))$\
As $F$ is fully faithful, there exists some $\beta' \in \text{Hom}_\mathcal{A}(G(f), G(g))$ such that $F(\beta')=\beta$. We let $F(\alpha) = \beta'$\
Didn't finish my goals, so time to do a penalty drawing 
Also, I intend to take a break from my usual communities for a while, so from now, I hereby absolve myself of all the accountabilities currently active in this thread. I'll probably make a new thread when I come back, unless this thread is somehow still alive by then XD
(EDIT: Note this is for after I've done the penalty drawing mentioned in this comment; I still have to do that, but I don't have to answer my 3 questions for next week etc :])
