A database of categories
| リビジョン | 0b0a9b4b3b18cbdf1a97e05e4cee46070740f4df (tree) |
|---|---|
| 日時 | 2023-12-09 02:25:43 |
| 作者 | Corbin <cds@corb...> |
| コミッター | Corbin |
Take notes about species.
catabase.db (diff)
todo.txt (diff)| @@ -6,16 +6,38 @@ | ||
| 6 | 6 | * Would handle opposite categories too... |
| 7 | 7 | * `all_groupoids`? Would include cores and homotopy categories? |
| 8 | 8 | * Random facts not yet encodable |
| 9 | - * https://en.wikipedia.org/wiki/Kappa_calculus is the internal language | |
| 10 | - for "contextually complete" categories | |
| 11 | - * https://en.wikipedia.org/wiki/Esakia_duality | |
| 9 | + * Functor categories | |
| 10 | + * Topoi <-> categories of sheaves on spaces | |
| 11 | + * Species ≈ [Bij, Set] | |
| 12 | + * SSet = [Op(Δ), Set] | |
| 13 | + * Note that categories of presheaves involve Op(-) | |
| 14 | + * Note that these are dagger-functor categories | |
| 15 | + * CFT = [2Cob, Hilb] | |
| 16 | + * TQFT = [nCob, FinVect] | |
| 17 | + * Variations on species | |
| 18 | + * LSpecies ≈ [L, Set] | |
| 19 | + * VecSpecies_k ≈ [Bij, Vect_k] | |
| 20 | + * CatSpecies ≈ [Bij, Cat] | |
| 21 | + * Internal languages | |
| 22 | + * CCCs have lambda calculus, famously | |
| 23 | + * https://en.wikipedia.org/wiki/Kappa_calculus is the internal language | |
| 24 | + for "contextually complete" categories | |
| 25 | + * Esakia duality | |
| 26 | + * category of Heyting algebras ≈ Op(Esa), category of Esakia spaces | |
| 27 | + * This is a dual equivalence; one side needs Op(-) | |
| 28 | + * Esa is the category of Esakia spaces and Esakia morphisms | |
| 29 | + * Subcategory: Every Esakia space is a Stone space | |
| 12 | 30 | * Cohen topos |
| 13 | 31 | * AoC holds but CH fails |
| 14 | 32 | * Simplex topos |
| 15 | 33 | * Two-valued but LEM fails |
| 16 | 34 | * 2-subcategories (probably should be sub-2-categories) |
| 17 | 35 | * Johnstone's topological topos |
| 36 | + * Needs a shortname and characterizations of its arrows | |
| 37 | + * Is CCC (is a topos, duh) | |
| 38 | + * Built from N+∞, the one-point compactification of N | |
| 18 | 39 | * Some categories are the "finite" versions of other categories |
| 40 | + * FinVect, FinSet, FinRel | |
| 19 | 41 | * Categorical axiomatic descriptions? |
| 20 | 42 | * https://golem.ph.utexas.edu/category/2019/11/total_maps_of_turing_categorie.html |
| 21 | 43 | * https://golem.ph.utexas.edu/category/2021/09/axioms_for_the_category_of_hil_1.html |
| @@ -38,6 +60,11 @@ | ||
| 38 | 60 | * Needs polymorphism |
| 39 | 61 | * Categorical product -> Cartesian closed |
| 40 | 62 | * Many other cases to handle |
| 63 | + * Species has 6 monoidal structures!! | |
| 64 | + * Distinguished by different joins and units | |
| 65 | + * Generalized derivatives | |
| 66 | + * Given functor category [L, G] and a monoid (L, +, I), | |
| 67 | + the K'th derivative of functor F(l) is F(K + l) | |
| 41 | 68 | * Cartesian closed categories should have exponential objects labeled |
| 42 | 69 | * Free categories on one object: Relationship to logical completeness? |
| 43 | 70 | * Braid -> BrMonCat |
| @@ -49,8 +76,6 @@ | ||
| 49 | 76 | * Shelf |- Grp "free group on a shelf" !? |
| 50 | 77 | * The Euler characteristic AKA groupoid cardinality of P is Euler's |
| 51 | 78 | constant e ~ 2.718 |
| 52 | - * Functor categories: structure types, categories of simplicial objects, ... | |
| 53 | - * Presheaf categories: FinSet, Species, ... | |
| 54 | 79 | * The span construction: Span(X) for FinSet, Set, Grpd, ... |
| 55 | 80 | * Those all have pullbacks, so they're 2-categories |
| 56 | 81 | * Spans give double categories; the other arrows are from the underlying |
| @@ -66,8 +91,6 @@ | ||
| 66 | 91 | * PROP for commutative bialgebras is Span(FinSet) |
| 67 | 92 | * and PROP for special commutative Frobenius algebras is Cospan(FinSet) |
| 68 | 93 | * Arrow categories: Sierpinski topos, ... |
| 69 | - * Topoi <-> categories of sheaves on spaces | |
| 70 | - * CCCs: DagCat, ... | |
| 71 | 94 | * Ring/group completion: Banach rings, complete normed groups, etc. |
| 72 | 95 | * Note that Cauchy completion/Karoubi envelopes are already handled |
| 73 | 96 | * Lawvere theories: "X is equivalent to the Lawvere theory of Ys" |
| @@ -76,18 +99,72 @@ | ||
| 76 | 99 | * Sub-2-categories |
| 77 | 100 | * Reflective sub-2-categories: Pos in Cat, ... |
| 78 | 101 | * 2-posets: listing at https://ncatlab.org/nlab/show/2-poset |
| 102 | + * Pos, Rel, Δ, Lat, DistLat, Frm, Loc, HeytAlg, BoolAlg | |
| 103 | + * All allegories and bicategories of relations | |
| 79 | 104 | * 2-vector spaces |
| 80 | 105 | * Categorification: |
| 81 | 106 | * Set -> Cat |
| 82 | - * Topoi which classify spaces should say *which* space is classified | |
| 107 | + * Topoi which classify theories should say *which* theory is classified | |
| 108 | + * Set classifies the empty theory | |
| 109 | + * Also classifies any theory whose model is unique | |
| 110 | + * Theories of initial objects, terminal objects, NNOs | |
| 111 | + * 1 classifies any theory without models | |
| 112 | + * SSet classifies the theory of linear orders | |
| 113 | + * More at https://ncatlab.org/nlab/show/classifying+topos | |
| 114 | + * Grothendieck topoi | |
| 115 | + * Must be complete categories | |
| 116 | + * 1 is the initial object? Also 1 is inconsistent as a topos? | |
| 117 | + * Set is the terminal object | |
| 118 | + * Notable non-Grothendieck topoi: | |
| 119 | + * FinSet doesn't have all small limits, so is not complete | |
| 120 | + * [-, FinSet] sheaf topoi aren't big enough | |
| 121 | + * Eff isn't Grothendieck either | |
| 122 | + * Strict objects | |
| 123 | + * Strict initial object: no incoming arrows | |
| 124 | + * Always the case in posets and topoi | |
| 125 | + * Set, Cat, SSet, ... | |
| 126 | + * Also the case in Top, Grpd, ... | |
| 127 | + * Strict terminal object: no outgoing arrows | |
| 128 | + * All theories with 0 and 1 s.t. only the trivial model has 0=1 | |
| 129 | + * Ring in particular | |
| 130 | + * Also BoolAlg, absorbtion monoids | |
| 131 | + * Absorbtion monoids | |
| 132 | + * Monoid objects in Set* | |
| 133 | + * Initial object is 2, terminal object is 1 | |
| 134 | + * Semicategories | |
| 135 | + * Categories without identity | |
| 136 | + * SemiCat: objects are semicategories, arrows are semifunctors | |
| 137 | + * Free-forgetful adjunction: every category is a semicategory | |
| 138 | + * Ategories | |
| 139 | + * Categories without composition | |
| 140 | + * At: objects are ategories, arrows are functors | |
| 141 | + * Not clear whether functors need to respect composition, | |
| 142 | + probably not though | |
| 143 | + * Free-forgetful: Every category is an ategory | |
| 144 | + * Gabriel-Ulmer duality | |
| 145 | + * Op(Lex) ≈ LFP | |
| 146 | + * Lex: objects are finitely complete categories, | |
| 147 | + arrows are finite-limit-preserving functors, NTs are NTs | |
| 148 | + * LFP: objects are locally finitely presentable categories, | |
| 149 | + arrows are finitary right adjoints, NTs are NTs | |
| 150 | + * Also, Op(V-Lex) ≈ V-LFP for any enriching V | |
| 151 | + * V must be symmetric monoidal closed, complete, cocomplete, | |
| 152 | + locally finitely presentable | |
| 153 | + * e.g. when V is 2, Op(SemiLat) ≈ AlgLat | |
| 154 | + * NNOs | |
| 155 | + * Not all infinite categories have NNO | |
| 156 | + * In general, infinite topoi have NNO | |
| 157 | + * Set, Cat, ... | |
| 83 | 158 | * Chu spaces and the Stone gamut |
| 84 | 159 | * Row template for (n,r)-categories |
| 160 | + * Triple categories: internal categories in DblCat | |
| 85 | 161 | * Way of the Dagger |
| 86 | 162 | * Monoidal dagger-categories |
| 87 | 163 | * Rig dagger-categories |
| 88 | 164 | * Compact dagger-categories |
| 89 | 165 | * Only zero objects, not initial or terminal |
| 90 | 166 | * Only biproducts, not products or sums |
| 167 | + * DagCat is CCC | |
| 91 | 168 | * Compact categories: nCob, FinVect, FinHilb, ... |
| 92 | 169 | * Adjunctions!!! |
| 93 | 170 | * Should include centers |
| @@ -95,6 +172,10 @@ | ||
| 95 | 172 | * Categories of interesting arrows |
| 96 | 173 | * Categories of monics, epics, etc. |
| 97 | 174 | * Categories of retracts, sections, etc. |
| 175 | + * Parameterized categories | |
| 176 | + * M-Act, S-Act: left actions of monoids M or semigroups S | |
| 177 | + * R-Mod: left modules over a ring/semiring R | |
| 178 | + * G-Set: sets acted on by a group G | |
| 98 | 179 | * Dismantle `enrichments` |
| 99 | 180 | * Already banned: Set, Cat |
| 100 | 181 | * Manage Evil |
| @@ -108,9 +189,8 @@ | ||
| 108 | 189 | `is_element_of` and `is_not_element_of` |
| 109 | 190 | * Original stuff: |
| 110 | 191 | * [[Tomb]], my precious child, the e.w. subcategory of TurCat whose arrows are compilers. Can't write this page until I know what a compiler is. |
| 111 | - * [[Metaverse theory]], my pipe dream where literary analysis is categorified | |
| 112 | - * [[Perfectoid dramatic analysis]], the number-theoretic version of metaverse theory | |
| 113 | - * [[Compositional tool-assisted speedrunning]], a categorified approach to TAS | |
| 192 | + * [[Fictional-universe theory]], my pipe dream where literary analysis is categorified | |
| 193 | + * [[Perfectoid dramatic analysis]], the number-theoretic version of fictional-universe theory | |
| 114 | 194 | * Conjectures |
| 115 | 195 | * Does every allegory give a double category when we consider taking the |
| 116 | 196 | maps of the allegory as our second class of arrows? |
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