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bbdbe1c12c9bbe5970ea717e1759144a1221f0cd | subsection | 313 | 319 | Deriving discontinuities for | Taking expression (REF ) and replacing L_{N|(v)w}\rightarrow -\Lambda _N we find directly that we can write the discontinuities for l\ge 1 as&\left[ \log Y_{M} \right]_{\tau (M+2l)} = \sum _{J=1}^{l+1}\sum _{Q=1+J}^{M+J-1} \left[L_{vw|Q-1}^{(\alpha )}\right]_{\tau (Q+2l-2J+1)} + \left[L_-^{(\alpha )}\right]_{\tau 2l} -... | {
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e29806b38aa78a2cdf963797e80fc93b4967b9fd | subsection | 314 | 319 | Deriving discontinuities for | The discontinuity of Y_Q can now be written as follows for all l\ge 0:\left[ \log Y_{M} \right]_{\tau (M+2l)}(u) &= \left[D_{\tau (M+2l)}^Q\right]_0 (u) -\delta _{l,0}\left[\log Y_{1}\right]_{-\tau 1}(u).This will allow us to continue the derivation of the Y_Q TBA-equation in section (REF ). | {
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f2118726f7cf25f484e633fd1a708221c5561d71 | subsection | 315 | 319 | Two lemmas | In this appendix we will consider a few technical lemmas that are necessary for the correctness of the argumentation in the main text. We will show that \mathbf {P} functions can be expanded in terms of the x function and that periodic analytic functions with exponential growth must be trigonometric polynomials. | {
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3900242d2b793ff46b777011217e87bd4a7db36f | subsection | 316 | 319 | Undeformed case. | \mathbf {P} functions have short cuts and it seems natural to expect an expansion in the x_s function, which we will denote simply by x from now on. As a function x: it is not surjective, its image is given by
\begin{equation}
{>1} :=\left\lbrace z \in | \, |z| >1 \right\rbrace .
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28690904d2e3349dc0839ad8405b1ec13deb77f3 | subsection | 317 | 319 | Deformed case. | In the deformed case the u domain takes the form of a cylinder. \mathbf {P} functions are periodic or anti-periodic, and in the latter case multiplication by an anti-periodic prefactor such as e^{iu/2} makes it periodic. We will consider only this case. The deformed x-function is surjective as a function x:i \mathbb {R... | {
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2b2eaff0da77a5a1bc49296270c04d0f5aa96cc7 | subsection | 318 | 319 | Lemma. | Let f: i \mathbb {R}\times [-\pi ,\pi ) \rightarrow be analytic and have infinitely many non-zero coefficients an on the positive side (thus for positive n) of its Fourier series
\begin{equation}f(u) = \sum _{n\in \mathbb {Z}} a_n e^{-inu}.
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2ffad420ebd36dabc77c21fdfd3e34a84a94dc2c | abstract | 0 | 95 | Abstract | We outline a definition of accessible and presentable objects in a 2-category
$\mathcal K$ endowed with a "KZ context", that is to say a pair of
lax-idempotent monads interacting in a prescribed way; this perspective
suggests a unified treatment of many "Gabriel-Ulmer like" theorems, asserting
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4eeff24d4c7d965bb3feed22f48aeafb9ead493b | subsection | 1 | 95 | Introduction | “The theory of categories enriched in some base closed category , is couched in set-theory; some of the interesting results even require a hierarchy of set-theories. Yet, there is a sense in which the results themselves are of an elementary nature.” The present work stems from the conjoint desire of both authors to det... | {
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e57b8168837c89fde36e9ea3176dd05381bd5a7d | subsection | 2 | 95 | Introduction | A well-known Leibnizian principle in 2-category theory asserts that we can probe the internal structure of an object only via external universal properties; this forces us to the choice of as a formal incarnation of (say) \lambda -filtered colimit completion for A\in ; it is clear how the recent as well as the very rec... | {
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5024fa4e6e119f95c3e84b85ad368a77fc8d82fb | subsection | 3 | 95 | Introduction | We define a Gabriel
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bc9ae219d6161d8591d9c4efee0881c5ad2e4d34 | subsection | 4 | 95 | Introduction | Classical Gabriel
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189300e4b7be69bf8e39073bad3f92d2dcbe4442 | subsection | 5 | 95 | Introduction | It is then clear that left extensions in are left liftings in ^\mathrm {op}, right liftings in are left liftings in ^\mathrm {co}, and right extensions are left liftings in ^\mathrm {coop}.
The situation is conveniently depicted in the following array of universal objects:
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0abb0ce5e8c6e90a99eecb354a2e27b8bcc94d82 | subsection | 6 | 95 | Introduction | A useful lemma of gluing together extensions and lifts is the following:
Lemma
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0458061f0f59cb39d605755f1755a1bcd8b22b0d | subsection | 7 | 95 | Introduction | For each admissible A, in the diagram
{
&A[dr]^{y_A} [dl]_{y_A}@{}[d]|(.6){\rotatebox [origin=c]{225}{\Rightarrow }\chi ^{y_A}}&\\
A && A @{=}[ll]
}
the 2-cell \chi ^{y_A} exhibits \langle 1_{A}, 1_{y_A}\rangle as a left extension.
For admissible A, B and f,g as below,
the diagram
\begin{}[column sep=large, row s... | {
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00c06a90db1c846cf2fb4a542baf65c10e74d408 | subsection | 8 | 95 | Introduction | This claim becomes evident as soon as we rephrase REF as follows: given a pair of composable the universal property of \chi ^{gf} entails that there is a unique 2-cell \theta ^{gf} : filling the diagram
\begin{}[column sep=huge, row sep=huge] A [dashed,Rightarrow,dr,
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cfa11424cb89ad6ef42230d4ba8da80f8c994e5b | subsection | 9 | 95 | Introduction | Moreover, we will need an additional weakening, in that the archetypal example of such a structure (the correspondence taking an object of \protect {\underline{\text{\fontseries {b}\selectfont {\upshape CAT}}}} to its free cocompletion under a prescribed class of colimits) fails to be a pseudomonad for size reasons, si... | {
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6ab6e87dfbc75a77ddfaeaff180f00f252a5ab88 | subsection | 10 | 95 | Introduction | As for the first notion, since the canonical example of a Yoneda structure is free cocompletion under colimits, must “recognize cocomplete objects” exhibiting a similar universal property as that of \protect {\underline{\text{\fontseries {b}\selectfont {\upshape Cat}}}}; briefly, one can define a cocomplete category as... | {
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2d0ea79bf0ea1a24a4cdb542b36923dcdcbe8058 | subsection | 11 | 95 | Introduction | Definition (The 2-category of relative kz
Hypothesis doctrines on )
Let be a 2-category; we define a 2-category \protect {\underline{\text{\fontseries {b}\selectfont {\upshape RKZ}}}}() whose 0-cells are the relative kz
Hypothesis doctrines on , and whose hom-category is the one of pseudonatural transformations \text{... | {
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847c0c49a02b0915794db756ffec67ddef713a93 | subsection | 12 | 95 | Introduction | Remark
The definition of accessible object in depending on the above discussion will be given right away at the beginning of the next section; it is easy to motivate REF as follows: there is a standard choice for a Yoneda context on the 2-category \protect {\underline{\text{\fontseries {b}\selectfont {\upshape CAT}}}}... | {
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d8be5b3a2c3505d569430ffc19381f0d86794e99 | subsection | 13 | 95 | Introduction | Remark
The 2-category \protect {\underline{\text{\fontseries {b}\selectfont {\upshape RKZ}}}}(\protect {\underline{\text{\fontseries {b}\selectfont {\upshape Cat}}}})/ is visibly populated by many other objects: a slight variation on the above theme gives a Yoneda context \text{\begin{}{UTF8}{min}よ\end{}}_{\textsc {c}... | {
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7637cedbae868c9bae31acccabecf599a4f0ff61 | subsection | 14 | 95 | Introduction | For the simple reason that in general one cannot expect that always dominates the Yoneda structure , we will favour the first definition of presentability. However, since it has been an instructive challenge to devise an abstract definition encompassing REF , and in particular to define a “functor preserving -filtered ... | {
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82f29b69bfec159aee82b21d0656a7c0f2fab865 | subsection | 15 | 95 | Introduction | We shall focus on the following diagram.
{
\bar{G} [r]^\ell [dr]_{y^{}_{\bar{G}} }& (G) [r]^i & G^{\prime } \\
& \bar{G} [u]_{L} [ur]_{\text{Yan}_G(\ell i)}&
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1edf02dfb3fe81f60e0ebc3d04aec7ec7687c80d | subsection | 16 | 95 | Introduction | To show that A is cocomplete, we can appeal the easy result that reflective subobjects of -cocomplete objects remain -cocomplete. This concludes the first implication.
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ee5c814889637b5509113a0354c60254129e8910 | subsection | 17 | 95 | Introduction | In particular this means that the diagram
@R=5mm@C=5mm{
A[rr]^{y^{}_A}[dd]_{\iota _A} && A \\
&(\widehat{A})@{=}[ur]\\
\widehat{A} [ur]_{y^{}_{\widehat{A}}}
}
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5a9ccd3f449a85afd2e23089ce474391d500f691 | subsection | 18 | 95 | Introduction | Formal Gabriel-Ulmer duality
(the present terse but very clear introduction to Gabriel-Ulmer duality comes from ) Gabriel-Ulmer duality builds an a biequivalence
\textsf {Mod} : \underline{\text{Lex}}^\mathrm {op}\leftrightarrows \underline{\text{LFP}} : \textsf {Th}
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b6d5c9799998fc2bd5fc6641946fe18d5c0abd7a | subsection | 19 | 95 | Introduction | Theorem (Gabriel
Hypothesis Ulmer duality)
Let \text{\begin{}{UTF8}{min}よ\end{}}: \Rightarrow be a context on , with climbable and assume that there exist a gu envelope relative to \text{\begin{}{UTF8}{min}よ\end{}}. There is a biadjunction
\text{\begin{}{UTF8}{min}よ\end{}}\textsf {-Mod}: \text{L\hspace{-1.66702pt}Ê}... | {
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8fdae324e87514d178371c22eeb0f2c5c83ad5fc | subsection | 20 | 95 | Introduction | We start defining the action of the functors \text{\begin{}{UTF8}{min}よ\end{}}\textsf {-Mod} and \text{\begin{}{UTF8}{min}よ\end{}}\textsf {-Th}; the first is defined “applying ”, in the sense that its action on 0- and 1-cells is as follows:
{
G @[lightgray]@{^{(}->}@<-4pt>[r]_{\color {lightgray} i_G} & \color {lightg... | {
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0fdeae8bf6062104085e07be7617b23ca9ef6d18 | subsection | 21 | 95 | Introduction | We have to check that this really defines a functor taking values in \textsc {Pr}(\text{\begin{}{UTF8}{min}よ\end{}}); this is easily seen, as every G is -cocomplete (it's a reflection of X for a suitable X), and each F is an -cell since in the diagram
{
G @{}[drr]|\rotatebox [origin=c]{225}{\Rightarrow }[dr]^F @/^1pc/... | {
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d0e711dbe35133302328fca483eafda171ae08d7 | subsection | 22 | 95 | Introduction | We need to check that this is a \widehat{(\,\rule {}{}\,)}-cell, and in order to do this we consider the diagram
{
H[r]^\ell @/_1.5pc/[ddr]_{y_H} & G^{\prime }[r]^F @{}[d]|\dashv @<4pt>[d] & G@{}[d]|\dashv @<4pt>[d] \\
& \widehat{G}^{\prime }[r]_{\widehat{F}}@<4pt>[u] & \widehat{G} @<4pt>[u]\\
& \widehat{H}[u]@/_1pc/... | {
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60f699a2d90c2c21afa81d64a7e7b2b1b1442198 | subsection | 23 | 95 | Introduction | In the diagram
{
G^{\prime }@/_2pc/[dd]_{y_{G^{\prime }}} [d]_\ell & G[l]_q [d]^{y_G}\\
A@<-4pt>@{^{(}->}[r]_i & G@<-4pt>[l]_L@{}[l]|\perp @{.>}@/^1pc/[dl]^\Sigma \\
G^{\prime }@{.>}[u]^\Lambda }
-cocompleteness properties imply the existence of 1-cells \Lambda = \text{Yan}_{G^{\prime }}(\ell ) : G^{\prime }\rightar... | {
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066be93014427ca0bc2bc636830e7fc35a8957d6 | subsection | 24 | 95 | Introduction | Examples
Example (Categories and the \lambda -Pres doctrine)
In the 2-category of categories, functors, and natural transformations, the canonical Yoneda structure of REF having A = [A^\mathrm {op},\text{\fontseries {b}\selectfont {\upshape Set}}] yields notions of accessible and presentable that coincide with the cla... | {
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28ca0e8d836708fd2f6ad1fd26e834b39d522de4 | subsection | 25 | 95 | Introduction | The notion of accessible and presentable object in the 2-category of -enriched categories, -enriched functors and -natural transformations, with its natural Yoneda structure having A = [A^\mathrm {op},] was first outlined in , , ; more in detail, the first two papers establish the theory of accessibility, and the last ... | {
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b9df9b9b9b88516f2e61b5dd9737a2fca96f72cf | subsection | 26 | 95 | Introduction | It is an interesting fact that the whole context \text{\begin{}{UTF8}{min}よ\end{}}_{\textsc {c},\lambda } : {I\hspace{-1.49994pt}n\hspace{-1.00006pt}d}_\lambda \Rightarrow restricts to this framework, as the convolution of two functors that are \lambda -filtered colimits of representables (say, on filtered categories I... | {
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7d45b8cbb1bda47503e1dbb4a27e2cbb6c830801 | subsection | 27 | 95 | Introduction | In particular
For a \infty -category, or more generally a simplicial set, filteredness is captured as a lifting property against certain cofibrations (T.5.3.1.7) and moreover this property is equivalent (T.5.3.1.13) to a property of topologically enriched categories.
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0970fac5cc73170fdb9a30c35c381499e7a67a22 | subsection | 28 | 95 | Introduction | Our conjecture here is that derivator theory homes a theory of accessibility and presentability mainly because the 2-category \protect {\underline{\text{\fontseries {b}\selectfont {\upshape PDer}}}} (and its subcategory \protect {\underline{\text{\fontseries {b}\selectfont {\upshape Der}}}} of derivators) possesses a s... | {
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408b0a3f01961ddbb5f2a6355e97335a5d306975 | subsection | 29 | 95 | Introduction | The Cisinski-Tabuada Yoneda structure, (following ; there, the authors prove that for every stable derivator there is an action \text{\fontseries {b}\selectfont {\upshape Sp}}\times \rightarrow that can be promoted to a two-variable adjunction, using Brown representability). The internal-hom part of this two-variable a... | {
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0bf518d4c0430d1f7a175cc1bf6fc14f9c0cbbc7 | subsection | 30 | 95 | Yoneda Structures | The notion of Yoneda structure was given in in order to capture a formal framework in which one can develop abstract category theory; the definition encompasses different facets of the Yoneda lemma, recognized as the backbone of categorical algebra. Roughly speaking, a Yoneda structure on provides a “presheaf construct... | {
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7523fa1e4fa427a045e9348d396ab2062a959177 | subsection | 31 | 95 | Yoneda Structures | Remark In order to avoid a certain notational pedantry, and clumsy accumulation of super- and subscripts, we will usually shorten y_A^{}, B(f,1)_{}, and \chi ^f_{} to the simpler y_A, \chi ^f, B(f,1); we will nevertheless explicitly tell from which Yoneda structure we are taking these maps from, when working with two d... | {
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2bf474862b0e4467df605ed743825bb1667a80ac | subsection | 32 | 95 | Yoneda Structures | This special kind of left extension is so ubiquitous in our discussion that it deserved a special name: in such a situation, we shortly denote L=\text{Lan}_{y^{}_A}\ell as \text{Yan}_A(\ell ).
This concludes our introduction on Yoneda structures; now we move to the second relevant definition of this introductory sect... | {
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"raw": "M. Fiore, N. Gambino, M. Hyland, and G. Winskel, Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures, Selecta ... | 1804.08710 | Accessibility and presentability in 2-categories | [
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c0a3a9050fc0632e44f1e2af50e5b4039adc7a18 | subsection | 33 | 95 | Yoneda Structures | We end this subsection recalling the definitions of -cocomplete object and introducing the notion of a -cell.
Both notions are not new to formal category theory, as the first appears in the theory of pro-arrow equipments , , and the second is called under the name “-homomorphism”.
As for the first notion, since the can... | {
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"doi": "",
"end": 282,
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"raw": "R.J. Wood, Abstract pro arrows I, Cahiers de topologie et géometrie différentielle categoriques 23 (1982), no. 3, 279–290.",
"source_ref_id": "0f1d55c98c4632427... | 1804.08710 | Accessibility and presentability in 2-categories | [
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6acbb643da4739091f6a4952568832a7da85d250 | subsection | 34 | 95 | Yoneda Structures | Definition (The 2-category of relative kz
Hypothesis doctrines on )
Let be a 2-category; we define a 2-category \protect {\underline{\text{\fontseries {b}\selectfont {\upshape RKZ}}}}() whose 0-cells are the relative kz
Hypothesis doctrines on , and whose hom-category is the one of pseudonatural transformations \text{... | {
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266e61b7bfc63e76fb5d4cc844e90c5d222f1561 | subsection | 35 | 95 | Yoneda Structures | Remark
The definition of accessible object in depending on the above discussion will be given right away at the beginning of the next section; it is easy to motivate REF as follows: there is a standard choice for a Yoneda context on the 2-category \protect {\underline{\text{\fontseries {b}\selectfont {\upshape CAT}}}}... | {
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"end": 1201,
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"raw": "M. Bunge and J. Funk, On a bicomma object condition for KZ-doctrines, J. Pure Appl. Algebra 143 (1999), no. 1-3, 69–105.",
"source... | 1804.08710 | Accessibility and presentability in 2-categories | [
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edd87850fe12fdc3ffb538ee06e10149d5c63876 | subsection | 36 | 95 | Yoneda Structures | Remark
The 2-category \protect {\underline{\text{\fontseries {b}\selectfont {\upshape RKZ}}}}(\protect {\underline{\text{\fontseries {b}\selectfont {\upshape Cat}}}})/ is visibly populated by many other objects: a slight variation on the above theme gives a Yoneda context \text{\begin{}{UTF8}{min}よ\end{}}_{\textsc {c}... | {
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"doi": "10.1016/s0022-4049(02)00126-3",
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"raw": "J. Adámek, F. Borceux, S. Lack, and J. Rosický, A classification of accessible categories, Theory Appl. Categ. (2002), 7–30.",
"sou... | 1804.08710 | Accessibility and presentability in 2-categories | [
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a1101a79cb6fdb09ae95d7c955437670bfd16d2d | subsection | 37 | 95 | Yoneda Structures | For the simple reason that in general one cannot expect that always dominates the Yoneda structure , we will favour the first definition of presentability. However, since it has been an instructive challenge to devise an abstract definition encompassing REF , and in particular to define a “functor preserving -filtered ... | {
"cite_spans": []
} | 1804.08710 | Accessibility and presentability in 2-categories | [
"Ivan Di Liberti",
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1bc8c2560d6f892c59bafa74849bf44a4caa5dd6 | subsection | 38 | 95 | Yoneda Structures | We shall focus on the following diagram.
{
\bar{G} [r]^\ell [dr]_{y^{}_{\bar{G}} }& (G) [r]^i & G^{\prime } \\
& \bar{G} [u]_{L} [ur]_{\text{Yan}_G(\ell i)}&
}
By assumption, i\cong \text{Yan}_G(i y_G), and this extension is pointwise; since has finite limits, we can build the comma object (\ell /y_G), and the resul... | {
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242e897092005e17ea55f85e969562cc2f42411e | subsection | 39 | 95 | Yoneda Structures | To show that A is cocomplete, we can appeal the easy result that reflective subobjects of -cocomplete objects remain -cocomplete. This concludes the first implication.
Now assume that A is accessible; we shall show that the axioms REF –REF hold. Since A is -cocomplete, i has a left adjoint; REF is part of the assumptio... | {
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} | 1804.08710 | Accessibility and presentability in 2-categories | [
"Ivan Di Liberti",
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c694e0a11f7d7cd3fa274dfeab6a3e22205c1908 | subsection | 40 | 95 | Yoneda Structures | In particular this means that the diagram
@R=5mm@C=5mm{
A[rr]^{y^{}_A}[dd]_{\iota _A} && A \\
&(\widehat{A})@{=}[ur]\\
\widehat{A} [ur]_{y^{}_{\widehat{A}}}
}
commutes. Moreover, for every \widehat{(\,\rule {}{}\,)}-cell f : G\rightarrow G^{\prime } between \widehat{(\,\rule {}{}\,)}-cocomplete objects, there exists... | {
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} | 1804.08710 | Accessibility and presentability in 2-categories | [
"Ivan Di Liberti",
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1a8e243498bd13b5cfa215ef5d00196b0a7acb3f | subsection | 41 | 95 | Yoneda Structures | Formal Gabriel-Ulmer duality
(the present terse but very clear introduction to Gabriel-Ulmer duality comes from ) Gabriel-Ulmer duality builds an a biequivalence
\textsf {Mod} : \underline{\text{Lex}}^\mathrm {op}\leftrightarrows \underline{\text{LFP}} : \textsf {Th}
between the 2-category of small finitely complete ... | {
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dc7bffdc0ecc8695ced334d95dfb7b3f9c9e7f95 | subsection | 42 | 95 | Yoneda Structures | Theorem (Gabriel
Hypothesis Ulmer duality)
Let \text{\begin{}{UTF8}{min}よ\end{}}: \Rightarrow be a context on , with climbable and assume that there exist a gu envelope relative to \text{\begin{}{UTF8}{min}よ\end{}}. There is a biadjunction
\text{\begin{}{UTF8}{min}よ\end{}}\textsf {-Mod}: \text{L\hspace{-1.66702pt}Ê}... | {
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... | 1804.08710 | Accessibility and presentability in 2-categories | [
"Ivan Di Liberti",
"Fosco Loregian"
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28fb37a1d83ca1f8b77f04816b2b666d742c70b1 | subsection | 43 | 95 | Yoneda Structures | We start defining the action of the functors \text{\begin{}{UTF8}{min}よ\end{}}\textsf {-Mod} and \text{\begin{}{UTF8}{min}よ\end{}}\textsf {-Th}; the first is defined “applying ”, in the sense that its action on 0- and 1-cells is as follows:
{
G @[lightgray]@{^{(}->}@<-4pt>[r]_{\color {lightgray} i_G} & \color {lightg... | {
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9c0d2c7cd45511b4a3b8d903499acbe81640a837 | subsection | 44 | 95 | Yoneda Structures | We have to check that this really defines a functor taking values in \textsc {Pr}(\text{\begin{}{UTF8}{min}よ\end{}}); this is easily seen, as every G is -cocomplete (it's a reflection of X for a suitable X), and each F is an -cell since in the diagram
{
G @{}[drr]|\rotatebox [origin=c]{225}{\Rightarrow }[dr]^F @/^1pc/... | {
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9c80f91fb597434710de3063d157562832d1ddf4 | subsection | 45 | 95 | Yoneda Structures | We need to check that this is a \widehat{(\,\rule {}{}\,)}-cell, and in order to do this we consider the diagram
{
H[r]^\ell @/_1.5pc/[ddr]_{y_H} & G^{\prime }[r]^F @{}[d]|\dashv @<4pt>[d] & G@{}[d]|\dashv @<4pt>[d] \\
& \widehat{G}^{\prime }[r]_{\widehat{F}}@<4pt>[u] & \widehat{G} @<4pt>[u]\\
& \widehat{H}[u]@/_1pc/... | {
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7f7ec8c4e398430882572aa5500fe4d58e7d3562 | subsection | 46 | 95 | Yoneda Structures | In the diagram
{
G^{\prime }@/_2pc/[dd]_{y_{G^{\prime }}} [d]_\ell & G[l]_q [d]^{y_G}\\
A@<-4pt>@{^{(}->}[r]_i & G@<-4pt>[l]_L@{}[l]|\perp @{.>}@/^1pc/[dl]^\Sigma \\
G^{\prime }@{.>}[u]^\Lambda }
-cocompleteness properties imply the existence of 1-cells \Lambda = \text{Yan}_{G^{\prime }}(\ell ) : G^{\prime }\rightar... | {
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2fdf9e7a7d133a014e0e282d902e3444fed6b958 | subsection | 47 | 95 | Yoneda Structures | Examples
Example (Categories and the \lambda -Pres doctrine)
In the 2-category of categories, functors, and natural transformations, the canonical Yoneda structure of REF having A = [A^\mathrm {op},\text{\fontseries {b}\selectfont {\upshape Set}}] yields notions of accessible and presentable that coincide with the cla... | {
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12d194c9a7a25e2b9059ad7cc170c443349beda7 | subsection | 48 | 95 | Yoneda Structures | The notion of accessible and presentable object in the 2-category of -enriched categories, -enriched functors and -natural transformations, with its natural Yoneda structure having A = [A^\mathrm {op},] was first outlined in , , ; more in detail, the first two papers establish the theory of accessibility, and the last ... | {
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cd95526a55b1054895fb60be0ed57c2289a8f5d4 | subsection | 49 | 95 | Yoneda Structures | It is an interesting fact that the whole context \text{\begin{}{UTF8}{min}よ\end{}}_{\textsc {c},\lambda } : {I\hspace{-1.49994pt}n\hspace{-1.00006pt}d}_\lambda \Rightarrow restricts to this framework, as the convolution of two functors that are \lambda -filtered colimits of representables (say, on filtered categories I... | {
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39b16165ca14529b710d5691bdcb390303b27f5f | subsection | 50 | 95 | Yoneda Structures | In particular
For a \infty -category, or more generally a simplicial set, filteredness is captured as a lifting property against certain cofibrations (T.5.3.1.7) and moreover this property is equivalent (T.5.3.1.13) to a property of topologically enriched categories.
\lambda -accessible and \lambda -presentable \inft... | {
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2a18283effc962c06e3190c591328746500e20fd | subsection | 51 | 95 | Yoneda Structures | Our conjecture here is that derivator theory homes a theory of accessibility and presentability mainly because the 2-category \protect {\underline{\text{\fontseries {b}\selectfont {\upshape PDer}}}} (and its subcategory \protect {\underline{\text{\fontseries {b}\selectfont {\upshape Der}}}} of derivators) possesses a s... | {
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d185719b0fd410d4632eafc023a9b71df53f1957 | subsection | 52 | 95 | Yoneda Structures | The Cisinski-Tabuada Yoneda structure, (following ; there, the authors prove that for every stable derivator there is an action \text{\fontseries {b}\selectfont {\upshape Sp}}\times \rightarrow that can be promoted to a two-variable adjunction, using Brown representability). The internal-hom part of this two-variable a... | {
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25176d7903e3286aee12fe6cd4670e73380f6f47 | subsection | 53 | 95 | Accessibility and presentability | In light of the previous two remarks, the following definition appears straighforward.Definition (\text{\begin{}{UTF8}{min}よ\end{}}-accessible object)
Let \text{\begin{}{UTF8}{min}よ\end{}} be a context on the 2-category ; A\in is \text{\begin{}{UTF8}{min}よ\end{}}-accessible if there exists a -small object G\in such th... | {
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1fbe413c9a557f3591aa9f851e1b2fae1420cb73 | subsection | 54 | 95 | Accessibility and presentability | Definition (\text{\begin{}{UTF8}{min}よ\end{}}-presentable object)
Let \text{\begin{}{UTF8}{min}よ\end{}} be a context; A\in is \text{\begin{}{UTF8}{min}よ\end{}}-presentable if
p
A is a left split subobject of G for some G;
A is \text{\begin{}{UTF8}{min}よ\end{}}-accessible;
The fully faithful functor i : A\rightarr... | {
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c4a667ccd181e612eb4ab06177c63a2dbc0ef865 | subsection | 55 | 95 | Accessibility and presentability | We shall focus on the following diagram.
{
\bar{G} [r]^\ell [dr]_{y^{}_{\bar{G}} }& (G) [r]^i & G^{\prime } \\
& \bar{G} [u]_{L} [ur]_{\text{Yan}_G(\ell i)}&
}
By assumption, i\cong \text{Yan}_G(i y_G), and this extension is pointwise; since has finite limits, we can build the comma object (\ell /y_G), and the resul... | {
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70d3378a6f5b7d518c5c234704995b8a617abc30 | subsection | 56 | 95 | Accessibility and presentability | To show that A is cocomplete, we can appeal the easy result that reflective subobjects of -cocomplete objects remain -cocomplete. This concludes the first implication.
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0.106201171875,
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0.1932373046875,
0.27392578125,
0.049072265625,
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0.0919189453125,
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0.2059326171875,
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0.167236328125,
0.2181396484375,
0.057586669921875,
0.1385498046875,
0.1915283203125,... | |
29d3a2a4c293deae2150613472e3f19216010ab0 | subsection | 57 | 95 | Accessibility and presentability | In particular this means that the diagram
@R=5mm@C=5mm{
A[rr]^{y^{}_A}[dd]_{\iota _A} && A \\
&(\widehat{A})@{=}[ur]\\
\widehat{A} [ur]_{y^{}_{\widehat{A}}}
}
commutes. Moreover, for every \widehat{(\,\rule {}{}\,)}-cell f : G\rightarrow G^{\prime } between \widehat{(\,\rule {}{}\,)}-cocomplete objects, there exists... | {
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} | 1804.08710 | Accessibility and presentability in 2-categories | [
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d2ce1393964aec99b027ecaf8ca3fc9c6a73fb10 | subsection | 58 | 95 | Accessibility and presentability | Formal Gabriel-Ulmer duality
(the present terse but very clear introduction to Gabriel-Ulmer duality comes from ) Gabriel-Ulmer duality builds an a biequivalence
\textsf {Mod} : \underline{\text{Lex}}^\mathrm {op}\leftrightarrows \underline{\text{LFP}} : \textsf {Th}
between the 2-category of small finitely complete ... | {
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... | 1804.08710 | Accessibility and presentability in 2-categories | [
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cf1abfce04c943bfe0e4aa467e4a33f0e6de5728 | subsection | 59 | 95 | Accessibility and presentability | Theorem (Gabriel
Hypothesis Ulmer duality)
Let \text{\begin{}{UTF8}{min}よ\end{}}: \Rightarrow be a context on , with climbable and assume that there exist a gu envelope relative to \text{\begin{}{UTF8}{min}よ\end{}}. There is a biadjunction
\text{\begin{}{UTF8}{min}よ\end{}}\textsf {-Mod}: \text{L\hspace{-1.66702pt}Ê}... | {
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... | 1804.08710 | Accessibility and presentability in 2-categories | [
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"Fosco Loregian"
] | [
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a2d6d9873413a93b72b9abbf33e91e22efab5c8c | subsection | 60 | 95 | Accessibility and presentability | We start defining the action of the functors \text{\begin{}{UTF8}{min}よ\end{}}\textsf {-Mod} and \text{\begin{}{UTF8}{min}よ\end{}}\textsf {-Th}; the first is defined “applying ”, in the sense that its action on 0- and 1-cells is as follows:
{
G @[lightgray]@{^{(}->}@<-4pt>[r]_{\color {lightgray} i_G} & \color {lightg... | {
"cite_spans": []
} | 1804.08710 | Accessibility and presentability in 2-categories | [
"Ivan Di Liberti",
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e5d80714192b148146bd1561c11293cf3b49024b | subsection | 61 | 95 | Accessibility and presentability | We have to check that this really defines a functor taking values in \textsc {Pr}(\text{\begin{}{UTF8}{min}よ\end{}}); this is easily seen, as every G is -cocomplete (it's a reflection of X for a suitable X), and each F is an -cell since in the diagram
{
G @{}[drr]|\rotatebox [origin=c]{225}{\Rightarrow }[dr]^F @/^1pc/... | {
"cite_spans": []
} | 1804.08710 | Accessibility and presentability in 2-categories | [
"Ivan Di Liberti",
"Fosco Loregian"
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01e29697aad9aad91e7766ca02a4ac9e704bc1db | subsection | 62 | 95 | Accessibility and presentability | We need to check that this is a \widehat{(\,\rule {}{}\,)}-cell, and in order to do this we consider the diagram
{
H[r]^\ell @/_1.5pc/[ddr]_{y_H} & G^{\prime }[r]^F @{}[d]|\dashv @<4pt>[d] & G@{}[d]|\dashv @<4pt>[d] \\
& \widehat{G}^{\prime }[r]_{\widehat{F}}@<4pt>[u] & \widehat{G} @<4pt>[u]\\
& \widehat{H}[u]@/_1pc/... | {
"cite_spans": []
} | 1804.08710 | Accessibility and presentability in 2-categories | [
"Ivan Di Liberti",
"Fosco Loregian"
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ec3a3bdee7f9fde58067e7449fa3d4e0f8992754 | subsection | 63 | 95 | Accessibility and presentability | In the diagram
{
G^{\prime }@/_2pc/[dd]_{y_{G^{\prime }}} [d]_\ell & G[l]_q [d]^{y_G}\\
A@<-4pt>@{^{(}->}[r]_i & G@<-4pt>[l]_L@{}[l]|\perp @{.>}@/^1pc/[dl]^\Sigma \\
G^{\prime }@{.>}[u]^\Lambda }
-cocompleteness properties imply the existence of 1-cells \Lambda = \text{Yan}_{G^{\prime }}(\ell ) : G^{\prime }\rightar... | {
"cite_spans": []
} | 1804.08710 | Accessibility and presentability in 2-categories | [
"Ivan Di Liberti",
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e0500f00a0abcc6ea4f28163da55abf80ca3032f | subsection | 64 | 95 | Accessibility and presentability | Examples
Example (Categories and the \lambda -Pres doctrine)
In the 2-category of categories, functors, and natural transformations, the canonical Yoneda structure of REF having A = [A^\mathrm {op},\text{\fontseries {b}\selectfont {\upshape Set}}] yields notions of accessible and presentable that coincide with the cla... | {
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aa689b603be5b3ba98a1aa0d3c0f2c180e84e2b5 | subsection | 65 | 95 | Accessibility and presentability | The notion of accessible and presentable object in the 2-category of -enriched categories, -enriched functors and -natural transformations, with its natural Yoneda structure having A = [A^\mathrm {op},] was first outlined in , , ; more in detail, the first two papers establish the theory of accessibility, and the last ... | {
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... | 1804.08710 | Accessibility and presentability in 2-categories | [
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a8e509164a18e29baac3995ed510330871658496 | subsection | 66 | 95 | Accessibility and presentability | It is an interesting fact that the whole context \text{\begin{}{UTF8}{min}よ\end{}}_{\textsc {c},\lambda } : {I\hspace{-1.49994pt}n\hspace{-1.00006pt}d}_\lambda \Rightarrow restricts to this framework, as the convolution of two functors that are \lambda -filtered colimits of representables (say, on filtered categories I... | {
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ead803f7059aa510aadafcd39e7b1506ad0bf8f5 | subsection | 67 | 95 | Accessibility and presentability | In particular
For a \infty -category, or more generally a simplicial set, filteredness is captured as a lifting property against certain cofibrations (T.5.3.1.7) and moreover this property is equivalent (T.5.3.1.13) to a property of topologically enriched categories.
\lambda -accessible and \lambda -presentable \inft... | {
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99548176da2403ef5f5002191de0cc08b98c076b | subsection | 68 | 95 | Accessibility and presentability | Our conjecture here is that derivator theory homes a theory of accessibility and presentability mainly because the 2-category \protect {\underline{\text{\fontseries {b}\selectfont {\upshape PDer}}}} (and its subcategory \protect {\underline{\text{\fontseries {b}\selectfont {\upshape Der}}}} of derivators) possesses a s... | {
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b058a123d894e372ecd91e29ed2fe29a10f1eac3 | subsection | 69 | 95 | Accessibility and presentability | The Cisinski-Tabuada Yoneda structure, (following ; there, the authors prove that for every stable derivator there is an action \text{\fontseries {b}\selectfont {\upshape Sp}}\times \rightarrow that can be promoted to a two-variable adjunction, using Brown representability). The internal-hom part of this two-variable a... | {
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e3e5f487d042c82653540cf1d7c10d2f8d3786d9 | subsection | 70 | 95 | Gabriel-Ulmer duality | We start with the simple observation that the closure under finite colimits of A\in \protect {\underline{\text{\fontseries {b}\selectfont {\upshape Cat}}}} is the subcategory of [A^\mathrm {op},\text{\fontseries {b}\selectfont {\upshape Set}}] generated by finite colimits of representables; we call this category \wideh... | {
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} | 1804.08710 | Accessibility and presentability in 2-categories | [
"Ivan Di Liberti",
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39435a8d9681aeca2217d000aecb26354116955d | subsection | 71 | 95 | Gabriel-Ulmer duality | Moreover, for every \widehat{(\,\rule {}{}\,)}-cell f : G\rightarrow G^{\prime } between \widehat{(\,\rule {}{}\,)}-cocomplete objects, there exists a 1-cell f in diagram{
G[dr]^{y_G^{}}[rrr]^f [dd]_{y_G^{}} &&& G^{\prime } [dd]^{y_{G^{\prime }}^{}}[dl]_{y_{G^{\prime }}^{}}\\
& G[dl]^{\text{\begin{}{UTF8}{min}よ\end{}}_... | {
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} | 1804.08710 | Accessibility and presentability in 2-categories | [
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def61d4aff52bc045aed808bf50d9c607f5d4c8b | subsection | 72 | 95 | Formal Gabriel-Ulmer duality | (the present terse but very clear introduction to Gabriel-Ulmer duality comes from ) Gabriel-Ulmer duality builds an a biequivalence\textsf {Mod} : \underline{\text{Lex}}^\mathrm {op}\leftrightarrows \underline{\text{LFP}} : \textsf {Th}between the 2-category of small finitely complete categories, finite limit preservi... | {
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fde67848ada1c079e274f7fb70ae3b5692ef688b | subsection | 73 | 95 | Formal Gabriel-Ulmer duality | We define the 2-category \text{L\hspace{-1.66702pt}Ê}\hspace{-1.25pt}X(UTF8minよ) having 0-cells the ( )-cocomplete objects, 1-cells the ( )-cells, and all 2-cells between these.Definition (The category \textsc {Pr}({\text{\begin{}{UTF8}{min}よ\end{}}}))
The objects of the 2-category \textsc {Pr}({\text{\begin{}{UTF8}... | {
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} | 1804.08710 | Accessibility and presentability in 2-categories | [
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870b459d8146b0aa7ffce1742fee45f54c273e17 | subsection | 74 | 95 | Formal Gabriel-Ulmer duality | There is a biadjunction\text{\begin{}{UTF8}{min}よ\end{}}\textsf {-Mod}: \text{L\hspace{-1.66702pt}Ê}\hspace{-1.25pt}X(UTF8minよ) Pr(UTF8minよ) : UTF8minよ-Th
which is in fact a biequivalence of 2-categories.Our argument will be identical in spirit to the one offered in .We start defining the action of the functors \text{\... | {
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... | 1804.08710 | Accessibility and presentability in 2-categories | [
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f159ea2a2854c6dd206e662016ecb13cab48c9bc | subsection | 75 | 95 | Formal Gabriel-Ulmer duality | We have to check that this really defines a functor taking values in \textsc {Pr}(\text{\begin{}{UTF8}{min}よ\end{}}); this is easily seen, as every G is -cocomplete (it's a reflection of X for a suitable X), and each F is an -cell since in the diagram{
G @{}[drr]|\rotatebox [origin=c]{225}{\Rightarrow }[dr]^F @/^1pc/[d... | {
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} | 1804.08710 | Accessibility and presentability in 2-categories | [
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7afde3665282b185347372c9d03427f31433d249 | subsection | 76 | 95 | Formal Gabriel-Ulmer duality | We need to check that this is a \widehat{(\,\rule {}{}\,)}-cell, and in order to do this we consider the diagram{
H[r]^\ell @/_1.5pc/[ddr]_{y_H} & G^{\prime }[r]^F @{}[d]|\dashv @<4pt>[d] & G@{}[d]|\dashv @<4pt>[d] \\
& \widehat{G}^{\prime }[r]_{\widehat{F}}@<4pt>[u] & \widehat{G} @<4pt>[u]\\
& \widehat{H}[u]@/_1pc/[ur... | {
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9799d90e832608d11e421ca75573c2ec7ab49039 | subsection | 77 | 95 | Exchange and domination | GabrielHypothesis Ulmer contexts are richer structures than it can appear at first sight: it is indeed possible to show that the existence of a gu envelope relative to a context \text{\begin{}{UTF8}{min}よ\end{}} is really near to a necessary and sufficient condition for the distinction between presentabilities REF and ... | {
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} | 1804.08710 | Accessibility and presentability in 2-categories | [
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61ead03bd9094b2f3b89cb25a195517a1297e573 | subsection | 78 | 95 | Exchange and domination | We claim that A \cong G^{\prime }.In the diagram{
G^{\prime }@/_2pc/[dd]_{y_{G^{\prime }}} [d]_\ell & G[l]_q [d]^{y_G}\\
A@<-4pt>@{^{(}->}[r]_i & G@<-4pt>[l]_L@{}[l]|\perp @{.>}@/^1pc/[dl]^\Sigma \\
G^{\prime }@{.>}[u]^\Lambda }-cocompleteness properties imply the existence of 1-cells \Lambda = \text{Yan}_{G^{\prime }}... | {
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d9b86d8dfbb8975f738f49fb0c2793387a350c92 | subsection | 79 | 95 | Examples | Example (Categories and the \lambda -Pres doctrine)
In the 2-category of categories, functors, and natural transformations, the canonical Yoneda structure of REF having A = [A^\mathrm {op},\text{\fontseries {b}\selectfont {\upshape Set}}] yields notions of accessible and presentable that coincide with the classical no... | {
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5398d2c6acdf1f50c6074e865b6d46fcb5f43f02 | subsection | 80 | 95 | Examples | The notion of accessible and presentable object in the 2-category of -enriched categories, -enriched functors and -natural transformations, with its natural Yoneda structure having A = [A^\mathrm {op},] was first outlined in , , ; more in detail, the first two papers establish the theory of accessibility, and the last ... | {
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4978fd63f71c4a36360d76f3ebdb9763bf5388a3 | subsection | 81 | 95 | Examples | It is an interesting fact that the whole context \text{\begin{}{UTF8}{min}よ\end{}}_{\textsc {c},\lambda } : {I\hspace{-1.49994pt}n\hspace{-1.00006pt}d}_\lambda \Rightarrow restricts to this framework, as the convolution of two functors that are \lambda -filtered colimits of representables (say, on filtered categories I... | {
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8fe3083c8f35bfc454baed09a6fe64afac05ea31 | subsection | 82 | 95 | Examples | Then a Yoneda structure on can be transferred to using the coreflector 2-functor: under suitable assumptions on the 2-categories and the monad, this yields that (for example) a Yoneda structure, and a Yoneda context on lift to the category of algebras = T\text{-Alg} of an idempotent 2-monad on .Example (Categories and ... | {
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3f88fc1fb23d7ccddcec4132423e44ad533c34f5 | subsection | 83 | 95 | Examples | The locally presentable objects in this Yoneda structure are the algebraic lattices in the sense of , while the accessible objects are “accessible posets” (there does not seem to be a name for these categories, but they are simply posetal categories that are accessible in \protect {\underline{\text{\fontseries {b}\sele... | {
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292ab69282387023ee88c0f6b2f79209f879363a | subsection | 84 | 95 | Examples | The former example specializes to this context, but -enriched Ind-completion, the gu envelope and the representation theorem do not seem (to the best of our knowledge) to admit a topological characterization.
given an additive category , regarded as a particular preadditive (=\text{\fontseries {b}\selectfont {\upshape... | {
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0e9d5fd5e6131c40bd62a7cb431306c6d5135664 | subsection | 85 | 95 | Examples | It is enticing to conjecture how slight variations on this theme can lead to unexpected (or even new) results, unveiling a deeper and wider pattern for accessibility and presentability phenomena: among many other cases of interest we recordthe case of finitely complete categories, left exact left adjoints and natural t... | {
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d6f865e0af92cff9a61c8c3c07438954604c4c0b | subsection | 86 | 95 | The case of | An extensive treatment of accessibility and presentability in the realm of quasicategories occupies ( is called simply “T” from now on): a general motif of the chapter is that the classical theory carries over with slight or no change at all in the \infty -categorical setting. In particularFor a \infty -category, or mo... | {
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01749bb191b2ffac328b581af45a061b27ab06d2 | subsection | 87 | 95 | The case of | T.5.3.5.10 shows that there exists a gu envelope given by the equivalence between A (=\infty -functors to the \infty -category of spaces) and the \lambda -filtered colimit completion of A\in \text{\fontseries {b}\selectfont {\upshape QCat}}.The onerous exercise is style of re-reading REF with \infty -goggles (not an ex... | {
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5c87e598543d1ef9f495c56f604c2b41579f25af | subsection | 88 | 95 | The case of | In view of 's claim that the category theory of \text{\fontseries {b}\selectfont {\upshape QCat}} is faithfully captured by a 2-dimensional theory, this structure can be now transported to a honest Yoneda structure on this 2-category \text{\fontseries {b}\selectfont {\upshape QCat}}_2.Passed under this correspondence, ... | {
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a60e625bffa2f7ea4ad1369ede776929b5609811 | subsection | 89 | 95 | The case of derivators | The preprint ends with a short statement of purpose drawing a connection with the present work. As the main purpose of was to lay down the theory of co/reflective localizations of derivators, it has been natural to surmise that there exists a notion of locally presentable and accessible derivator allowing to restate th... | {
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a3ba3cffe4979e872dc024fd52ad4a827ea429d7 | subsection | 90 | 95 | The case of derivators | In fact, Street's one is presumably only one among a complicated web of Yoneda structures: notably, we can record/conjecture the existence ofThe Muro-Raptis Yoneda structure, built taking the represented derivator on \text{\fontseries {b}\selectfont {\upshape sSet}}; this is again representable, and the canonical map \... | {
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c2494bfebaa6f5f629fb5e70809866c67672f59a | subsection | 91 | 95 | The case of derivators | This evidently echoes our REF .The preprint ends with a short statement of purpose drawing a connection with the present work. As the main purpose of was to lay down the theory of co/reflective localizations of derivators, it has been natural to surmise that there exists a notion of locally presentable and accessible d... | {
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929643866455e0439380f62862f10783ad660090 | subsection | 92 | 95 | The case of derivators | In fact, Street's one is presumably only one among a complicated web of Yoneda structures: notably, we can record/conjecture the existence ofThe Muro-Raptis Yoneda structure, built taking the represented derivator on \text{\fontseries {b}\selectfont {\upshape sSet}}; this is again representable, and the canonical map \... | {
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ad00206a20bfafaeb08e775e27412fa9fedbcddd | subsection | 93 | 95 | The case of derivators | This evidently echoes our REF .The preprint ends with a short statement of purpose drawing a connection with the present work. As the main purpose of was to lay down the theory of co/reflective localizations of derivators, it has been natural to surmise that there exists a notion of locally presentable and accessible d... | {
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