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87012884498fb48826d12895df5cffe420c5fccc | subsection | 17 | 31 | Cutoff Frequency and Diffusion Coefficient | If the cutoff frequency is about 10^{18} Hz
instead of 10^{17} Hz, then \bar{\kappa } \le (0.5–2.3)
\bar{\kappa }_{\rm B}. In each of these cases, the highest-energy electrons
have diffusion coefficients nearly as small as the Bohm coefficient (i.e.,
are being accelerated about as fast as possible).Similar results for... | {
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... | 10.1086/589628 | 0807.1702 | Evidence of a Curved Synchrotron Spectrum in the Supernova Remnant SN
1006 | [
"G. E. Allen",
"J. C. Houck",
"S. J. Sturner"
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cf55c0576a08a5cb69c94cb1ed8e4b2edc11ae6c | subsection | 18 | 31 | Cutoff Frequency and Diffusion Coefficient | There are at least three explanations for this apparent
dilemma. One explanation is that, near the shock, B_{1} may be amplified
by cosmic-ray streaming , , . For example,
equation (15) of yields B_{1} = 30 \mu G (see Ksenofontov
et al. 2005) and, hence, \kappa _{1} \sim \kappa _{\rm B,1}, if the upstream
mass density ... | {
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1006 | [
"G. E. Allen",
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"S. J. Sturner"
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06cef986696944fe2cb1c79a46a76bc38000b19c | subsection | 19 | 31 | Cutoff Frequency and Diffusion Coefficient | Therefore, the limit only applies to electrons at the cutoff momentum [i.e.,
\bar{\kappa } = \bar{\kappa }(p = p_{m})]. An advantage to using this
momentum is that the limit does not depend on the functional form of
\bar{\kappa }(p).4. The limit is based on the assumption that the cutoff is due to
synchrotron losses. I... | {
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"... | 10.1086/589628 | 0807.1702 | Evidence of a Curved Synchrotron Spectrum in the Supernova Remnant SN
1006 | [
"G. E. Allen",
"J. C. Houck",
"S. J. Sturner"
] | [
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6db48c40da24391a113d176a627d0f9d4b1f1948 | subsection | 20 | 31 | Conclusions | We have performed a joint spectral analysis of some Chandra ACIS X-ray
data and MOST radio data for 13 small regions along the bright northeastern
rim of the supernova remnant SN 1006. The data were fitted with a model that
includes a synchrotron emission component. This component is based on an
electron spectrum that ... | {
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"do... | 10.1086/589628 | 0807.1702 | Evidence of a Curved Synchrotron Spectrum in the Supernova Remnant SN
1006 | [
"G. E. Allen",
"J. C. Houck",
"S. J. Sturner"
] | [
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9d9c77d80f66afdda6326c7cdd0784f8e6f8d36c | subsection | 21 | 31 | Conclusions | This result
implies that at least some of the highest-energy electrons in SN 1006 diffuse
close to the Bohm limit (i.e., are accelerated about as fast as possible),
which provides additional support for the idea that Galactic cosmic rays are
predominantly accelerated by the shocks of supernova remnants.We gratefully ac... | {
"cite_spans": []
} | 10.1086/589628 | 0807.1702 | Evidence of a Curved Synchrotron Spectrum in the Supernova Remnant SN
1006 | [
"G. E. Allen",
"J. C. Houck",
"S. J. Sturner"
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834e5e0dec91edbccc8352b06161e97b6e5a1235 | subsection | 22 | 31 | Diffusion coefficient | This appendix describes how measurements of or inferences about the shock
velocity and cutoff frequency can be used to place an upper limit on the
electron diffusion coefficient. Since the mean rate of synchrotron losses
cannot exceed the mean rate of energy gains at momenta below the cutoff of
the electron spectrum,\l... | {
"cite_spans": []
} | 10.1086/589628 | 0807.1702 | Evidence of a Curved Synchrotron Spectrum in the Supernova Remnant SN
1006 | [
"G. E. Allen",
"J. C. Houck",
"S. J. Sturner"
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ccf4a016f27f2fb94bc0488359f9ec7b63b24e8e | subsection | 23 | 31 | Diffusion coefficient | If \sin \theta in equation (REF ) is replaced by the isotropic mean
value of \pi / 4 and if \sin ^{2} \theta _{1} and \sin ^{2} \theta _{2} in
equation (REF ) are replaced by the isotropic mean value of
{2}{3}, then a combination of equations (REF ),
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} | 10.1086/589628 | 0807.1702 | Evidence of a Curved Synchrotron Spectrum in the Supernova Remnant SN
1006 | [
"G. E. Allen",
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f4e778d7af36c0696035f1249d057c230f90f3a1 | subsection | 24 | 31 | Diffusion coefficient | Equation () is similar to equation (22) of ,
equation (12) of , equation (A.4) of , and
equation (22) of , except these authors use the peak frequency
instead of the critical frequency. The latter three also use \sin \theta =
1 instead of \pi / 4.f1b.eps
[f1b.eps]
Color-coded ACIS image of the northeastern rim of SN 10... | {
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1006 | [
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34f9e43bfb56c59904519bf3ec4b49e37d3b77dc | subsection | 25 | 31 | Diffusion coefficient | Bottom: Differences between
the data points and the solid line, divided by the uncertainties in the data
points.f5.eps
[f5.eps]
X-ray emission profiles along a 49 -wide strip passing through
region 6 toward the center of SN 1006. The black histogram in the top panel is
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} | 10.1086/589628 | 0807.1702 | Evidence of a Curved Synchrotron Spectrum in the Supernova Remnant SN
1006 | [
"G. E. Allen",
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"S. J. Sturner"
] | [
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5d8fc7ede1a3fb620e3c7b54d9534ed6ceefd21e | subsection | 26 | 31 | Diffusion coefficient | The
red data points, which have similar confidence intervals, are the results
obtained using the peak radio fluxes (Table ). The dashed line is
the weighted mean value (4.98 \times 10^{16} Hz) of the black points. The
corresponding 90% confidence level interval [(4.31–5.65) \times 10^{16} Hz] lies between the two dotte... | {
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} | 10.1086/589628 | 0807.1702 | Evidence of a Curved Synchrotron Spectrum in the Supernova Remnant SN
1006 | [
"G. E. Allen",
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0b68a52e758035513a88e2d8db13dfd5214b4b8c | subsection | 27 | 31 | Diffusion coefficient | 1988; (4) Gardner & Milne 1965; (5) Kundu 1970; (6) Milne
& Dickel 1975.ccccccccccc
Best-Fit Parameters Using the Curved Model and “Cospatial”
Radio Fluxes
0pt
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(J2000)
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"cite_spans": []
} | 10.1086/589628 | 0807.1702 | Evidence of a Curved Synchrotron Spectrum in the Supernova Remnant SN
1006 | [
"G. E. Allen",
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"S. J. Sturner"
] | [
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cb6027c4fe1e1a592a78003f76bf5779419b64ee | subsection | 28 | 31 | Diffusion coefficient | The uncertainties, which include only
the statistical contributions, are reported at the 90% confidence level.
These uncertainties were used to compute the weighted mean values of
\Gamma and a. Units of right ascension are hours, minutes, and seconds,
and units of declination are degrees, arcminutes, and arcseconds.a
A... | {
"cite_spans": []
} | 10.1086/589628 | 0807.1702 | Evidence of a Curved Synchrotron Spectrum in the Supernova Remnant SN
1006 | [
"G. E. Allen",
"J. C. Houck",
"S. J. Sturner"
] | [
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628094d12063015581fee4c27bd28467a409328c | subsection | 29 | 31 | Diffusion coefficient | Units of
right ascension are hours, minutes, and seconds, and units of declination
are degrees, arcminutes, and arcseconds.aThe curvature parameter was fixed at zero.ccccccccccc
Best-Fit Parameters Using the Curved Model and Peak Radio
Fluxes
0pt
\alpha _{R}
\delta _{R}
\Delta \Psi
\zeta
\nu _{m}Region
(J200... | {
"cite_spans": []
} | 10.1086/589628 | 0807.1702 | Evidence of a Curved Synchrotron Spectrum in the Supernova Remnant SN
1006 | [
"G. E. Allen",
"J. C. Houck",
"S. J. Sturner"
] | [
"astro-ph"
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67d88d3aeeb58f7f94823df28488c879ce7bb4cf | subsection | 30 | 31 | Diffusion coefficient | The uncertainties, which include only the
statistical contributions, are reported at the 90% confidence level. These
uncertainties were used to compute the weighted mean values of \Gamma and
a. Units of right ascension are hours, minutes, and seconds, and units of
declination are degrees, arcminutes, and arcseconds.a
A... | {
"cite_spans": []
} | 10.1086/589628 | 0807.1702 | Evidence of a Curved Synchrotron Spectrum in the Supernova Remnant SN
1006 | [
"G. E. Allen",
"J. C. Houck",
"S. J. Sturner"
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69e328111dfe7e3eb1f518345e411bfc2fb84dd4 | abstract | 0 | 23 | Abstract | (abridged) MHD turbulence is known to exist in shearing boxes with either
zero or nonzero net magnetic flux. However, the way turbulence survives in the
zero-net-flux case is not explained by linear theory and appears as a purely
numerical result. Aims: We look for a nonlinear mechanism able to explain the
persistence ... | {
"cite_spans": []
} | 10.1051/0004-6361:200810152 | 0807.1703 | On Self-Sustained Dynamo Cycles in Accretion Discs | [
"G. Lesur",
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7d53e41e450eb28eb1fdbc20e80b93cec3b19d50 | subsection | 1 | 23 | Introduction | The problem of angular momentum transport is a central issue of accretion disc theory. Following , angular momentum transport is often modelled assuming the disc is turbulent, using a kind of turbulent viscosity (the so-called \alpha model). However, the way discs may become turbulent is still a highly debated subject.... | {
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6cc6b6ef49dea7ae8d9e13d43e5bad6b9a1b6a51 | subsection | 2 | 23 | Introduction | Note also that studied this dynamo process in discs, using boundary conditions allowing for mean flux variations. Although a azimuthal field was generated in their simulations, no physical understanding of the underlying process was provided.In this paper, we describe a possible mechanism able to sustain MHD turbulence... | {
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d768db86525bea37497ab03609ae40a521a8f196 | subsection | 3 | 23 | Shearing-box equations and numerical method | MRI-related turbulence has been extensively studied in the literature. Therefore, we will recall here briefly the basic equations for the shearing-box model. The reader may consult , and for an extensive discussion of the properties and limitations of this model. Since MHD turbulence in discs is subsonic, we will work ... | {
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3d2aaddf77f6886dd28a4ffa1023a06dce87f483 | subsection | 4 | 23 | Shearing-box equations and numerical method | These may be written as {v}={u}-Sy{e_x}, leading to the following equations for {v}:\partial _t {v}+{\nabla \cdot }({v \otimes v})&=&-{\nabla } \Pi +{\nabla \cdot }({ B \otimes B})-Sy\partial _x {v}\\
& & +(2\Omega -S) v_y{e_x}-2\Omega v_x {e_y}+\nu {\Delta v},\\
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a54663f5c832ea7be8d894b6135a1ab1a4367a82 | subsection | 5 | 23 | Shearing-box equations and numerical method | One orbit corresponds to T_\mathrm {orb}=3\pi S^{-1}.
[Figure: Fourier analysis of \widehat{B_x}(k_x=0,k_y=0,k_z=2\pi /L_z): amplitude (top panel) and phase (middle panel). This mode exhibits long-timescale (T\sim 50\, S^{-1}) cycles during which the phase is approximately constant. The transport coefficient (\alpha , ... | {
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c79dde1324f23a3503e438bbc53f2d02dbd9581e | subsection | 6 | 23 | Long-timescale cycle in zero-net-flux MHD turbulence | The first zero-net-flux turbulent flow was computed by in the context of a stratified compressible shearing box. In this section we consider the simpler unstratified and incompressible case. The aspect ratio is set to L_y=L_z/2 and L_x=2L_z, which corresponds to the box used by elongated twice in the vertical direction... | {
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17cc98c057dc417290332c9017d480cb4e6fe370 | subsection | 7 | 23 | Long-timescale cycle in zero-net-flux MHD turbulence | One easily observes the large-scale B_x(z) on the t=260 snapshot whereas strong nonaxisymmetric structures destroy the large scale structures at t=280.][Figure: Amplitude projected on \widehat{B_x}(k_0,t) (left) and phase (right) of the terms involved in equation (). The amplitude clearly shows one cycle comparable to ... | {
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6e17d305dce9b66f66a4cc9d36fa3be15e079e0a | subsection | 8 | 23 | Cycle analysis | Naturally, one may wonder what mechanism generates this magnetic field structure, and whether this mechanism is related to some turbulent transport properties. To investigate these questions, we first reduce the Reynolds number of the simulation, keeping Pm constant. This allows us to have “cleaner” flows to work with,... | {
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1f95b177192978eea8690cf213d2a31bf54d8805 | subsection | 9 | 23 | Cycle analysis | The shear term is positive in the beginning of the cycle, but becomes negative for t>270, whereas the EMF has a systematic resistive effect, with a clear phase correlation between the EMF and the resistive term. According to these results, the long-timescale cycle in the azimuthal field comes from the behaviour of the ... | {
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} | 10.1051/0004-6361:200810152 | 0807.1703 | On Self-Sustained Dynamo Cycles in Accretion Discs | [
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20fdc1fc31ac9c0690042989670d1ac51c82208c | subsection | 10 | 23 | Modal analysis of non axisymmetric structures | The EMFs described previously are obviously nonlinear terms, involving a coupling between {v} and {B}. An interesting question is therefore which modes contribute most to the EMFs observed in Figs REF –REF . In particular, one may wonder whether nonaxisymmetric structures play a role and if so, which structures are dom... | {
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6b6e3b67b15470262a41aec28f4bfd2f5a14505c | subsection | 11 | 23 | Modal analysis of non axisymmetric structures | REF . During one cycle, we note that the helicity associated with the vertical modes changes sign when B_y is reversed (t\simeq 270). Moreover, this reversal seems to be associated with an exchange of magnetic helicity between the nonaxisymmetric and the axisymmetric vertical modes. Note however that the total magnetic... | {
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5f66709375fb1bf6c3937bf76c665df24c779c84 | subsection | 12 | 23 | Model and equations | To understand the nonaxisymmetric origin of the EMF described previously, we consider a linear model including shearing waves in the presence of a background azimuthal magnetic field with a vertical structure, B_x^0(z). This vertical structure is required in order to compare the EMF generated by the perturbation with t... | {
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8e6cb6e7118b56b91607d8d276c69b34f47d83ea | subsection | 13 | 23 | Model and equations | Using these solutions in the evolution equations (REF )–(), one eventually finds:\partial _t {\bar{v}}&=&-(i{k}+{e_z}\partial _z)\bar{\Pi }+(2\Omega -S)\bar{v}_y{e_x}-2\Omega \bar{v}_x {e_y}\\
& &+ik_xB_x^0(z){\bar{b}} +\bar{b}_z\partial _zB_x^0(z){e_x}+\nu (\partial _z^2-k^2){\bar{v}},\\
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0464249241903e79972139014aa75eeb0c84292f | subsection | 14 | 23 | Numerical solution | In the following, we will assume a large-scale field similar to the oscillating Fourier mode studied in the previous section. We therefore set:B_x^0(z)=B_0\cos (k_0z).The resistive diffusion of this non-uniform field may be neglected on the timescales of interest here. To solve the equations (REF ) and (), we consider ... | {
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5b4596d4df1975fd804eedb8785491238353e868 | subsection | 15 | 23 | Numerical solution | In a real system, this excitation is a highly nonlinear process which depends on the small-scale properties of the turbulence.Note that these properties may depend in turn on the large-scale field and the amplitudes of the shearing waves. Therefore, our random excitation is not a precise enough turbulence model and we ... | {
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b56a6741c8abadbe6cf79050d98fcd054ba7e9f0 | subsection | 16 | 23 | Phenomenological properties | The waves described in this section are clearly inhomogeneous in the vertical direction. However, we can understand them as a version of the magnetorotational instability in the presence of a varying azimuthal field , . As one would expect, the transport coefficients \langle b_x b_y\rangle and \langle v_x v_y\rangle as... | {
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89ee2a6bee1aaed60d8465f4995259634a6898e5 | subsection | 17 | 23 | A toy model | In this section, we provide a toy model reproducing the basic linear properties exhibited in the previous section. This toy model does not pretend to be an accurate set of closure relations for equations (REF )–() but it includes the main physical ingredients required to reproduce qualitatively the cycle behaviour desc... | {
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68f258c57ef4cd39b036d6aae9101a2e5ed3f890 | subsection | 18 | 23 | A toy model | The resulting evolution of B_x and B_y is plotted in Fig. REF , where the “mean shear” curve corresponds to the first term on the right-hand side of equation (REF ), and the EMFs are the \beta and \gamma terms. When comparing with the fully nonlinear cycle (Figs REF –REF ), we find essentially the same time history for... | {
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e55fd38536766a9d3d05973f9515852ce0c75517 | subsection | 19 | 23 | Summary | In this paper, we have investigated the behaviour of the large-scale magnetic field in zero-net-flux simulations of the magnetorotational instability. We have first shown that the large-scale azimuthal field B_x(z) is subject to a long-timescale oscillation when the flow is turbulent. Studying the induction equation, w... | {
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3535799db61982100a87fe21406a02152ea43c38 | subsection | 20 | 23 | Comparison with previous works | We would like to stress that the model presented here does not constitute a full description of a sustaining mechanism for MHD turbulence in discs. Indeed, we have assumed in the linear analysis and in the closure model that the flow is able to generate continuously small-amplitude shearing waves. As mentioned previous... | {
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13cc8eba290a530132a149bf9433a109c306bcaf | subsection | 21 | 23 | Comparison with previous works | If this picture is correct, we would then expect a minimum transport of the order of a few times 10^{-3}, independently of the Reynolds numbers. Note however that this conclusion relies on the assumption that the large-scale field does not depend on the dissipation coefficients. We have shown that this assumption is pl... | {
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df1d08f259998abb77aaf0d9675e6dab3bc25c6f | subsection | 22 | 23 | Future work | As discussed previously, the main issue raised by our findings is how the turbulence is able to excite shearing waves. Interestingly, the same kind of problem arises in the case of the shearing box with a mean azimuthal field . Therefore, a simpler way to study this effect is to investigate the way turbulence is sustai... | {
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58ff8b89ce8534200190105fbea17f7d7476002b | abstract | 0 | 64 | Abstract | A "Chen space" is a set X equipped with a collection of "plots" - maps from
convex sets to X - satisfying three simple axioms. While an individual Chen
space can be much worse than a smooth manifold, the category of all Chen spaces
is much better behaved than the category of smooth manifolds. For example, any
subspace ... | {
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db42c29e4b42b0ec3ff658778fef7b3dcd93254e | subsection | 1 | 64 | Introduction | Algebraic topologists have become accustomed to working in a category
of spaces for which many standard constructions have good formal
properties: mapping spaces, subspaces and quotient spaces,
limits and colimits, and so on. In differential geometry the
situation is quite different, since the most popular category,
th... | {
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c693e0f8273e6cf8b1093c5c6628c9aac914c19b | subsection | 2 | 64 | Introduction | Despite a superficial resemblance to charts in the
theory of manifolds, plots are very different: we should think of a
plot in X as an arbitrary smooth map to X from a convex
subset of a Euclidean space of arbitrary dimension. So instead
of ensuring that Chen spaces look nice locally, plots play a different
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af62660aa05a4f0268602f83bf701933368cbf20 | subsection | 3 | 64 | Introduction | It is nice having the solution set of an equation between smooth maps
be a smooth space, but the price we pay is that a smooth space can be
locally as bad as the Cantor set.So, we should not expect the theory of smooth spaces to support the
wealth of fine-grained results familiar from the theory of smooth
manifolds. In... | {
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f56f9dfd2d69c4aab017644f97fef77bbeafb222 | subsection | 4 | 64 | Introduction | Our work covers
a wide class of definitions that take the `maps in' approach.The structure of the paper is as follows. In Section ,
we define Chen spaces and diffeological spaces and give some examples. We
also discuss the relation between these two formalisms, focusing on manifolds
with corners and the work of Stacey ... | {
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b2a2148ed00fb0bcdae4a1d777633080b425d024 | subsection | 5 | 64 | Smooth Spaces | Souriau's notion of a `diffeological space' is very simple:Definition 1 An open set is an open subset of {\mathbb {R}}^n. A function
f\colon U\rightarrow U^{\prime } between open sets is called smooth if it has
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273ea66f3cc465b31700cacfc0210bf22d22fcad | subsection | 6 | 64 | Smooth Spaces | This marks an important realization, emphasized by Stacey
: we can give a space a smooth structure without
first giving it a topology. Indeed, we shall see that a smooth structure
determines a topology!The notion of a smooth function f \colon C \rightarrow C^{\prime } between convex sets
is a bit subtle, particularly f... | {
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4b28ffa1c91aead15745ce7ceb0527327c293e83 | subsection | 7 | 64 | Smooth Spaces | If \gamma \colon {\mathbb {R}}\rightarrow C is a smooth curve in C, then
f \gamma is a smooth curve in C^{\prime }.The equivalence of conditions 1 and 2 is not hard; the equivalence
of 2 and 3 was proved by Kriegl , and appears
as Theorem 24.5 in Kriegl and Michor's book .Since most of our results apply both to Chen sp... | {
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2096ba5132d903aacbce6d22b70e771067ac4d88 | subsection | 8 | 64 | Examples | Next we give some examples. For these it is
handy to call the set of plots in a smooth space its smooth
structure. So, we may speak of taking a set and putting a smooth
structure on it to obtain a smooth space.Any domain D becomes a smooth space, where the plots
\varphi \colon D^{\prime } \rightarrow D are just the smo... | {
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21c1d5c87d236bc446685c18581e2c388ea762f1 | subsection | 9 | 64 | Comparison | We should also say a bit about how Chen spaces and diffeological
spaces differ, and how they are related. To begin with, let us
compare their treatment of manifolds with boundary, or more generally
manifolds with corners , .An n-dimensional manifold with corners M has charts of the form
\varphi \colon X_k \rightarrow M... | {
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aef7e2766180858995d5b4ed7fbf615488c88f07 | subsection | 10 | 64 | Comparison | These take advantage of
the fact that every open subset of \mathbb {R}^n becomes a Chen space
with its subspace smooth structure, and conversely, every convex subset
of \mathbb {R}^n becomes a diffeological space.Using this, Stacey defines for any Chen space X a diffeological space
{\rm So}X with the same underlying se... | {
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} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
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8dc8aec7185770285df5853fbb4f61c0969dd47b | subsection | 11 | 64 | Comparison | To see this,
he takes I to be [0,1] \subset {\mathbb {R}} made into a Chen space with its
subspace smooth structure. If I were isomorphic to a Chen space in
the image of {\rm Ch}^\flat , say I \cong {\rm Ch}^\flat X, we would then
have {\rm Ch}^\flat {\rm So}I = {\rm Ch}^\flat {\rm So}{\rm Ch}^\flat X = {\rm Ch}^\flat ... | {
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"John C. Baez",
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a3b3b2d9f27393eb2e5c8858d1b2e09eacbfd3c3 | subsection | 12 | 64 | Convenient Properties of Smooth Spaces | Now we present some useful properties shared by Chen spaces and
diffeological spaces. Following Def. REF , we call either
kind of space a `smooth space', and we use \mathcal {C} ^\infty to denote either
the category of Chen spaces or the category of diffeological spaces.
Most of the proofs are straightforward diagram c... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
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de7b04d77a99cc067e49d9a2a1e9bcd28d68b8da | subsection | 13 | 64 | Convenient Properties of Smooth Spaces | Not every
epimorphism is of this form: for example, the natural map from {\mathbb {R}} with
its standard smooth structure to {\mathbb {R}} with its indiscrete smooth structure
is also an epimorphism. In Prop. REF , we show that a smooth
map p \colon X \rightarrow Y comes from taking a quotient space precisely when p
is... | {
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"John C. Baez",
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4c283622316d9aeb5c2c9974b072d65b5e4bf33e | subsection | 14 | 64 | Convenient Properties of Smooth Spaces | Moreover, for any other smooth space Q with smooth maps
f_X \colon X \rightarrow Q and f_Y \colon Y \rightarrow Q, there exists a unique smooth
maps f \colon X + Y \rightarrow Q such that
{
& Q & \\
X[r]_{i_X}[ur]^{f_X} & X + Y[u]_{f} & Y[l]^{i_Y}[ul]_{f_Y}\\
}
commutes. So, X+Y is indeed the coproduct of X and Y in ... | {
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853f032d84aa1d3fbedea89467afda304890dfb6 | subsection | 15 | 64 | Convenient Properties of Smooth Spaces | The natural functions
p_X \colon X \times _Z Y \rightarrow X, \qquad p_Y \colon X \times _Z Y \rightarrow Y
are then smooth, and it is easy to check this diagram is a pullback square:
{
X \times _Z Y [d]_{p_Y} [r]^{p_X} & X[d]^f \\
Y[r]_g & Z \\
}
In other words, given any commutative square of smooth maps
like thi... | {
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ba92805891df8357f473d2422d9ecad99647a813 | subsection | 16 | 64 | Convenient Properties of Smooth Spaces | Mapping spaces
Given smooth spaces X and Y, the set
\mathcal {C} ^\infty (X,Y) =\lbrace f\colon X\rightarrow Y\colon f\text{ is smooth}\rbrace
becomes a smooth space where a function \tilde{\varphi } \colon D\rightarrow \mathcal {C} ^\infty (X,Y) is a plot if and only if the corresponding function
\varphi \colon D \t... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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82233aaa4f12e8e9c4b76610b0cd55222efef329 | subsection | 17 | 64 | Convenient Properties of Smooth Spaces | This allows us to define the smooth structure on \mathcal {C} ^\infty _B(Y,Z):
for any domain D, a function
\tilde{\varphi } \colon D \rightarrow C_B^\infty (Y,Z)
is a plot if and only if p \tilde{\varphi } is smooth and
the corresponding function
\varphi \colon D \times _B Y \rightarrow Z
is smooth. With this smoo... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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6bb043d27be41a98153d7617abda0c89259f4386 | subsection | 18 | 64 | Smooth Spaces as Generalized Spaces | The concept of a `generalized space' was developed in the context of
quasitopos theory by Antoine , Penon
, and Dubuc , .
Generalized spaces form a natural framework for studying Chen spaces,
diffeological spaces, and even simplicial complexes. For us, a category
of generalized spaces will be a category of `concrete sh... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 122,
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"raw": "P. Antoine, Etude élémentaire des catégories d'ensembles structurés, Bull. Soc. Math. Belgique, 18 (1966), 142–164, 387–414.",
"source_ref_id": "2cd3854a9b285706e4684ea7bb17f7cf79c93c54",
... | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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6a3428e969d19eee076e6fdf1457e0f9425df576 | subsection | 19 | 64 | Smooth Spaces as Generalized Spaces | Axiom 1 in the definition of a Chen space is what gives us a
contravariant functor from {\sf Chen} to {\rm Set}: it says that given
any morphism f \colon C \rightarrow D in {\sf Chen}, we get a functionX(f) \colon X(D) \rightarrow X(C)sending any plot \varphi \colon D \rightarrow X to the plot \varphi f \colon C \right... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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... | |
af8a5eae6fc62f747e33bb8f532c247bf6fc285a | subsection | 20 | 64 | Smooth Spaces as Generalized Spaces | \end{}Quite generally, any object D in a category with a
terminal object has an underlying set hom(1,D), often
called its set of `points^{\prime }. The requirement
that hom(1,-) be faithful says that two morphisms
f,g C D in are equal when they induce
the same functions from points of C to points of D.
In other words: ... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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... | |
417191e50b8a0dea1b7fb5341b2ec0307c3476ab | subsection | 21 | 64 | Smooth Spaces as Generalized Spaces | The category
{\sf Chen} has a subcanonical coverage where (i_j\colon C_j\rightarrow C|j\in J)
is a covering family if and only if the convex sets C_j \subseteq C form an open covering of the convex
set C \subseteq {\mathbb {R}}^n with its usual subspace topology, and
i_j \colon C_j \rightarrow C are the inclusions.Give... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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... | |
feee3ee7163f8dc98480a06fc0d8df39c8ec30ea | subsection | 22 | 64 | Smooth Spaces as Generalized Spaces | Then, let \varphi \in {\mathbf {X}}(C) and compute
\underline{\varphi } \colon {\rm hom}(1,C) \rightarrow X(1) \cong X:\underline{\varphi }(c) = X(c)(\varphi ) = \varphi (c)where at the last step we identify the smooth function c \in {\rm hom}(1,C)
with the one point in its image. So, \underline{\varphi } is the underl... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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... | |
b24d87d2f1d4cdc96350559685670ac26c1fe28d | subsection | 23 | 64 | Smooth Spaces as Generalized Spaces | When we turn a concrete sheaf {\mathbf {X}} into a Chen
space X and back into a concrete sheaf {\mathbf {X}}^{\prime }, we have{\mathbf {X}}^{\prime }(C) = \lbrace \underline{\varphi } \colon C \rightarrow X(1) \rbracebut the latter is naturally isomorphic to X(C) via the function\begin{array}{rcl}
X(C) &\rightarrow & ... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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0.00... | |
4a34b4ee58316875753f69016bfd827ae5076120 | subsection | 24 | 64 | Smooth Spaces as Generalized Spaces | For instance, take f_i to be the identity function on D. The coverage
is clearly subcanonical since each covering includes the identity morphism.Henceforth we make {\sf F} into a concrete site with the
above coverage. Since every covering family contains the identity,
this coverage is `vacuous': every presheaf is a she... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 494,
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"raw": "M. Grandis, Finite sets and symmetric simplicial sets, Th. Appl. Cat. bf 8 (2001), 244–252. Available at http://www.tac.mta.ca/tac/volumes/8/n8/8-08abs.html.",
"source_ref_id": "c5455f959b658a... | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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fe0999502bd0e19054b5f4fa5e29456e055a7b0b | subsection | 25 | 64 | Smooth Spaces as Generalized Spaces | Since a map {\mathbf {X}}\Rightarrow {\mathbf {Y}} of {\sf F} spaces is completely determined by the
function 1\colon {\mathbf {X}}(1)\rightarrow {\mathbf {Y}}(1) it is clear that this functor is
faithful. We see that the functor is full since given a map of
simplicial complexes f\colon (X,K)\rightarrow (Y,L) and a mor... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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... | |
4a90cd253558efe1cfc4795fad9d3ec0ac922d17 | subsection | 26 | 64 | Convenient Properties of Generalized Spaces | In this section we establish convenient properties of any category
of generalized spaces. We begin with some handy notation. In Section
we introduced three closely linked notions of
`underlying set' or `underlying function' in the context of a concrete site
. It will now be convenient, and we hope not confusing, to de... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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-0.01802104711532592... | |
2e963887528018ace1ca994619685cd7c9407ebf | subsection | 27 | 64 | Convenient Properties of Generalized Spaces | Recall that
f and g are natural transformations between the functors
X,Y opSet, so given D the
following squares commute for each d\in \underline{D}:{
X(D)[r]^{f_D}[d]_{X(d)} & Y(D)[d]^{Y(d)}
& & X(D)[r]^{g_D}[d]_{X(d)} & Y(D)[d]^{Y(d)} \\
X(1)=\underline{X}[r]_{\underline{f}} & \underline{Y}=Y(1)
& & X(1)=\underline{X... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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-0... | |
15bca24e02ddfaed9c8dc2ec86b91be4617258ea | subsection | 28 | 64 | Convenient Properties of Generalized Spaces | Henceforth we denote this
functor by an underline:
\underline{\;\;} \colon {\rm Set}.
So, any domain D has an underlying set \underline{D}, and any morphism
f \colon C \rightarrow D in has an underlying function f C D. The concreteness condition on says that this underlying set
functor is faithful.
The underlying se... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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064bd90e75e09ac1a7e54447a8a8e014e70a039a | subsection | 29 | 64 | Convenient Properties of Generalized Spaces | Since the natural transformation\underline{\;\;} \colon Y(D) \rightarrow \underline{Y}^{\underline{D}}is one-to-one, it suffices to show that\underline{f_D(\varphi )}(d) = \underline{g_D(\varphi )}(d)for all d\in D, or in other words,Y(d) f_D(\varphi ) = Y(d) g_D(\varphi ) .By the above commuting squares, this amounts ... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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... | |
2907a6f474c79ddd9e35ce858f13cca79d47d507 | subsection | 30 | 64 | Subspaces, Quotient Spaces, and Limits | With these preliminaries in hand, we now study subspaces and
quotient spaces of spaces, and show that the category of
spaces has a weak subobject classifier, \Omega . In the process
we will show that has limits.For Chen spaces or
diffeological spaces, \Omega is just the 2-element set 2 = \lbrace 0,1\rbrace
equipped w... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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c1b63f31929b571ac3f7524b9334a4a0f449449e | subsection | 31 | 64 | Subspaces, Quotient Spaces, and Limits | Finally, since plots \varphi \in \Omega (D) are in one-to-one correspondence with functions
\underline{\varphi } \colon \underline{D} \rightarrow 2, the sheaf \Omega is concrete.Proposition 31
A monomorphism (resp. epimorphism) in is a map
f\colon X\rightarrow Y for which the underlying function \underline{f} is injec... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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e37bab10938c275baa6005a1871385b8b4e74153 | subsection | 32 | 64 | Subspaces, Quotient Spaces, and Limits | Note that for any plot \varphi \in B(D), the plot
g_D(\varphi ) \in X(D) has\underline{g_D (\varphi )}(\underline{D}) = \underline{g} \underline{\varphi } (\underline{D}) \subseteq \underline{i}(\underline{A}) ,where in the first step we use the naturality of the map sending a plot to
its underlying function, and in th... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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deacafdd1bc395b482d8dfe2f82f62c965f2a4fe | subsection | 33 | 64 | Subspaces, Quotient Spaces, and Limits | Since
A' is a subspace of X and (D)=A', there
exists a unique plot A'(D) such that jD()=.
Thus we have tD()A(D) and by commutativity of the diagram
iDtD()=jD()=. For any other 'A(D) with
iD(')=, we have '=tD() since i is a
monomorphism, and thus tD() is unique as desired.Definition 35
In any category, an epimorphism p... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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c1286d4c5b542600c4d9b2449935e18c1b66fd89 | subsection | 34 | 64 | Subspaces, Quotient Spaces, and Limits | To
show that \underline{t} induces a map of spaces, we need to check that
the following naturality square commutes for every map d D' D in :{
B(D) [r]^{t_D}@{-> >}[d]_{B(d)} & A(D) @{ >->}[d]^{A(d)}\\
B(D^{\prime }) [r]_{t_{D^{\prime }}} & A(D^{\prime })\\
}Since B and A are concrete sheaves, we can check that this dia... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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... | |
e2e3d37312579ae8935eb51643ab32bcf157d72a | subsection | 35 | 64 | Subspaces, Quotient Spaces, and Limits | Either \varphi came from a plot in E(D),
in which case we consider the covering family with just the identity
map (1 \colon D \rightarrow D), and there exists a plot \hat{\varphi }\in E(D) which maps to \varphi , or \varphi arose from sheafification.
In the latter case, there exists a family (f_i \colon D_i \rightarrow... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1427,
"openalex_id": "",
"raw": "S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer, Berlin, 1992.",
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... | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.04378775507211685,
0.004870815668255091,
-0.02409089356660843,
-0.008582094684243202,
0.006850418634712696,
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-0.03179570659995079,
0.015051087364554405,
-0.... | |
9d3de5f3b5351e72a67df912e1c2287a2563ea9b | subsection | 36 | 64 | Subspaces, Quotient Spaces, and Limits | So, taking the limits of both diagrams, we get a one-to-one function\lim _{\alpha \in C} \, F(\alpha )(D) \rightarrow \lim _{\alpha \in C} \, \underline{F(\alpha )}^{\underline{D}}which by the properties of limits can be reinterpreted
as a one-to-one function\lim _{\alpha \in C} \, F(\alpha )(D) \rightarrow (\lim _{\al... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.04494926705956459,
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0.00004249515768606216,
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-0.02331380918622017,
0.04140947386622429,
-0... | |
ada572b020c5ab08fd015f7ef5135befb34ec075 | subsection | 37 | 64 | Subspaces, Quotient Spaces, and Limits | So, (D) will be the power set of D:Proposition 30
There is a space such that for any object D ,
\Omega (D) = 2^{\underline{D}}, and for any morphism f \colon C \rightarrow D in ,
(f) 2D 2C sends any plot D 2 to the plot f C 2.\Omega is clearly a presheaf. To show that it is a sheaf, we
suppose (f_i \colon D_i \rightar... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.0349399596452713,
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-0.009406325407326221,
0.0017794205341488123,
... | |
2d90d17bc960bec9dd9dbc92635b87853fb0ac72 | subsection | 38 | 64 | Subspaces, Quotient Spaces, and Limits | Then the map from {\rm hom}(1,X) to
{\rm hom}(1,Y) given by composing with f is injective, but this
says precisely that \underline{f} is injective.Next, suppose f is epic. Then the map from {\rm hom}(Y,\Omega ) to
{\rm hom}(X,\Omega ) given by composing with f is injective, but
this says that the map from 2^{\underline... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.034879665821790695,
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-0.01225213147699833,
0.030180219560861588,
-0... | |
57e2c250b212c955242dee607492ec0117f012f9 | subsection | 39 | 64 | Subspaces, Quotient Spaces, and Limits | We sett_D(\varphi ) = \psi .We can check that t is a natural transformation by considering a morphism
f\colon D^{\prime }\rightarrow D in and the following diagram:{
&&&
\underline{A}@{>->}[d]^{\underline{i}}\\
\underline{D^{\prime }}@{-->}[urrr][r]_{\underline{f}} & \underline{D}[r]_{\underline{\varphi }}@{-->}[urr]|-... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.033970754593610764,
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-0.01823677122592926,
-0.03317718580365181,
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-0.03195631504058838,
0.014894634485244751,
0.... | |
212c00880350223b26ecc113e62d60aa9488f0ba | subsection | 40 | 64 | Subspaces, Quotient Spaces, and Limits | For any other 'A(D) with
iD(')=, we have '=tD() since i is a
monomorphism, and thus tD() is unique as desired.Definition 35
In any category, an epimorphism p\colon E\rightarrow B is strong if
given any monomorphism i\colon A\rightarrow X and morphisms f,g making the
outer square here commute:{
E [r]^{f}@{-> >}[d]_{p} ... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.025053545832633972,
0.012709868140518665,
-0.01160366740077734,
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-0.01050509512424469,
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0.015029075555503368,
-0.029081642627716064,
-0.01353379711508751,
-0.005195330362766981,... | |
35ce16cba8dd11f5c26f701fb91b674d67e1743a | subsection | 41 | 64 | Subspaces, Quotient Spaces, and Limits | To
show that \underline{t} induces a map of spaces, we need to check that
the following naturality square commutes for every map d D' D in :{
B(D) [r]^{t_D}@{-> >}[d]_{B(d)} & A(D) @{ >->}[d]^{A(d)}\\
B(D^{\prime }) [r]_{t_{D^{\prime }}} & A(D^{\prime })\\
}Since B and A are concrete sheaves, we can check that this dia... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.04271157830953598,
0.0020574014633893967,
-0.03471841290593147,
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-0.019418515264987946,
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-0.011082129552960396,
0.009419429115951061,
... | |
9f4656764aab843475efa5cd86ee86bc53b88532 | subsection | 42 | 64 | Subspaces, Quotient Spaces, and Limits | Either \varphi came from a plot in E(D),
in which case we consider the covering family with just the identity
map (1 \colon D \rightarrow D), and there exists a plot \hat{\varphi }\in E(D) which maps to \varphi , or \varphi arose from sheafification.
In the latter case, there exists a family (f_i \colon D_i \rightarrow... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1427,
"openalex_id": "",
"raw": "S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer, Berlin, 1992.",
"source_ref_id": "b8026f5bc4c6627f9a2d7b94052bd2f2dc9153df",
... | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.04378775507211685,
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0.014623889699578285,
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-0.03179570659995079,
0.015051087364554405,
-0.... | |
ead758b05aef4e1178851315423e2981c9aa860a | subsection | 43 | 64 | Subspaces, Quotient Spaces, and Limits | So, taking the limits of both diagrams, we get a one-to-one function\lim _{\alpha \in C} \, F(\alpha )(D) \rightarrow \lim _{\alpha \in C} \, \underline{F(\alpha )}^{\underline{D}}which by the properties of limits can be reinterpreted
as a one-to-one function\lim _{\alpha \in C} \, F(\alpha )(D) \rightarrow (\lim _{\al... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.0348476879298687,
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-0.00847543403506279,
0.0349697470664978,
-0.011... | |
aad27fec1d8767fedc611566d8309eafc622aeb7 | subsection | 44 | 64 | Parametrized Mapping Spaces | We next turn to the existence of parametrized mapping spaces between
spaces over a fixed base B.Definition 41
Given an object B in a category \mathcal {C} , the category of
objects over B (sometimes called the slice category
of B), has morphisms f\colon E\rightarrow B in \mathcal {C} as objects and
commuting triangle... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.015933403745293617,
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-0.01994727924466133,
0.02067984826862812,
0.02951647713780403,
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0.015437392517924309,
0.04975373297929764,
... | |
caeac2191073a9041dfbe4b1464812abb49021e5 | subsection | 45 | 64 | Parametrized Mapping Spaces | Alternatively, the space
structure on XB Y can be quickly obtained by the following lemma:Lemma 44 The monomorphism m\colon X\times _A Y\rightarrow X\times Y in given by
inclusion of sets \underline{X\times _A Y}\hookrightarrow \underline{X\times Y} is a
strong monomorphism.Given C\in , then for any (XY)(C) such that
(... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.014255387708544731,
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0.027900267392396927,
0.016209017485380173,
-0.02347407676279545,
-0.013156471773982048,
0.04145357012748718,... | |
d4e32c432ddedab74b592ffb8be1dc83e0177527 | subsection | 46 | 64 | Parametrized Mapping Spaces | We say that \underline{\varphi }
determines a plot for \mathcal {D} _B(X,Y) if this composite function
q\underline{\varphi } underlies a map of spaces and
\underline{C}\times _{\underline{B}} \underline{X}\stackrel{\underline{\varphi }\times _{\underline{B}} 1}{\longrightarrow }\coprod _{b\in \underline{B}} \underline... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.0348210372030735,
0.002788582118228078,
-0.05957113206386566,
0.013862800784409046,
-0.007679091300815344,
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0.005325390491634607,
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0.014457901939749718,
0.01747155375778675,
-0.02658117562532425,
0.021896664053201675,
0.0228... | |
d474205e0363e7ed40a4ff61ec903a76d9c030f6 | subsection | 47 | 64 | Parametrized Mapping Spaces | We consider the following diagram for each
j \in J:{
D_j [r]^{g_j} [drr]_{\tau _{ij}} & C^{\prime } [r]^{\psi } & C \times _B
X [r]^<<<<<{\varphi \times _B 1} & \mathcal {D} _B(X,Y)
\times _B X [r]^>>>>>>{{\rm ev}} & Y \\ & & C_i \times _B X
[ur]_{\varphi _i \times _B 1} & & }It is easy to check that this diagram commu... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.02282775193452835,
0.027939580380916595,
-0.01426734495908022,
-0.00691622868180275,
-0.010635657235980034,
-0.03022846020758152,
0.010681434534490108,
0.02478092722594738,
0.022705677896738052,
0.05111829191446304,
-0.024124782532453537,
-0.031769637018442154,
0.014442824758589268,
0.0... | |
3cce0cd2c0302ed505ba2ff8b9e24bf2641c2675 | subsection | 48 | 64 | Parametrized Mapping Spaces | The equality of the first components
follows from \underline{f_i h_{ij}}(e_j) = \underline{f_l h_{lk}}(e_k), and that the
family of plots \lbrace \varphi _i \in \mathcal {D} _B(X,Y)(C_i) | i \in I
\rbrace is compatible. Since Y is a smooth space, we have by the
sheaf condition that {\rm ev}(\varphi \times _B 1_X)\psi i... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.011962197721004486,
0.005366967990994453,
-0.03304862231016159,
0.012618287466466427,
0.040372416377067566,
-0.012099518440663815,
0.0418371744453907,
-0.0023153871297836304,
0.051205530762672424,
0.035276275128126144,
0.01789752207696438,
-0.014311915263533592,
0.015730900689959526,
0.... | |
79af5fd0adafaced69d69f37f43973cdef5040e1 | subsection | 49 | 64 | Parametrized Mapping Spaces | One can check that the following
diagram commutes and it follows that f determines a
map:
(-40,0)*{\underline{C}\times _{\underline{B}} \underline{X}}="1";
(-15,0)*{\underline{Z}\times _{\underline{B}} \underline{X}}="2";
(15,0)*{\coprod \underline{\mathcal {D}}(X_b,Y_b)\times _{\underline{B}} \underline{X}}="3";
(... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.04351116344332695,
-0.021008018404245377,
-0.04366372525691986,
-0.005187165457755327,
-0.012250863946974277,
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0.03896476328372955,
0.007864657789468765,
0.02640877291560173,
0.04222962632775307,
-0.031672220677137375,
-0.02991773746907711,
0.01550047006458044,
-0.... | |
727423837467c86bd4c1fe013ad6fc2059f36473 | subsection | 50 | 64 | Parametrized Mapping Spaces | This induces a map Z^{\prime }\times _B X to
Z\times _B X which we compose with \tilde{f} to obtain the desired
map. It follows that (z^{\prime },x)\in Z^{\prime }\times _B X\mapsto (g(z^{\prime }),x)\in Z\times _B X\mapsto \tilde{f}(g(z^{\prime }),x)\in Y. Thus the diagram
commutes. Given a map h\colon Y \rightarrow Y... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.013921384699642658,
-0.006941622123122215,
-0.015408875420689583,
0.016766685992479324,
0.021526657044887543,
-0.0040581789799034595,
-0.011754988692700863,
0.003699655644595623,
0.020107818767428398,
0.04406023025512695,
-0.009886089712381363,
-0.0035165799781680107,
0.03521156683564186,... | |
f0a6afab9847709d176226e93ef8d87b355ee6f7 | subsection | 51 | 64 | Parametrized Mapping Spaces | Given objects X,Y over B, we call the internal hom \mathcal {C} _B(X,Y) a
parametrized mapping space.We want to show that the category of spaces is locally cartesian closed.
To do this we need to determine the product and internal hom in the
category of spaces over some space B. Given two spaces over B,
{
X[dr]_{p_X} ... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.014922825619578362,
0.002414659596979618,
-0.03521664813160896,
0.015022005885839462,
0.016860656440258026,
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0.010833543725311756,
0.009109332226216793,
0.029982976615428925,
-0.026320934295654297,
0.013023140840232372,
0.05248928442597389,
... | |
b7f308dce347759b8ead0bac5fd5f18ddcb98b89 | subsection | 52 | 64 | Parametrized Mapping Spaces | We say that \underline{\varphi }
determines a plot for \mathcal {D} _B(X,Y) if this composite function
q\underline{\varphi } underlies a map of spaces and
\underline{C}\times _{\underline{B}} \underline{X}\stackrel{\underline{\varphi }\times _{\underline{B}} 1}{\longrightarrow }\coprod _{b\in \underline{B}} \underline... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.0348210372030735,
0.002788582118228078,
-0.05957113206386566,
0.013862800784409046,
-0.007679091300815344,
-0.03356979787349701,
0.005325390491634607,
0.032410115003585815,
0.00397115433588624,
0.014457901939749718,
0.01747155375778675,
-0.02658117562532425,
0.021896664053201675,
0.0228... | |
82df8e38153b3fa7153b2a4c0262cfdd2caeba0c | subsection | 53 | 64 | Parametrized Mapping Spaces | We consider the following diagram for each
j \in J:{
D_j [r]^{g_j} [drr]_{\tau _{ij}} & C^{\prime } [r]^{\psi } & C \times _B
X [r]^<<<<<{\varphi \times _B 1} & \mathcal {D} _B(X,Y)
\times _B X [r]^>>>>>>{{\rm ev}} & Y \\ & & C_i \times _B X
[ur]_{\varphi _i \times _B 1} & & }It is easy to check that this diagram commu... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.02282775193452835,
0.027939580380916595,
-0.01426734495908022,
-0.00691622868180275,
-0.010635657235980034,
-0.03022846020758152,
0.010681434534490108,
0.02478092722594738,
0.022705677896738052,
0.05111829191446304,
-0.024124782532453537,
-0.031769637018442154,
0.014442824758589268,
0.0... | |
ff5f94621fe37d355da790f311a574dbd00b9df6 | subsection | 54 | 64 | Parametrized Mapping Spaces | The equality of the first components
follows from \underline{f_i h_{ij}}(e_j) = \underline{f_l h_{lk}}(e_k), and that the
family of plots \lbrace \varphi _i \in \mathcal {D} _B(X,Y)(C_i) | i \in I
\rbrace is compatible. Since Y is a smooth space, we have by the
sheaf condition that {\rm ev}(\varphi \times _B 1_X)\psi i... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
-0.011962197721004486,
0.005366967990994453,
-0.03304862231016159,
0.012618287466466427,
0.040372416377067566,
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0.0418371744453907,
-0.0023153871297836304,
0.051205530762672424,
0.035276275128126144,
0.01789752207696438,
-0.014311915263533592,
0.015730900689959526,
0.... | |
312f3698681800bf67a1bc594ad9aea3a16b496b | subsection | 55 | 64 | Parametrized Mapping Spaces | One can check that the following
diagram commutes and it follows that f determines a
map:
(-40,0)*{\underline{C}\times _{\underline{B}} \underline{X}}="1";
(-15,0)*{\underline{Z}\times _{\underline{B}} \underline{X}}="2";
(15,0)*{\coprod \underline{\mathcal {D}}(X_b,Y_b)\times _{\underline{B}} \underline{X}}="3";
(... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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69203da627fc94ae3d254d12b836f9b940b506cd | subsection | 56 | 64 | Parametrized Mapping Spaces | This induces a map Z^{\prime }\times _B X to
Z\times _B X which we compose with \tilde{f} to obtain the desired
map. It follows that (z^{\prime },x)\in Z^{\prime }\times _B X\mapsto (g(z^{\prime }),x)\in Z\times _B X\mapsto \tilde{f}(g(z^{\prime }),x)\in Y. Thus the diagram
commutes. Given a map h\colon Y \rightarrow Y... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
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... | |
1384fee5db932d55419245ed69b9d6eb9317e693 | subsection | 57 | 64 | Colimits | In Prop. REF we showed that the category of spaces has
limits, which can be computed pointwise. To compute colimits in ,
we need some facts about `sheafification^{\prime } and also `concretization^{\prime }.
Given any site , sheafification is a functor
that takes presheaves on to sheaves on , but does not affect
preshe... | {
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"... | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
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96435034d7a5d1eaf664475d8afa750ed5203df5 | subsection | 58 | 64 | Colimits | On the other hand, there is an obvious inclusion functor
R\colon \mathrm {Conc}({\rm Set}^{{\rm op}}) \rightarrow {\rm Set}^{{\rm op}} .Lemma 47
L is left adjoint to R.Given a presheaf X, a concrete presheaf Y and a natural
transformation f\colon L(X)\rightarrow Y, we obtain a natural transformation
\tilde{f}\colon X... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
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] | 2,008 | en | Mathematics | [
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1a8bcaa262e49d41a40fe093390ac9ae8f0e3b51 | subsection | 59 | 64 | Colimits | This
function takes a compatible family \lbrace \varphi _i \rbrace to the
compatible family \lbrace \varphi _{j} \rbrace \subseteq \lbrace \varphi _i \rbrace for R^{\prime }.Since the sheafification process is pointwise a colimit of sets, given
plots \varphi ,\varphi ^{\prime } \in X^+(D) with the same underlying funct... | {
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... | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
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0.01... | |
6a83806ecb70bda1f7e1794cedd361ca4aeff75e | subsection | 60 | 64 | Colimits | Chen described a systematic process for improving any predifferentiable
space to a Chen space. This process is just the plus construction!
The point is that by Lemma REF , we can turn
a concrete presheaf into a concrete sheaf using
the plus construction.The following result is an easy spinoff of what we have done:Propo... | {
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{
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"... | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
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0.0... | |
bcfa47dffad4b77a45f577d7987d9e94946fd589 | subsection | 61 | 64 | Colimits | So, if X is a presheaf, X++ is a sheaf ---
and in fact it is the sheafification of X.
Next we turn to concretization, which makes presheaves
`concrete':Definition 46 Given a concrete site , we say
a presheaf X opSet is {\bf concrete} if for every
object D , the function sending plots \varphi \in X(D) to
functions \unde... | {
"cite_spans": []
} | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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... | |
4c4cf1f5c9dca16f228ca2ba0854ddce3fc67a0c | subsection | 62 | 64 | Colimits | Naturality in the second argument
follows similarly.Lemma 48 Any concrete presheaf on a concrete site is a separated presheaf.Clear.It follows that for any concrete presheaf, sheafification
is the same as one application of Grothendieck's plus
construction. This brings us to the following lemma:Lemma 49
Given a concre... | {
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... | 0807.1704 | Convenient Categories of Smooth Spaces | [
"John C. Baez",
"Alexander E. Hoffnung"
] | [
"math.DG",
"math.CT"
] | 2,008 | en | Mathematics | [
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