227 }; |
228 }; |
228 |
229 |
229 \draw[->] (A) -- (B); |
230 \draw[->] (A) -- (B); |
230 \draw[->] (A) -- (C); |
231 \draw[->] (A) -- (C); |
231 \end{tikzpicture} |
232 \end{tikzpicture} |
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233 \end{center} |
232 \caption{Producing weak identities from half pinched products} |
234 \caption{Producing weak identities from half pinched products} |
233 \label{fzo2} |
235 \label{fzo2} |
234 \end{figure} |
236 \end{figure} |
235 As suggested by the figure, these are two different reparameterizations |
237 As suggested by the figure, these are two different reparameterizations |
236 of a half-pinched version of $a\times I$. |
238 of a half-pinched version of $a\times I$. |
237 We must show that the two compositions of these two maps give the identity 2-morphisms |
239 We must show that the two compositions of these two maps give the identity 2-morphisms |
238 on $a$ and $a\bullet \id_x$, as defined above. |
240 on $a$ and $a\bullet \id_x$, as defined above. |
239 Figure \ref{fzo3} shows one case. |
241 Figure \ref{fzo3} shows one case. |
240 \begin{figure}[t] |
242 \begin{figure}[t] |
241 \begin{equation*} |
243 \begin{center} |
242 \mathfig{.83}{tempkw/zo3} |
244 \begin{tikzpicture} |
243 \end{equation*} |
245 |
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246 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
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247 \newcommand{\nsep}{1.8} |
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248 |
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249 \node(A) at (0,0) { |
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250 \begin{tikzpicture} |
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251 |
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252 \draw (0,0) coordinate (p1); |
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253 \draw (3.6,0) coordinate (p2); |
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254 \draw (2.3,1) coordinate (p3); |
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255 \draw (2.3,-1) coordinate (p4); |
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256 |
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257 \begin{scope} |
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258 \clip (p1) .. controls +(.5,-.5) and +(-.8,0) .. (p4) -- |
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259 (p2) -- (p3) .. controls +(-.8,0) and +(.5,.5) .. (p1); |
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260 \foreach \x in {0,0.26,...,4} { |
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261 \draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p3)}, second line={(0,0)--(0,1)}); |
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262 \draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p4)}, second line={(0,0)--(0,1)}); |
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263 } |
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264 \end{scope} |
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265 |
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266 \draw (p1) .. controls ($(p1) + (.5,.5)$) and ($(p3) + (-.8,0)$) .. node[red, above=3pt] {$a$} (p3); |
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267 \draw (p1) .. controls ($(p1) + (.5,-.5)$) and ($(p4) + (-.8,0)$) .. node[red, below=3pt] {$a$} (p4); |
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268 \draw (p3) -- node[red, above=7pt, right=1pt] {$x \times I$} (p2); |
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269 \draw (p4) -- node[red, below=7pt, right=1pt] {$x \times I$} (p2); |
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270 \draw (p1) -- (p2); |
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271 |
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272 \draw (p1) \vertex; |
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273 \draw (p2) \vertex; |
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274 \draw (p3) \vertex; |
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275 \draw (p4) \vertex; |
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276 |
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277 \end{tikzpicture} |
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278 }; |
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279 |
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280 \node[outer sep=\nsep](B) at (5.5,0) { |
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281 \begin{tikzpicture} |
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282 |
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283 \draw (0,0) coordinate (p1); |
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284 \draw (3.6,0) coordinate (p2); |
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285 \draw (2.3,1) coordinate (p3); |
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286 \draw (2.3,-1) coordinate (p4); |
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287 \draw (4.6,0) coordinate (p2b); |
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288 |
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289 \begin{scope} |
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290 \clip (p1) .. controls +(.5,-.5) and +(-.8,0) .. (p4) -- |
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291 (p2) -- (p3) .. controls +(-.8,0) and +(.5,.5) .. (p1); |
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292 \foreach \x in {0,0.26,...,4} { |
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293 \draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p3)}, second line={(0,0)--(0,1)}); |
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294 \draw[green!50!brown] (\x,0) -- (intersection cs: first line={(p2)--(p4)}, second line={(0,0)--(0,1)}); |
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295 } |
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296 \end{scope} |
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297 |
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298 \begin{scope} |
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299 \clip (p3)--(p2)--(p4)--(p2b)--cycle; |
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300 \draw[blue!50!brown, step=.23] ($(p4)+(0,-1)$) grid +(3,3); |
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301 \end{scope} |
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302 |
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303 \draw (p1) .. controls ($(p1) + (.5,.5)$) and ($(p3) + (-.8,0)$) .. node[red, above=3pt] {$a$} (p3); |
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304 \draw (p1) .. controls ($(p1) + (.5,-.5)$) and ($(p4) + (-.8,0)$) .. node[red, below=3pt] {$a$} (p4); |
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305 \draw (p3) -- (p2); |
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306 \draw (p4) -- (p2); |
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307 \draw (p3) -- node[red, above=7pt, right=1pt] {$x \times I$} (p2b); |
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308 \draw (p4) -- node[red, below=7pt, right=1pt] {$x \times I$} (p2b); |
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309 |
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310 \draw (p1) \vertex; |
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311 \draw (p2) \vertex; |
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312 \draw (p3) \vertex; |
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313 \draw (p4) \vertex; |
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314 \draw (p2b) \vertex; |
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315 |
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316 \end{tikzpicture} |
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317 }; |
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318 |
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319 \node[outer sep=\nsep](C) at (11,0) { |
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320 \begin{tikzpicture} |
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321 |
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322 \draw (0,0) coordinate (p1); |
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323 \draw (2.3,0) coordinate (p2); |
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324 \draw (2.3,1) coordinate (p3); |
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325 \draw (2.3,-1) coordinate (p4); |
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326 \draw (3.6,0) coordinate (p2b); |
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327 |
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328 \begin{scope} |
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329 \clip (p1) .. controls +(.5,-.5) and +(-.8,0) .. (p4) -- |
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330 (p2) -- (p3) .. controls +(-.8,0) and +(.5,.5) .. (p1); |
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331 \foreach \x in {0,0.26,...,4} { |
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332 \draw[green!50!brown] (\x,-1) -- (\x,1); |
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333 } |
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334 \end{scope} |
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335 |
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336 \begin{scope} |
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337 \clip (p3)--(p2)--(p4)--(p2b)--cycle; |
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338 \draw[blue!50!brown, step=.23] ($(p4)+(0,-1)$) grid +(3,3); |
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339 \end{scope} |
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340 |
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341 \draw (p1) .. controls ($(p1) + (.5,.5)$) and ($(p3) + (-.8,0)$) .. node[red, above=3pt] {$a$} (p3); |
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342 \draw (p1) .. controls ($(p1) + (.5,-.5)$) and ($(p4) + (-.8,0)$) .. node[red, below=3pt] {$a$} (p4); |
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343 \draw[green!50!brown] (p3) -- (p4); |
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344 \draw (p3) -- node[red, above=7pt, right=1pt] {$x \times I$} (p2b); |
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345 \draw (p4) -- node[red, below=7pt, right=1pt] {$x \times I$} (p2b); |
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346 |
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347 \draw (p1) \vertex; |
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348 \draw (p3) \vertex; |
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349 \draw (p4) \vertex; |
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350 \draw (p2b) \vertex; |
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351 |
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352 \end{tikzpicture} |
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353 }; |
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354 |
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355 \draw[->, thick, blue!50!green] (A) -- (B); |
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356 \draw[->, thick, blue!50!green] (B) -- node[black, above] {$=$} (C); |
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357 |
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358 \end{tikzpicture} |
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359 \end{center} |
244 \caption{Composition of weak identities, 1} |
360 \caption{Composition of weak identities, 1} |
245 \label{fzo3} |
361 \label{fzo3} |
246 \end{figure} |
362 \end{figure} |
247 In the first step we have inserted a copy of $(x\times I)\times I$. |
363 In the first step we have inserted a copy of $(x\times I)\times I$. |
248 Figure \ref{fzo4} shows the other case. |
364 Figure \ref{fzo4} shows the other case. |