141 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
141 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
142 to the intersection of the boundaries of $B$ and $B_i$. |
142 to the intersection of the boundaries of $B$ and $B_i$. |
143 If $k < n$ we require that $\gl_Y$ is injective. |
143 If $k < n$ we require that $\gl_Y$ is injective. |
144 (For $k=n$, see below.)} |
144 (For $k=n$, see below.)} |
145 |
145 |
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146 \xxpar{Strict associativity:} |
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147 {The composition (gluing) maps above are strictly associative. |
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148 It follows that given a decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
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149 into small $k$-balls, there is a well-defined |
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150 map from an appropriate subset of $\cC(B_1)\times\cdots\times\cC(B_m)$ to $\cC(B)$, |
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151 and these various $m$-fold composition maps satisfy an |
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152 operad-type associativity condition.} |
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153 |
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154 \nn{above maybe needs some work} |
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155 |
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156 The next axiom is related to identity morphisms, though that might not be immediately obvious. |
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157 |
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158 \xxpar{Product (identity) morphisms:} |
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159 {Let $X$ be homeomorphic to a $k$-ball and $D$ be homeomorphic to an $m$-ball, with $k+m \le n$. |
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160 Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. |
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161 If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram |
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162 \[ \xymatrix{ |
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163 X\times D \ar[r]^{\tilde{f}} \ar[d]^{\pi} & X'\times D' \ar[d]^{\pi} \\ |
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164 X \ar[r]^{f} & X' |
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165 } \] |
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166 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
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167 |
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168 \nn{Need to say something about compatibility with gluing (of both $X$ and $D$) above.} |
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169 |
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170 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
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171 The last axiom (below), concerning actions of |
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172 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
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173 |
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174 We start with the plain $n$-category case. |
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175 |
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176 \xxpar{Isotopy invariance in dimension $n$ (preliminary version):} |
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177 {Let $X$ be homeomorphic to the $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
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178 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
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179 Then $f(a) = a$ for all $a\in \cC(X)$.} |
146 |
180 |
147 |
181 |
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182 |
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183 |
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184 |
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185 \medskip |
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186 |
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187 \hrule |
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188 |
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189 \medskip |
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190 |
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191 \nn{to be continued...} |
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192 |