definition of 'finite simplicial complexes - simplicial maps' category to Top functor
Topics
About: category
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of 'finite simplicial complexes - simplicial maps' category.
- The reader knows a definition of Top category.
- The reader knows a definition of covariant functor.
Target Context
- The reader will have a definition of 'finite simplicial complexes - simplicial maps' category to Top functor.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( K\): \(= \text{ the 'finite simplicial complexes - simplicial maps' category }\)
\( Top\): \(= \text{ the Top category }\)
\(*\vert \bullet \vert\): \(: Obj (K) \to Obj (Top), Mor (K) \to Mor (Top)\), \(\in \{\text{ the covariant functors }\}\)
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Conditions:
\(\forall C \in Obj (K) (\vert C \vert = \vert C \vert)\), where the right hand side is the underlying space
\(\land\)
\(\forall f \in Mor (C_1, C_2) (\forall S \in C_1 (\vert f \vert_{\vert_{S}} = \text{ the affine map }))\)
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2: Natural Language Description
For the 'finite simplicial complexes - simplicial maps' category, \(K\), and the \(Top\) category, the covariant functor, \(\vert \bullet \vert\), \(: Obj (K) \to Obj (Top), Mor (K) \to Mor (Top)\), such that for each \(C \in Obj (K)\), \(\vert C \vert = \vert C \vert\), where the right hand side is the underlying space, for each \(f \in Mor (C_1, C_2)\), for each \(S \in C_1\), \(\vert f \vert_{\vert_{S}} = \text{ the affine map }\)
3: Note
\(\vert \bullet \vert\) is indeed a covariant functor: \(\vert C \vert\) is indeed a topological space; 1) \(\vert f_1 \vert \in Mor (\vert C_1\vert, \vert C_2 \vert)\), because \(\vert f \vert\) is indeed a continuous map from \(\vert C_1 \vert\) into \(\vert C_2 \vert\), because each simplex in \(C_1\) is closed on \(\vert C_1 \vert\), by the proposition that any affine simplex on any finite-dimensional real vectors space is closed and compact on the canonical topological superspace and each affine map is continuous, by the proposition that any affine map from the affine or convex set spanned by any possibly-non-affine-independent set of base points on any finite-dimensional real vectors space into any finite-dimensional real vectors space is continuous with respect to the canonical topologies, and the proposition that any map between topological spaces is continuous if the domain restriction of the map to each closed set of a finite closed cover is continuous can be applied; 2) \(\vert id_{C_1} \vert = id_{\vert O_1 \vert}\), obviously; 3) \(\vert f_2 \circ f_1 \vert = \vert f_2 \vert \circ \vert f_1 \vert\), because \(\vert f_2 \circ f_1 \vert_{\vert_S}\) is \(\sum_{j \in \{0, ..., n\}} t^j p_j \mapsto \sum_{j \in \{0, ..., n\}} t^j f_2 \circ f_1 (p_j)\) while \(\vert f_2 \vert \circ \vert f_1 \vert_{\vert_S}\) is \(\sum_{j \in \{0, ..., n\}} t^j p_j \mapsto \sum_{j \in \{0, ..., n\}} t^j f_1 (p_j) \mapsto \sum_{j \in \{0, ..., n\}} t^j f_2 \circ f_1 (p_j)\), by the proposition that any affine map from the affine or convex set spanned by any possibly-non-affine-independent set of base points on any real vectors space is linear
