%0 Journal Article %1 forssell2011topological %A Forssell, Henrik %D 2011 %K geometry mathematics topology %R 10.1002/malq.201100080 %T Topological Representation of Geometric Theories %U http://arxiv.org/abs/1109.0699 %X Using Butz and Moerdijk's topological groupoid representation of a topos with enough points, a `syntax-semantics' duality for geometric theories is constructed. The emphasis is on a logical presentation, starting with a description of the semantical topological groupoid of models and isomorphisms of a theory and a direct proof that this groupoid represents its classifying topos. Using this representation, a contravariant adjunction is constructed between theories and topological groupoids. The restriction of this adjunction yields a contravariant equivalence between theories with enough models and semantical groupoids. Technically a variant of the syntax-semantics duality constructed in Awodey and Forssell, arXiv:1008.3145v1 for first-order logic, the construction here works for arbitrary geometric theories and uses a slice construction on the side of groupoids---reflecting the use of `indexed' models in the representation theorem---which in several respects simplifies the construction and allows for an intrinsic characterization of the semantic side.

@article{forssell2011topological, abstract = {Using Butz and Moerdijk's topological groupoid representation of a topos with enough points, a `syntax-semantics' duality for geometric theories is constructed. The emphasis is on a logical presentation, starting with a description of the semantical topological groupoid of models and isomorphisms of a theory and a direct proof that this groupoid represents its classifying topos. Using this representation, a contravariant adjunction is constructed between theories and topological groupoids. The restriction of this adjunction yields a contravariant equivalence between theories with enough models and semantical groupoids. Technically a variant of the syntax-semantics duality constructed in [Awodey and Forssell, arXiv:1008.3145v1] for first-order logic, the construction here works for arbitrary geometric theories and uses a slice construction on the side of groupoids---reflecting the use of `indexed' models in the representation theorem---which in several respects simplifies the construction and allows for an intrinsic characterization of the semantic side.}, added-at = {2020-01-13T23:05:37.000+0100}, author = {Forssell, Henrik}, biburl = {https://www.bibsonomy.org/bibtex/2062c4aec9b02229d5af315f92a47a2f3/kirk86}, description = {[1109.0699] Topological Representation of Geometric Theories}, doi = {10.1002/malq.201100080}, interhash = {a73480b188c8afb8ef5ec6d481b1c818}, intrahash = {062c4aec9b02229d5af315f92a47a2f3}, keywords = {geometry mathematics topology}, note = {cite arxiv:1109.0699Comment: 32 pages. This is the first pre-print version, the final revised version can be found at http://onlinelibrary.wiley.com/doi/10.1002/malq.201100080/abstract (posting of which is not allowed by Wiley). Changes in v2: updated comments}, timestamp = {2020-01-13T23:05:37.000+0100}, title = {Topological Representation of Geometric Theories}, url = {http://arxiv.org/abs/1109.0699}, year = 2011 }