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One example that I've seen is the use of the word "synthetic," which has multiple uses in differential geometry.

1. There is a field called synthetic differential geometry, which studies differential geometry from the viewpoint of topos theory. This is based off work of Lawvere, and popular among the more categorically minded; the ncat lab describes it here.

2. There is also a field of synthetic differential geometry, mentioned by Matt F, "in a totally different tradition more closely connected to foundations of math and Finsler geometry." In that tradition Herbert Busemann is the founding figure; here are some sample results.

3. There is a separate idea known as synthetic curvature. This approach is based in analysis and uses ideas from convex analysis to understand curvature for spaces which are not necessarily smooth. This usage I'm a bit more familiar with and can give a few more details.

The analogy is that we can define convexity for a smooth function in terms of its Hessian being non-negative-definite. However, for less smooth functions, we can define convexity by saying the function lies below all of its secant lines. The latter is a "synthetic" definition of convexity, and is more general.

Following this analogy, we can use the same approach in differential geometry. For instance, it's possible to give synthetic definitions for sectional curvature bounds (e.g. the $CAT(\kappa)$ inequality) which make sense for geodesic spaces. Furthermore, one interesting insight of Villani's work on optimal transport is that it provides synthetic versions of Ricci lower bounds that make sense on metric-measure spaces.

In my experience, there aren't too many collisions between the first and third definitions because one originates from a categorical viewpoint and the other from an analytic perspective. In Matt F's experience, there aren't too many collisions with the second definition because Busemann's overall approach, despite coming earlier, never attracted many followers.

One example that I've seen is the use of the word "synthetic," which has multiple uses in differential geometry.

1. There is a field called synthetic differential geometry, which studies differential geometry from the viewpoint of topos theory. This is based off work of Lawvere, and popular among the more categorically minded; the ncat lab describes it here.

2. There is also a field of synthetic differential geometry, mentioned by Matt F, "in a totally different tradition more closely connected to foundations of math and Finsler geometry." In that tradition Herbert Busemann is the founding figure; here are some sample results.

3. There is a separate idea known as synthetic curvature. This approach is based in analysis and uses ideas from convex analysis to understand curvature for spaces which are not necessarily smooth. This usage I'm a bit more familiar with and can give a few more details.

The analogy is that we can define convexity for a smooth function in terms of its Hessian being non-negative-definite. However, for less smooth functions, we can define convexity by saying the function lies below all of its secant lines. The latter is a "synthetic" definition of convexity, and is more general.

Following this analogy, we can use the same approach in differential geometry. For instance, it's possible to give synthetic definitions for sectional curvature bounds (e.g. the $CAT(\kappa)$ inequality) which make sense for geodesic spaces. Furthermore, one interesting insight from optimal transport is that it provides synthetic versions of Ricci lower bounds that make sense on metric-measure spaces. One good reference is this paper. Another good reference is Villani's survey paper

In my experience, there aren't too many collisions between the first and third definitions because one originates from a categorical viewpoint and the other from an analytic perspective. In Matt F's experience, there aren't too many collisions with the second definition because Busemann's overall approach, despite coming earlier, never attracted many followers.

One example that I've seen is the use of the word "synthetic," which has at least two different uses in differential geometry. Edit: @Matt F pointed out that there is a third use of synthetic in differential geometry, so I've edited my answer to acknowledge him.

1. There is a field called synthetic differential geometry, which seems to study differential geometry from the viewpoint of topos theory. This is based off work of Lawvere, and popular among the more categorically minded; the ncat lab describes it here.

2. Matt F mentions there is a second usage which is "in a totally different tradition more closely connected to foundations of math and Finsler geometry." In that tradition Herbert Busemann is the founding figure; here are some sample results.

3. There is a separate idea known as synthetic curvature. This approach is based in analysis and uses ideas from convex analysis to understand curvature for spaces which are not necessarily smooth. This usage I'm a bit more familiar with and can give a few more details.

The analogy is that we can define convexity for a smooth function in terms of its Hessian being non-negative-definite. However, for less smooth functions, we can define convexity by saying the function lies below all of its secant lines. The latter is a "synthetic" definition of convexity, and is more general.

Following this analogy, we can use the same approach in differential geometry. For instance, it's possible to give synthetic definitions for sectional curvature bounds (e.g. the $CAT(\kappa)$ inequality) which make sense for geodesic spaces. Furthermore, one interesting insight of Villani's work on optimal transport is that it provides synthetic versions of Ricci lower bounds that make sense on metric-measure spaces.

In my experience, there aren't too many collisions between the first and third definitions because one originates from a categorical viewpoint and the other from an analytic perspective. There aren't too many collisions with the second definition because Busemann's overall approach, despite coming earlier, never attracted many followers.

One example that I've seen is the use of the word "synthetic," which has multiple uses in differential geometry.

1. There is a field called synthetic differential geometry, which studies differential geometry from the viewpoint of topos theory. This is based off work of Lawvere, and popular among the more categorically minded; the ncat lab describes it here.

2. There is also a field of synthetic differential geometry, mentioned by Matt F, "in a totally different tradition more closely connected to foundations of math and Finsler geometry." In that tradition Herbert Busemann is the founding figure; here are some sample results.

3. There is a separate idea known as synthetic curvature. This approach is based in analysis and uses ideas from convex analysis to understand curvature for spaces which are not necessarily smooth. This usage I'm a bit more familiar with and can give a few more details.

The analogy is that we can define convexity for a smooth function in terms of its Hessian being non-negative-definite. However, for less smooth functions, we can define convexity by saying the function lies below all of its secant lines. The latter is a "synthetic" definition of convexity, and is more general.

Following this analogy, we can use the same approach in differential geometry. For instance, it's possible to give synthetic definitions for sectional curvature bounds (e.g. the $CAT(\kappa)$ inequality) which make sense for geodesic spaces. Furthermore, one interesting insight of Villani's work on optimal transport is that it provides synthetic versions of Ricci lower bounds that make sense on metric-measure spaces.

In my experience, there aren't too many collisions between the first and third definitions because one originates from a categorical viewpoint and the other from an analytic perspective. In Matt F's experience, there aren't too many collisions with the second definition because Busemann's overall approach, despite coming earlier, never attracted many followers.

One example that I've seen is the use of the word "synthetic," which has at least two different uses in differential geometry. Edit: @Matt F pointed out that there is a third use of synthetic in differential geometry, so I've edited my answer to acknowledge him.

1. There is a field called synthetic differential geometry, which seems to study differential geometry from the viewpoint of topos theory. This seems to be based off work of Lawvere, and seems popular among the more categorically minded. I'm very much a non-expert, so I'm happy to be corrected.

2. Matt F mentions there is a second usage which is "in a totally different tradition more closely connected to foundations of math and Finsler geometry."

3. There is a separate idea known as synthetic curvature. This approach is based in analysis and uses ideas from convex analysis to understand curvature for spaces which are not necessarily smooth. This usage I'm a bit more familiar with and can give a few more details.

The analogy is that we can define convexity for a smooth function in terms of its Hessian being non-negative-definite. However, for less smooth functions, we can define convexity by saying the function lies below all of its secant lines. The latter is a "synthetic" definition of convexity, and is more general.

Following this analogy, we can use the same approach in differential geometry. For instance, it's possible to give synthetic definitions for sectional curvature bounds (e.g. the $CAT(\kappa)$ inequality) which make sense for geodesic spaces. Furthermore, one interesting insight of Villani's work on optimal transport is that it provides synthetic versions of Ricci lower bounds that make sense on metric-measure spaces.

In my experience, there aren't too many collisions between the first and third definitions because one originates from a categorical viewpoint and the other from an analytic perspective. I wasn't aware of the third but Matt F mentions there is some overlap in the literature.

One example that I've seen is the use of the word "synthetic," which has at least two different uses in differential geometry. Edit: @Matt F pointed out that there is a third use of synthetic in differential geometry, so I've edited my answer to acknowledge him.

1. There is a field called synthetic differential geometry, which seems to study differential geometry from the viewpoint of topos theory. This is based off work of Lawvere, and popular among the more categorically minded; the ncat lab describes it here.

2. Matt F mentions there is a second usage which is "in a totally different tradition more closely connected to foundations of math and Finsler geometry." In that tradition Herbert Busemann is the founding figure; here are some sample results.

3. There is a separate idea known as synthetic curvature. This approach is based in analysis and uses ideas from convex analysis to understand curvature for spaces which are not necessarily smooth. This usage I'm a bit more familiar with and can give a few more details.

The analogy is that we can define convexity for a smooth function in terms of its Hessian being non-negative-definite. However, for less smooth functions, we can define convexity by saying the function lies below all of its secant lines. The latter is a "synthetic" definition of convexity, and is more general.

Following this analogy, we can use the same approach in differential geometry. For instance, it's possible to give synthetic definitions for sectional curvature bounds (e.g. the $CAT(\kappa)$ inequality) which make sense for geodesic spaces. Furthermore, one interesting insight of Villani's work on optimal transport is that it provides synthetic versions of Ricci lower bounds that make sense on metric-measure spaces.

In my experience, there aren't too many collisions between the first and third definitions because one originates from a categorical viewpoint and the other from an analytic perspective. There aren't too many collisions with the second definition because Busemann's overall approach, despite coming earlier, never attracted many followers.